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I've been reading up on the notion of lottery propositions. It seems like there are two notions of knowledge one can subscribe to in relation to probability: either you're skeptical and think you only know stuff that has probability 1, or you think you can know stuff that has a sufficiently high probability that is less than 1. In the first view, you obviously can't know you'll lose the lottery. But apparently, there is a widespread intuition that even taking the second view - that you can know stuff that isn't 100% certain - you can't know you will lose the lottery, no matter how large the lottery is.

I honestly don't understand this intuition at all - when using the word "knowledge" in a way that allows for knowing things that have a probability less than 1, I think I simply know that I'll lose the lottery. I've been trying to read Hawthorne's Knowledge and Lotteries, and in the first chapter, he has a list of normal propositions and lottery propositions that he mentions in passing, and he acts like the difference between them is intuitively really obvious. I honestly just don't see it, I don't know what the fuss about lottery propositions is, and I can't even tell what the hell they are - they just seem like regular old low probability propositions to me.

I know that intuition is a personal thing and there may not be a way to convey one's intuition to someone that doesn't have it, but can anyone shed light on the topic of lottery propositions? This whole lottery proposition thing has me completely mystified.

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  • Tag suggested. Cleaned up some typos and added a link to SEP on the apparent topic.
    – J D
    Commented Dec 11, 2021 at 16:24
  • The lottery paradox isn't quite the same thing, though it is perhaps related. It is concerned with whether knowledge is closed under conjunction, i.e. with whether it is always the case that if you know A and know B then you always know "A and B". The lottery paradox and the preface paradox raise problems for this assumption.
    – Bumble
    Commented Dec 11, 2021 at 16:53
  • @Bumble I'm not talking about the lottery paradox. On a separate note though, a possible resolution is to use bayesian conditionalization so that once you know that a certain number of individual tickets won't win, you can't keep concluding that about the rest, because the pool of tickets left that can win is now much smaller. I don't think I buy into conjunction closure just because conjunctions can have lower probabilities than their individual conjuncts.
    – Tsvi Benschar
    Commented Dec 11, 2021 at 19:37
  • You're rightly confused trying to mix epistemic logic with probability stuffs. Since knowledge is usually defined as justified true belief, so the epistemic modal operator K already contains the other probability modality for those non 100% certain (lottery) propositions during one's own justification process. On the other hand epistemic logic is obviously non-normal (Nec rule fails at the start) otherwise we'll have logical omniscience paradox. Part of your confusion lies at you're used to normal thinking mode...
    – Double Knot
    Commented Dec 12, 2021 at 1:52
  • This is also similar to the preface paradox, as in your wikipedea source, Kyburg wants to reject the 3rd epistemic closure principle which will cause much confusion since it's really counterintuitive for most people. You can take the traditional probabilists's view to instead reject 1st principle which is essentially a normal modal principle which as I commented above is rather vulnerable. Even if a proposition p is very likely, we cannot logically arrive at Kp, we can only arrive at K(p is very likely). So our actual epistemic logic is not normal in practice, not idealized normal theories.
    – Double Knot
    Commented Dec 12, 2021 at 2:28

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The problem is that we seem to be faced with an unpleasant fork. If we say there must be 100% certainty that something is true in order to claim we know it, then most of the kinds of things that we ordinarily claim to know do not qualify as knowledge. After all, as Descartes pointed out, we can coherently doubt almost anything. But if we allow an error threshold, how low should it be? 99%? 99.9%? There is no obvious answer.

Lottery propositions are a thought experiment that aims to make this problem particularly sharp. Suppose you were inclined to think 99.9% should be enough. Then if you bought a lottery ticket with a 1 in 10,000 chance of winning, the probability that it won't win is above your threshold, so you would be allowed to say you know your ticket won't win. But this is misleading. If you show me a lottery ticket that you bought and tell me that you know it won't win, I would be entitled to assume that you are saying something over and above the obvious fact that lottery tickets by their nature have a low probability of winning. I would reasonably assume that you mean you know the lottery is fixed, or that the selection mechanism is broken, or something.

Also, consider why you bought the ticket in the first place. Presumably you didn't buy it because you knew it wouldn't win, but because you knew there was a chance of it winning, albeit small.

In practice, a threshold of 99.9% might be suitable for some purposes, but the exact value will depend on the proposition it is being applied to, and in any case, the value might be vague and impossible to put a precise number to. If you tell me you know some proposition A, the appropriate degree of credence with which you entertain A is going to depend on all sorts of factors, including whether A is the sort of thing it is feasible to be highly sure of, what degree of expertise you have regarding A, what common knowledge there is between us about A, and quite possibly other things too.

Thinking about lottery tickets is just a way to make the point that there is no single fixed value of ε such that you know A iff P(A) > 1 - ε. Though on the whole I would say that this is hardly a remarkable result, and does not tell us anything particularly interesting about epistemology.

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  • So are you saying the ability to use a binary logic in the evaluation of an event's occurrence is technically inaccurate but justified largely because it essentially presumes the likely outcome to allow reasoning to continue in the light of other evidence, particularly in the context of a non-monotonic logic or a system of defeasible reasoning? In essence, it charaterizes two propositional contradictory attitudes simultaneously: I know I won't win (despite the tiny odds of doing so), but I hope I will win.
    – J D
    Commented Dec 11, 2021 at 16:32
  • I'm just saying that minimally, ε is not a constant but a function of lots of factors. It is incorrect to say I know the lottery ticket will not win. The appropriateness of saying "I know it will not win" must be assessed against the background knowledge that it is very unlikely to. So, given a 1 in 10,000 chance of winning under normal circumstances, the claim "I know it will not win" implies a much lower value of ε than 1/10000. Maybe 1/1000000 would suffice. For a lottery with 1000000 tickets, a much smaller number would be needed. But for other propositions 1/1000 might be good enough.
    – Bumble
    Commented Dec 11, 2021 at 16:51
  • You're endorsing a contextualist view, where context dictates what kind of possibilities we need to consider? And lotteries call attention to the stark possibility of winning, while other scenarios may not call attention to the possibilities where you're wrong, even if they're more likely than winning the lottery? I can sort of get behind this, but it seems like at some point, the lottery would get so large that I would just know (assuming a non-skeptical view of knowledge) that I won't win. For instance, imagine a lottery with a googolplex tickets.
    – Tsvi Benschar
    Commented Dec 11, 2021 at 19:28
  • Also, actual lottery propositions, like "I won't win the lottery," are apparently not the only lottery propositions - there are also things like "I know I won't have a heart attack in the next 3 seconds". I can identify the ones mentioning lotteries as lottery propositions easily enough, the issue is when lottery propositions start having nothing to do with lotteries. I guess these also call attention more starkly to possibilities when you're wrong, lowering the epsilon value?
    – Tsvi Benschar
    Commented Dec 11, 2021 at 19:30
  • Yes, calling this position contextualist is OK. If a lottery had a googleplex of tickets, then presumably most of them will go unsold, so you might well be able to say in such circumstances that you know your ticket won't win, but the example is rather artificial. I don't see how I would justify saying, "I know I won't have a heart attack in 3 seconds". If I'm in excellent health, it is highly unlikely, but that's as far as I would go.
    – Bumble
    Commented Dec 12, 2021 at 1:42

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