Abstract
One thousand fair causally isolated coins will be independently flipped tomorrow morning and you know this fact. I argue that the probability, conditional on your knowledge, that any coin will land tails is almost 1 if that coin in fact lands tails, and almost 0 if it in fact lands heads. I also show that the coin flips are not probabilistically independent given your knowledge. These results are uncomfortable for those, like Timothy Williamson, who take these probabilities to play a central role in their theorizing.
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Notes
Indeed, some aspects of Williamson’s framework are completely unnecessary for this discussion. Most notably, his assumption that there is exactly one theoretically significant ur-prior.
This definition makes sense even if the agent doesn’t know the conjunction of the propositions she knows. Thus, although evidential probabilities so defined are available even to anti-closure epistemologists, it may be an uninteresting notion. For example, theorists who claim that we know that that each ticket in a fair lottery won’t win, bar the winning one, will assign an evidential probability of 1 to the winning ticket and 0 to the others according to the above definition. I will therefore exclude these kinds of views from consideration in what follows—such views would have little use for evidential probabilities anyway.
Those who find the idea of a primordial ur-prior too fantastic, even when conceived as an abstraction from present credences, can make do with the notion of credence conditional on their knowledge. In order for this to be well defined one must assume (quite plausibly) that the agent is not certain that her knowledge is false. One ought to be able to reconstruct everything I say about evidential probabilities using this notion.
The following observations might lead one to question this assumption, indeed I suspect myself that it is probably false. The use of this assumption does not diminish the interest of this thought experiment however, which, among other things, is intended to demonstrate that assumptions like this are not true in general to those who would initially accept the assumption.
If our knowledge about politics, say, could make it more or less likely that that the first coin will land heads then these simplifications are not innocent.
If there are multiple ur-priors among which it is indeterminate which represents the agents uninformed opinions, they will all agree on this partition conditional on this area of logical space provided each candidate ur-prior satisfies the Principal Principle. Multiplicity in the representation of our initial credences therefore does not affect our conclusions about evidential probabilities.
This fact is much more general: since many of us know that we live in a quantum mechanical universe where most propositions about the future have a non-zero chance of being false, evidential probabilities will frequently come apart from known chances (see Hawthorne and Lasonen-Aarnio 2009).
See the discussion of closure in chapter 1 of Hawthorne (2004).
I owe this example to Kenny Easwaran.
It is worth noting that, for all I’ve said, it’s still possible that the coins are pairwise independent of one another.
This argument is structurally analogous to the puzzle in Dorr et al. (2013). It was this puzzle that first led me to the above observation about evidential probabilities.
Note that this order is isomorphic to a lattice of subsets of 1,000 (i.e. sets of three digit numbers.) The reader may find it easier, mathematically speaking, to think of K as a set of subsets of 1,000 that is ordered by inclusion.
I think that if we are to have a complete picture of what is going on in these cases we should treat our epistemic state and our evidence as accessibility relations. In this setting monotonicity would state that one’s posterior epistemic state is just the result of intersection your prior accessibility relation with the evidence accessibility relation. This has the same effect regarding which worlds are compatible with your knowledge before and after acquiring the evidence.
I owe this particularly simple presentation of this puzzle to an anonymous reviewer. Note that similar puzzles can be raised for epistemic independence.
See also the discussion of the related principle in Dorr et al. (2013) (there labelled Inferential Anti-Dogmatism).
We can also stipulate that my source for the information about the fairness of the coins is different for each coin. If I had one source who told me they were all fair, and I observed 100 came up heads, I might be reasonable to doubt my source. Whether this means that I’m no longer in a position to know that the coins are fair is more contentious (see Lasonen-Aarnio 2010.) But at any rate, it is much harder to push this kind of analogy in the case where one has independent, reliable sources for each your beliefs about the fairness of the coins. This marks an important disanalogy between cases with one coin being flipped many times and cases with many coins being flipped once.
The discs don’t need to have perfectly indistinguishable sides—they might, for example, have distinctive scratches on each side. The crucial point is that there is no way to uniformly classify two discs as having landed ‘the same way up.’ There would be no such way if the scratches on each coin were different.
Kenny Easwaran points out to me that if the universe actually had perfect rotational symmetry we would find out that every coin landed heads after learning that one of the coins landed heads, so we may have to weaken this condition slightly. This issue doesn’t arise in the variant puzzle using blank metal discs.
Of course, a difference manifests itself in terms of the first personal factive mental states: while an ordinary person is in a position to know that their belief that they have hands constitutes knowledge a brain in a vat isn’t. However they are exactly alike regarding their non-factive mental states like belief (and furthermore, their knowledge about what’s epistemically possible is identical, even if their knowledge about what isn’t differs.) The crucial point is that a third party would be able to have different beliefs about which of the two persons beliefs constituted knowledge (and would know that the ‘no hands’ proposition was a possibility for one person and not the other.)
Here is one worry you might have about Coin Indifference, particularly the iterated version. Suppose exactly 541 of the coins land heads. Let X denote those coins, and let Y denote a random set of 541 coins that have roughly as many heads and tails. If you were to have been shown only the outcomes of the coins in X then it might be reasonable to come to believe that all the coins will land heads, whereas it wouldn’t be reasonable to conclude this if you had only observed the outcomes of the coins in Y. Since there would certainly be a different number of doxastic possibilities after making these two sets of observations it’s natural to think the number of epistemic possibilities could also be constrained this way. I think there are three problems with this kind of objection. Firstly, while this thought may have something to it if it were a single coin being flipped repeatedly (for example one might reasonably conclude it was double headed), it is less clear when each of the coins are distinct and isolated in this way. Secondly, whilst the number of sequences compatible with your knowledge does depend on the number of sequences compatible with your beliefs, the number I have been focussing is the number of sequences compatible with what you are in a position to know and it is not obvious that this will depend on your beliefs in this way. Finally, this worry cannot even get off the ground in the variant puzzle with blank metal discs since there would be no relevant difference between any pair of sets, X and Y, that have the same cardinality. Thanks to Kenny Easwaran for discussion here.
Given Closeness it suffices to show that if σ ∈ K and r(σ) = r(τ) then τ ∈ K. If σ ∈ K then the result of learning the outcomes of all the flips except for those whose outcomes differ from σ will leave us with 2r(σ) open possibilities by Epistemic Monotonicity. If r(σ) = r(τ) then, by iterating Coin Indifference, the result of learning the outcomes of all the flips except for those whose outcomes differ from τ will also leave us with 2r(σ) (i.e. 2r(τ)) open possibilities and this can only happen if τ ∈ K.
Models with this structure were first suggested to me by John Hawthorne.
It is natural to think that ‘differing by at most k places’ this could serve as an accessibility relation in a model of epistemic logic. This is a reflexive, symmetric, non-transitive relation. These models have the following puzzling feature: out of all of the sequences of outcomes that might for all you know obtain, exactly one of them (the actual sequence of outcomes) is such that you know that it might for all you know obtain. You could consider non-symmetric models where the accessibility radius, k, can vary from world to world.
These numbers were calculated using http://www.wolframalpha.com/. To calculate Q(T n ) when k = 10, for example, just input: sum[binomial(999, r), {r, 0, 10}]/sum[binomial(1,000, r), {r, 0, 10}] (note: it seems to crash when you chose k ≥ 70.)
This observation seems to be pretty stable provided k does not depend on the number of coins in the set up: whatever k is, you can also modify the number of coins to be flipped (currently 1,000) appropriately to achieve a similar result. One could in principle avoid this result by allowing the structure of knowledge to depend on things like the number of coins flips to occur throughout all history, or on how many of those flips with unknown outcomes you know to be fair, but these would be drastic measures.
Articulating a measure of closeness to actuality relevant for an arbitrary lottery, without knowing the mechanism by which the winner is generated, is a tricky business. One would also expect there to be a certain degree of context sensitivity regarding which possible outcomes to regard as closer to actuality than others. However, even if one cannot give a completely explicit account of closeness, safety based theories will still impose structural constraints on knowledge that will distinguish them from other approaches to the lottery paradox.
Note that there is also a recent tradition of taking practical factors (such as how much is at stake if you were to act as if p) to impact what you know. Theories in this vein will not always be compatible with some of the principles I appealed to in this paper (see Coin Indifference and Epistemic Independence.) I take this to be a problem for these theories rather than a problem for the principles. However, for the sake of space, I shall simply bracket these views from consideration.
My rough gloss of this notion is: one is in a position to know that p if and only if there is some way of coming to believe that p that would constitute knowledge.
Thanks to Jeremy Goodman for this point. This thought is probably behind many supporters of ‘intelligent design’: that one can knowledgeably conclude that some sequence of outcomes is not the result of chance if the results are sufficiently ordered or surprising, even if that sequence is just as probable, conditional on being chancy, as a particular unsurprising random sequence of outcomes.
It doesn’t strictly speaking matter that they be blank discs. One might worry that it is impossible to assign distinct epistemic possibilities to the different ways the coin can land unless one can distinguish the sides. For the record I disagree, but it does not matter: one can assume that the sides are distinguishable, but not in a uniform way like with coins (for example, perhaps the first disc has a scratch on one side, the second disc has different coloured sides, and so on.)
This objection also applies to a naïve version of the second model. A more sophisticated version of the second model might set things up so that the more unusual things are, the less you know (see the ideas described in Goodman 2013 section III, which is based Williamson’s analysis of Gettier cases from 2013.)
A natural way to model belief on this operational understanding would be to use sets of propositions that are represented by some probability/threshold pair: that for some Pr and α you believe that p iff Pr(p) > α (there is a natural relation between k and α.) Your beliefs would be closed under single-premise closure, but not multi-premise closure. In this model they would also be closed under a recombination principle, which states that whenever you believe \(A_1\ldots A_n\), and when \(P_1\ldots P_n\) is a partition and \(A^* = \{X_1\cap \ldots\cap X_n\mid X_i = A_i\) or \(\neg A_i\}\), then there is a function \(g:A^* \rightarrow \mathcal{P}(P)\) such that (i) \(|g(X)| \leq |\{A_i \mid X \subseteq A_i\}|\) and (ii) you believe the proposition \(\bigcup_{X\in A^*} (X\cap (\bigcup g(X)))\).
Certainly there have been at least 1,000 coin flips throughout the course of history whose outcomes are unknown to us, thus it seems natural to think that such a partition ought already to be available to us.
The addition of the clause ‘conditional on any proposition’ is redundant if we simply take conditional probabilities to be defined from unconditional probabilities (although, for technical reasons, it would be inadvisable to adopt such a definition.)
By the disjunction of two attitude verbs, V and \(V^\prime\), I mean the complex attitude that obtains when one V’s that p or \(V^\prime\)’s that p.
References
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Acknowledgments
Many thanks to Kenny Easwaran, Jeremy Goodman and John Hawthorne for helpful discussions on earlier drafts of this paper.
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Bacon, A. Giving your knowledge half a chance. Philos Stud 171, 373–397 (2014). https://doi.org/10.1007/s11098-013-0276-6
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DOI: https://doi.org/10.1007/s11098-013-0276-6