Over at Fake Barn Country, John Turri presents some cases to question John Hawthorne’s explanation for why we seem not to know we’ve lost in standard lottery situations. Turri quotes Hawthorne’s explantion:

*Without pretending to be able to have a full account of the relevant psychological forces driving the relevant intuitions, we can nevertheless see that in the paradigm lottery situation, something like the following often goes on: The ascriber divides possibility space into a set of subcases, each of which, from the point of view of the subject is overwhelmingly likely to not obtain, but which are such that the subject’s grounds for thinking that any one of the subcases does not obtain is not appreciably different than his grounds for thinking that any other subcase does not obtain…. In general, what is often at the root of the relevant lottery intuition is a division of epistemic space into a set of subcases with respect to which one’s epistemic position seems roughly similar. Once such a division is effected, a parity of reasoning argument can kick in against the suggestion that one knows that a particular subcase does not obtain, namely: If one can know that that subcase does not obtain, one can know of each subcase that it does not obtain. But it is absurd to suppose that one can know of each subcase that it does not obtain.* (**Knowledge and Lotteries**, 14 – 15)

You can see Turri’s cases here. He argues against both the necessity and the sufficiency of Hawthorne’s hypothesis –“DIVISION”– for the no-knowledge phenomenon in lottery cases.

In an old paper (“Knowledge, Assertions, and Lotteries,” Austr. JP, 1996; provisional version on my web page), I have a lottery case that I was using against a proposal of David Lewis’s. I think it also shows the non-necessity of Hawthorne’s DIVISION. It’s the variant of the “billionaire’s lottery,” and it’s burried in note 6 of KAL.

A billionaire holds a one-time lottery, and you are one of the 1 million people who have each received a numbered ticket, each one corresponding to one of the 1 million numbered ball in a giant bin. The balls are stirred, and the mechanical “grabber” is dropped into the bin. Given the way the “grabber” works, it has only a one-in-a-hundred chance of succeeding in grabbing any of the balls at all, as past experience shows. If it does grab a ball, the holder of the ticket corresponding to that ball wins a fabulous fortune. Otherwise, nobody receives anything. The chances that you’ve won, then, are 1 in 100 million; the chances that somebody or other has won are 1 in 100. In all likelihood, then, there is no winner. Still you seem not to know that you don’t win. But there doesn’t seem to be any DIVISION going on, if I’m understanding DIVISION correctly. The possibility that you think obtains — that nobody wins — is MUCH more probable than any of the other possibilities, and isn’t at all on a par with those other possibilities.

Maybe DIVISION explains some lottery intutions of no-knowledge, even though it isn’t operative in this case. But the fact that we get the same non-knowledge phenomenon in this lottery case where DIVISION does not occur may cast a certain doubt on the explanation.

Keith, the way I understand DIVISION, I agree this is a counterexample. You could also give counterexamples in which a million tickets are issued, with each having an exponentially lower probability than the prior ticket, where this fact is known to everyone participating in the lottery. Some chisholming could fix these problems, but Hawthorne is wanting an explanation that is psychologically realistic, and the required chisholming is almost certain to make it less realistic. But of course John has formulated the point with multiple hedges, making it much harder to assess the significance of any particular counterexample.

I guess this is the same as the case Jon presents, but perhaps a bit more natural: If you set out to flip a fair coin until it comes up tails, is there any N such that you know that the coin will come up tails within N flips?

I think this case is on the same level as the lottery case (that is, whenever I’d be prepared to answer “yes,” I’d also be prepared to answer “yes” for analogous lottery cases), but it doesn’t seem to fall under DIVISION. For each N the proposition “The coin will come up tails for the first time on the Nth flip” is only half as likely as for N-1.

I blogged this case here a while ago; I lay out the case in more detail there.