Friday, September 14, 2018

A puzzle about knowledge in lottery cases

I am one of those philosophers who think that it is correct to say that I know I won’t win the lottery—assuming of course I won’t. Here is a puzzle about the view, though.

For reasons of exposition, I will formulate it in terms of dice and not lotteries.

The following is pretty uncontroversial:

  1. If a single die is rolled, I don’t know that it won’t be a six.

And those of us who think we know we won’t win the lottery will tend to accept:

  1. If ten dice are rolled, I know that they won’t all be sixes.

So, as I add more dice to setup, somewhere I cross a line from not knowing that they won’t all be six to knowing. It won’t matter for my puzzle whether the line is sharp or vague, nor where it lies. (I am inclined to think it may already lie at two dice but at the latest at three.)

Let N be the proposition that not all the dice are sixes.

Now, suppose that ten fair dice get rolled, and you announce to me the results of the rolls in some fixed order, say left to right: “Six. Six. Six. Six. Six. Six. Six. Six. Six. And five.”

When you have announced the first nine sixes, I don’t know N to be true. For at that point, N is true if and only if the remaining die is six, and by (1) I don’t know of a single die that it won’t be a six.

Here is what puzzles me. I want to know if in this scenario I knew N in the first place, prior to any announcements or rolls, as (2) says.

Here is a reason to doubt that I knew N in the first place. Vary the case by supposing I wasn’t paying attemption, so even after the ninth announcement, I haven’t noticed that you have been saying “Six” over and over. If I don’t know in the original scenario where I was paying attention, I think I don’t know in this case, either. For knowledge shouldn’t be a matter of accident. My being lucky enough not to pay attention, while it better positioned me with regard to the credence in N (which remained very high, instead of creeping down as the announcements were made), shouldn’t have resulted in knowledge.

But if I don’t know after the ninth unheard announcement, surely I also don’t know before any of the unheard announcements. For unheard announcements shouldn’t make any difference. But by the same token, in the original scenario, I don’t know N prior to any of the announcements. For it shouldn’t make any difference to whether I know at t0 whether I will be paying attention. When I am not paying attention, I have a justified true belief that N is true, but I am Gettiered. Further, there is no relevant epistemic difference between me before the die rolls and between the die rolls and the start of the announcements. If I don’t know N at the latter point, I don’t know N at the beginning.

So it seems that contrary to (2) I don’t know N in the first place.

Yet I am still strongly pulled to thinking that normally I would know that the dice won’t all be sixes. This suggests that whether I will know that the dice won’t all be sixes depends not only on whether it is true, but what the pattern of the dice will in fact be. If there will be nine sixes and one non-six, then I don’t N. But if it will be more “random looking” pattern, then I do know N. This makes me uncomfortable. It seems wrong to think the actual future pattern matters. Maybe it does. Anyway, all this raises an interesting question: What do Gettier cases look like in lottery situations?

I see four moves possible here:

A. Reject the move from not knowing in the case where you hear the nine announcements to not knowing in the case where you failed to hear the nine announcements.

B. Say you don’t know in lottery cases.

C. Embrace the discomfort and allow that in lottery cases whether I know I won’t win depends on how different the winning number is from mine.

D. Reject the concept of knowledge as having a useful epistemological role.

Of these, move B, unless combined with D, is the least plausible to me.


James Goetz said...

What is more uncontroversial follows: if I roll ten fair dice, then the odds are only 1 in 60 million that all ten dice will be sixes.

Martin Cooke said...

This is a lovely intuition-pump.

There is obviously nothing wrong with your intuition here:
My being lucky enough not to pay attention [...] shouldn’t have resulted in knowledge.
People have tended to concede that knowledge is compatible with some epistemic luck; but clearly, not that much!

I think that you are also right about this:
It seems wrong to think the actual future pattern matters.
That, of course, is in direct conflict with your original
I know I won’t win the lottery — assuming of course I won’t

Incidentally, one problem with your scenario as intuition-pump is that such a long string of '6's is considerable evidence for the dice not being fair, so that however good your hypothetical evidence that they were fair, there would actually be a real question over that.

Martin Cooke said...

The latter problem could be completely eliminated easily enough,
e.g. predict a string like 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1,
and then get the first ten of those, with just the 1 to be got.

Angra Mainyu said...


I'm not sure in which sense of "should" you mean that "knowledge shouldn’t be a matter of accident", but here's another possible move: reject that knowledge shouldn’t be a matter of accident, or more precisely, that knowleged is not a matter of accident ever - regardless of whether it should -, and hold that in this particular case, it is a matter of accident, so you lose your knowledge if you hear the numbers.

Here's an example:

Suppose Rob parks his car outside, and goes inside a building for a meeting. He know the car is parked outside. Now tells Alice that a car meeting such-and-such description was just towed away. As it happens, the description matches Rob's car, and those cars are pretty uncommon. Bob tells that to Alice because he suspects that that's Alice's car, and wants to play a practical joke. If Rob happens to overhear (a matter of luck), he will no longer know that his car is parked outside.

Granted, in this case, the information provided by Bob is false. But it's still a matter of luck, so I think this counterexample works. Or do you mean it's only for cases of true information? If so, I can construct a different example.

Martin Cooke said...

I think that (D) is on the right lines, but false as it stands. The concept of knowledge is obviously natural and important; it cannot simply be eliminated from epistemology. Justification is the concept that does the heavy lifting in epistemology, but it is only important because it gives us knowledge. How justified do you have to be? Enough for you to have knowledge. The concept of degrees of justification is more logical, but the concept of knowledge grounds it in our lives.

It is the concept of knowledge that shows us such facts as that we know that we will not win the lottery, unless we do win; that, in general, standards of justification do depend on outcomes. With the concept of justification alone it would, perhaps, be too easy to conclude from winning the lottery that we had not been sufficiently justified in our earlier false belief. But with the concept of knowledge we can 'see' that we really were. It is the logical flaw in the concept of knowledge that lets us do that. And because the flaw is in knowledge, we can have a logical concept of degrees of justification to work with.

It is the way that the knowledge was never there, when we win the lottery, that signals the need to revise our standards of justification; signals it incorrectly in the case of the lottery. We will normally want to modify our standards if we keep getting it wrong. There is a similar thing going on in language more generally, when contradictions cause us to implement clarification procedures. Our terms are only as precise as our uses have forced them to be. Anything else would be to put an infinitely large cart before a poor horse! And perhaps we can live with a few semantic paradoxes (much as we could live with winning the lottery).

Alexander R Pruss said...


You might be right that in my case the accidentality is not problematic. If so, that would be a nice solution to the problem.


Oops. You're right that the pattern of future rolls does matter insofar as for knowledge it has to fit the belief. But I would initially think that this is the only feature of it that should matter.

I had an intuition behind the "pattern doesn't matter" thesis that I didn't manage to articulate. I can articulate it now, making it possible for me to run my story regardless of how the dice come out, as long as they don't come out all sixes.

Fix any pattern Z of ten die rolls. Now, consider any sequence of subsets A_1,A_2,...,A_10 of the set of all patterns of ten die rolls such that:
(a) A_10 has only one member, which is Z
(b) A_9 is a set of 6 members, and A_10 is a subset of A_9
(c) A_8 is a set of 6x6=36 members, and A_9 is a subset of A_8
(d) A_7 is a set of 6^3 members, and A_8 is a subset of A_7
(j) A_1 is a set of 6^9 members, and A_2 is a subset of A_1.

Now imagine that we have a sequence of announcements:
- The observed die pattern is a member of A_1
- The observed die pattern is a member of A_2
- The observed die pattern is a member of A_9
- The observed die pattern is not a member of A_10

Before the first announcement, I knew that the observed die pattern wasn't Z (assuming I know in lottery cases). After the ninth announcement, I didn't know that, because my best information said that the observed die pattern was in A_9, which contains 6 members, one of which is Z. After the tenth announcement, I knew it again.

Moreover, regardless of what the observed sequence of die throws is, as long as it isn't Z, there is a sequence of sets A_1,...,A_10 satisfying conditions (a)-(j) such the observed die pattern is a member of A_1,...,A_9 but not of A_10.

So a version of the story I gave in the post works regardless of what the actual sequence of die rolls will be and regardless of what I guess it won't be, as long as my guess as to what it won't be is in fact correct.

Alexander R Pruss said...


I agree, of course, that a string of nine sixes is strong evidence for the die being loaded. As usual in these thought experiments, we need to suppose we have incredibly prior strong evidence that it is fair.

Alexander R Pruss said...

Trent Dougherty kindly pointed out some typos to me.