The Dart Board Case

From Sophie Horowitz’s paper “Epistemic Akrasia,” which we are talking about in our epistemology reading group today. Here’s the case:

“You have a large, blank dartboard. When you throw a dart at the board, it
can only land at grid points, which are spaced one inch apart along the horizontal and vertical axes. … Although you are pretty good at picking out where the dart has landed, you are rationally highly confident that your discrimination is not perfect: in particular, you are confident that when you judge where the dart has landed, you might mistake its position for one of the points an inch away (i.e. directly above, below, to the left, or to the right). You are also confident that, wherever the dart lands, you will know that it has not landed at any point farther away than one of those four. You throw a dart, and it lands on a point somewhere close to the middle of the board.” (p. 19)

So, it sounds to me as if this is the situation: it kinda looks like it hit in the center, but you can’t be sure. Horowitz then reports Williamson’s assessment of the case (we suppose the grid for the board is 1-5 along both axes (so it kinda looks like it landed at <3,3>)):

“So let’s suppose that when the dart lands at <3,3>, you should be highly confident in the proposition that it landed at either <3,2>, <2,3>, <3,3>, <4,3>, or <3,4>–so, you can rationally rule out every point except for those five. … Williamson agrees with this verdict, and supposes further that your credence should be equally distributed over <3,2>, <2,3>, <3,3>, <4,3>, and <3,4>. So, for each of those five points, you should have .2 credence that the dart landed at that point.” (p. 20)

This strikes me as exactly wrong. As I read the case, my guess as to where the dart landed is that it landed at <3,3>, but I can’t be sure. I might easily mistake its position for an adjacent one. But then I’m not egalitarian with respect to the 5 possible positions. I don’t make guesses without some evidential substance to support them as opposed to alternative hypotheses, so when I guess that it landed at <3,3>, that means that the look in question supports a greater degree of confidence in that hypothesis than the others. It may also mean that my probability for that hypothesis is greater than .5. In any case, I’m more confident in one of them than in the others, and this confidence is based on the indefinite look in question (and, we may suppose, rationally so).

Of course, we can change the case so that my powers of discrimination are indeterminate between the five regions, but then I also won’t be able to use my powers of discrimination to rule out areas out side of the five in question, since I’m no more confident that the dart landed in the center of the 5-region territory than at the edge.

In short, I don’t see how the case is supposed to get the Williamsonian indifference between the 5 regions, compatible with one’s perception putting one in a position to rule out all other regions on the board.


The Dart Board Case — 7 Comments

  1. If your discrimination power is as you described, and {2,3} (those brackets were meant to be pointy, sorry) remains an epistemic possibility, then shouldn’t {1,3} also be impossible for you to rule out? After all, that’s only one square away from where the dart may have landed ({2,3}), so {1,3} should be within your uncertainty zone.

    But then, if {1,3} is within your uncertainty zone, why isn’t {0,3}?

    This may seem like a cheap argument, because maybe you think it’s just luminously known that the dart didn’t hit {0,3}. But if you know with certainty that it didn’t hit {0,3}, then you can rule out {1,3}, because when you can rule something out with certainty, you know the dart didn’t land right next to it. But now if you iterate this argument, you can show that you can be certain that the dart hit {3,3} – so obviously the argument does not iterate.

    Basically, if there is a circular halo of certainty where you know the dart isn’t, then you should be able to find the dart at the center of that halo. I didn’t read this article, so maybe I’m only saying boring and obvious stuff, but the only solution I can think of is to have shrinking but non-zero rational credences at every point, even miles away.

  2. Hi Jon, thanks for reading my paper!

    Is the indifference point the main reason you disagree with (the Williamsonian reading of) this case? I agree with you that should intuitively get more weight (which I mention in fn 36). But the main point of the case is that it’s a 3d version of Williamson’s clock case — one in which the Williamsonian line yields a big discrepancy between the rational credence to have, and the credence you should think you should have.

  3. Yeah, I think the same thing of the clock case. The way looks function is there is a focal and a distal region, and the look in question always lends greater credence to the focal region and less to the distal region. Then, as we move farther away from the focal region, you get to the space that can rule out on the basis of the look. Depending on one’s visual acuity, the focal region will be larger or smaller, but there’s never a case where the focal and distal regions are the same (which is what would be needed for the indifference point together with the ruling out point to be true simultaneously). That’s my worry, and it applies to TW’s clock case as well: it can sorta look like it’s 3:00, or it can sorta look like it’s 2-4:00. In the latter case you get indifference with respect to 2, 3, and 4:00; but you lose the ruling out point, since a focal region of 2-4:00 will leave a distal region that can’t be ruled out (say, 1:00 or 5:00).

    • I agree. I think ignoring that just makes things simpler. (And doesn’t change the main point, since you can always go to a 3d or 4d case and you’ll end up with more of your credence apportioned to the distal region.)

  4. David, I think the issue between you and TW is maybe about vagueness itself, but the quick answer is this: your credences go to zero at some point, and that’s where the ruling out region begins. So, the author of the case gets to stipulate where your credences bleed out in this way.

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