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.