Absolute Value

In Absolute Value as Belief, Steven Daskal aims to save anti-Humeanism against Lewis’s attacks in the Desire as Belief papers by changing the connection between credences and values. I like the idea he’s trying to develop – trying to use the difference in value between \(A\) and \(\neg A\) to state the theory more carefully. But the particular way he does it isn’t quite working, and I don’t really know how to fix it.

Here is the equation he ends up wanting to defend.

$$\sum_y C(g(A) = y) \cdot y = \sum_w C(w) \cdot (V(w \bullet A) – V(w \bullet \neg A))$$

The sum on the left is over possible values. The sum on the right is over possible worlds. And the \(\bullet\) is an imaging operator; so \(w \bullet A\) is the nearest world to \(w\) where \(A\) is true. (The general form of this allows ties, but we won’t need that level of specificity.)

I don’t think this can be right in general as it stands. Here is a puzzle case for the view. Assume there are three equiprobable worlds, \(w_1, w_2, w_3\), and the first two have goodness 1, the third has goodness 0. Assume also that these goodness facts are known. Let \(A\) be the proposition that \(w_1\) obtains. So we have the following for the LHS of the equation.

$$\sum_y C(g(A) = y) \cdot y = C(g(A) = 1) \cdot 1 = 1$$

Assuming that strong centring obtains for the ‘nearness’ function, we get the following.

\(w_1 \bullet A = w_1\)
\(w_2 \bullet A = w_1\)
\(w_2 \bullet \neg A = w_2\)
\(w_3 \bullet A = w_1\)
\(w_3 \bullet \neg A = w_3\)

It isn’t clear what \(w_1 \bullet \neg A\) should be; let’s call it \(w_x\). Substituting all these into the RHS of the equation we get:

$$\frac{V(w_1) – V(w_x)}{3} + \frac{V(w_1) – V(w_2)}{3} + \frac{V(w_1) – V(w_3)}{3}$$

The second term equals 0, and the third term equals 1/3. The value of the first term is unknown, but it is either 0 or 1/3. So the sum equals either 1/3 or 2/3.

So we have LHS equals 1, and RHS equals either 1/3 or 2/3. So the equation doesn’t work.

As I said, I like the idea of using differences between values of propositions and their negations in the theory of motivation. But I don’t think this particular way of doing it is quite right.

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