 # Absolute Value

[mathjax]

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.