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In “Desire as Belief, Lewis Notwithstanding“, Ruth Weintraub argues that Lewis’s refutation of Desire as Belief rests on an implausible theory of desirability. In particular, she argues that the additivity principle Lewis uses in the proof is implausible. After a bit of algebra, she concludes with this (correct) theorem in the logic of desirability Lewis uses, and an argument against Lewis’s interpretation of it.

So if \(D(A) = D(B)\) and \(D(\neg A) = D(\neg B)\), \(A\) and \(B\) have the same probability. But this shows that D isn’t a measure of the goodness of propositions (prospects). For there can be equally good prospects whose probabilities differ. Suppose, for instance, I have two lottery tickets, for two different lotteries. The (single) prize in both lotteries is $1,000, but there are twice as many tickets in the second one. The two prospects (winning in the first lottery and winning in the second) seem equally good, as do the prospects of not winning. But if ‘I win in the first lottery’ has the same desirability as ‘I win in the second lottery’, the desirability of ‘I do not win in the first lottery’ cannot equal that of ‘I do not win in the second lottery’. The more probable outcome, not winning in the second lottery, is less undesirable. If, alternatively, not winning in the lotteries is equally (un)desirable, then winning in the second is more desirable. (Weintraub 2007, 120, notation slightly altered).

But this isn’t quite right. Let \(A\) be that I win the first lottery, and \(B\) be that I win the second lottery. Then although the prize for each lottery is the same, \(D(B) > D(A)\). That’s because \(B\) means that I win $1,000, and have a ticket in a lottery I have a good chance of winning. But \(A\) means that I win $1,000, and have a ticket in a lottery I have a bad chance of winning. And the first is better for me than the second.

Weintraub has a second argument against the additivity principle. Assume \(P(A) = 1\), and \(T\) is a tautology.

$$D(T) = \frac{D(A)P(A) + D(\neg A)P(\neg A)}{P(A) + P(\neg A)} = D(A)$$

She comments

But this is absurd. When I win the lottery and am certain that I have won, I find my winning as good as I did when it seemed very unlikely. But according to its desirability (that of a tautology), I am indifferent to it! (Weintraub 2007, 120, emphasis in original)

This assumes that I am indifferent to tautologies. And on the correct way to understand Jeffrey’s theory, this isn’t quite right. After all, if things are going very very well for me, then \(D(T)\) will be extremely high. In general, the value of \(D(T)\) at a time is a reflection of how well things are going for me at that time.

Of course, at any given time it is possible to use \(D(T)\) as a baseline, and say that how good something is is given by the difference between its desirability and \(D(T)\). But we should remember that across time on Jeffrey’s theory, \(D(T)\) can change dramatically. And those changes have meaning; they reflect changes in the value the agent assigns to their current overall state.

So there isn’t anything absurd about \(D(T) = D(A)\) after I win the lottery. Both of them are high. ‘Learning’ either of them doesn’t improve things for me. But that doesn’t mean I’m indifferent to them; I’m rather happy that both of them are true.

There is something very funny about this view, though it’s not that \(D(T) = D(A)\). Imagine that a good thing, \(A\), and a bad thing \(B\) both happen, and I know they’ve both happened. Then we have \(D(A) = D(B)\), which seems odd. This is why I’d like to have some kind of theory of value like the proposal by Stephen Daskall I discussed in the previous post. Things now, with \(A \wedge B\) true, are as they are. But they’d be worse if \(A\) were false, and better if \(B\) were false. And there should be a way of turning that into a theory of ‘absolute value’. But I don’t see how one could do that consistent with any kind of Desire as Belief theory.

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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.