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 isnâ€™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 â€˜I win in the first lotteryâ€™ has the same desirability as â€˜I win in the second lotteryâ€™, the desirability of â€˜I do not win in the first lotteryâ€™ cannot equal that of â€˜I do not win in the second lotteryâ€™. 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

indifferentto 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.