The absence of a logic of desire

Reinventing broken wheels is more fun than patching small punctures in functioning ones. So here are some thoughts on desires that are undistorted by knowing the relevant literature.

It seems that unlike for rational belief, there are very few formal constraints on rational desire. For instance, if you desire A & B, it doesn't follow that you should desire A: I'd like to be beaten up and get a billion dollars compensation for it, but I don't desire to be beaten up. By the same example, you may desire A without desiring the disjunction A v B. More generally, all these principles are invalid for rational desire:

1) $m[1]
2) $m[1]
3) $m[1]
4) $m[1]
5) $m[1]
6) $m[1]
7) $m[1]
8) $m[1]
9) $m[1]
10) $m[1]
11) $m[1]

Here is one of the few valid principles:

12) $m[1]

That may look a bit odd at first sight because it's hard to see how the left-hand side could be true. (If it can't, (6) is valid, too.) I'll give an example a bit further down. Anyway, even if the left-hand side is impossible, (12) remains valid. <update>I've inserted a missing negation in (12).</update>

What explains this strange (absence of) logical behaviour? It looks like D is not a modal operator of the usual kind: it cannot be interpreted as a quantifier over certain worlds. The reason is that the desirability of a logically complex proposition is not in general a function of the desirability of its parts. The other thing that enters into desirability is credence.

I don't desire being beaten up partly because I believe that I won't get any compensation (or not enough). I assign low hedonic value to the proposition of being beaten up because I assign low credence to one of its parts (subpropositions): the one where I get a lot of compensation. If I were certain that I would get a billion dollars compensation, I would desire to be beaten up. Likewise, I desire going to the party partly because I believe it will be fun. If I came to believe that I'll be sent to hell or tortured by space aliens if I go there -- that is, if I assigned high credence to these parts of the proposition --, I would rather stay at home.

Credences become irrelevant when the proposition at issue is maximally specific, so that it doesn't have any relevant parts: a maximally specific proposition will include all the consequences and preconditions I otherwise fill in with my beliefs. It tells me everything about whether I get compensations, go to hell, etc.

Assume you've assigned hedonic values (degrees of desirability) to all such maximal propositions. Then the hedonic value of a disjunction A v B of such propositions is, I think, its normalized expected utility

*) $m[1]

For example, suppose there are only two ways for me to be beaten up, one (A) in which I get compensation and one (B) in which I don't. I value A, but not B. Then whether or not I want to be beaten up is determined by how probable I take A to be as opposed to B. The total probability of being beaten up is irrelevant; that's why the whole thing gets divided by this probability. If I value A and B equally well, I will value the disjunction to the same degree. And if I take A and B to be equally probable, the value of the disjunction will be the average of the values of the disjuncts.

(*) holds whenever A and B are mutually exclusive. So given a distribution of values over maximally specific propositions, it determines a value for all propositions.

But there are very few shortcuts. For example, on this picture, the hedonic value of a conjunction is not a function of the value of the conjuncts together with any fact about credences. The distribution of the values inside the conjuncts also matters. Similarly, the value of a disjunction with non-exclusive disjuncts is not a function of the value of these disjuncts. And obviously the value of a negation is not a function of the value of the negatum. This explains why the logic of desire is so poor.

(*) fits nicely with how we compute expected utilities: the expected utility of an action with two possible outcomes A and B is V(A)*P(A) + V(B)*P(B), which equals V(A v B) * P(A v B) by (*). (As Weng Hong pointed out to me, (*) says nothing if P(A) and P(B) are both zero. This is a special case of propositions having equal probability, so I want to say that the value of A v B here, too, is the average. I don't quite see how to make (*) say that.)

If (*) determines the distribution of hedonic value, and desirability is something like having positive hedonic value, then one can prove that principle (12) above is valid whereas all of (1)-(11) are invalid. I actually had to write a little script to come up with a counterexample to (6) (after Magdalena assured me that it can't be proven from (*)). Here is one:

Your hedonic value for Fred and Anna both coming to the party is -3, but for each coming without the other it is 1. You believe to degree 0.4 that Fred will come without Anna, to 0.4 that Anna will come without Fred, and to 0.2 that they will both come. Then your hedonic value for Fred coming should be -0.33, for Anna coming -0.33, for both of them coming -3, but for (at least) one of them coming 0.2. So in this case, you may rationally desire that Fred or Anna come while not desiring that Fred comes and not desiring that Anna comes (and not desiring that both come).

The hedonic values are quite close together here, so this isn't expected to be an intuitively clear case (and according to my script, all counterexamples to (6) are like this).

The picture also gives an answer to the desirability of a tautology, A v ~A: its hedonic value equals the expected utility of the tautology, which is high if you're optimistic about the world (if you assign high credence to valuable possibilities) and low if you're not. So if you want to know whether somebody is happy or not, just ask if they desire the tautology.


# on 09 April 2007, 21:51

looks like the community is shocked, no responses yet. Can you perhaps elaborate a little on the point by you think that we do not manage to get D controlled and why (12) above is still valid?
Just an idea, to get the discussion starting, or maybe everybody is busy finetunig his/her D-logic?

# on 12 April 2007, 13:45

Hey M! I've googled a bit after posting and found very little on a logic of desire (compared to doxastic and epistemic logics). So I suppose in the relevant community it is sort of known that D doesn't behave well.

As for the validity of (12): let n be the hedonic threshold for desire. Then (12) says that if V(A v B) >= n and V(A) < n and V(B) < n, then V(A & B) < n. Now A is equivalent to the exclusive disjunction (A & ~(A & B)) v (A & B), and B to the exclusive disjunction (B & ~(A & B)) v (A & B), and A v B to (A & ~(A & B)) v (B & ~(A & B)) v (A & B). Since these are exclusive disjunctions, (*) applies. Let x1 = V(A & ~(A & B)), x2 = V(B & ~(A & B)), x3 = V(A & B), y1 = P(A & ~(A & B)), y2 = P(B & ~(A & B)), y3 = P(A & B). Then by (*), the hedonic value of A, V(A) = (x1*y1 + x3*y3) / (y1+y3); and V(B) = (x2*y2 + x3*y3) / (y2+y3); and V(A v B) = (x1*y1 + x2*y2 + x3*y3) / (y1+y2+y3). So (12) says that whenever (x1*y1 + x2*y2 + x3*y3) / (y1+y2+y3) >= n, and (x1*y1 + x3*y3) / (y1+y3) < n and (x2*y2 + x3*y3) / (y1+y3) < n, then x3 < n. And this can be proven mathematically.

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