So I was reading the Hájek and Pettit paper in “Lewisian Themes”:http://www.amazon.com/exec/obidos/redirect?tag=caoineorg-20&camp=14573&creative=327641&link_code=am1&path=http%3A//www.amazon.com/gp/product/offer-listing/0199274568%3Fcondition%3Dall/ASIN/0199274568 and I was very very confused. This may be a reflection of something wrong with me, or maybe something confusing is going on. (Warning – this is written with effectively _zero_ knowledge of the actual literature, so I might just be reinventing the sled.)
The paper is on Lewis’s attacks on the Desire-as-Belief thesis, which somehow gets transformed into a general-purpose anti-Humean thesis. Here’s Lewis’s statement of the thesis, slightly reworded to allow for formatting in HTML.
bq. there is a certain function (call it the ‘star’ function) that assigns to any proposition A a proposition A* (‘A-star’) such that, necessarily, for any credence distribution C, V(A) = C(A*).
V here is the (ideal?) agent’s (normalised) valuation function and C her credence function. This thesis is shown to be false on the ground that it implies credences in starred propopsitions don’t change, which is I guess implausible. Hájek and Pettit point out that for many purposes an anti-Humean can get by with the following claim, which is invulnerable to Lewis’s arguments.
bq. For any credence function C, there is a star function that assigns to any proposition A a proposition A* (‘A-star’) such that V(A) = C(A*).
For instance, if the star function maps A onto the proposition that A maximises expected utility, then there is no _formal_ argument against their weaker claim. Why is this relevant? Well because Lewis seems to make rather bold claims about what follows from the falsity of _his_ thesis. He considers something like Hájek and Pettit’s thesis and says that its truth would not make us think that A* is the proposition that A is objectively good. As I said, Hájek and Pettit argue fairly convincingly that that is incorrect.
But they agree with Lewis about one thing. They think the claim that A is objectively good will have to be in a way indexical to avoid Lewis’s argument. (If the objectively good maximises _expected_ utility according to some salient credence function it will be indexical, so the decision-theoretic utilitarian is off the hook here.) And I’m just confused about why we should think that is the case.
It might make things easier if we consider a particular kind of objectivist about ethics, the ethical actualist. The actualist says that _A is good_ is necessarily equivalent to A. (Perhaps what is good is what is part of God’s plan, and God’s plan is revealed by what is true.) Now the actualist is an objectivist about ethics if anyone is. And the actualist is an anti-Humean – they certainly think you can infer an ought from an is. (The actualist theory is obviously _ethically_ flawed, but Lewis’s argument doesn’t seem to turn on _ethical_ considerations, so that should be set to one side.) But does the actualist accept Lewis’s anti-Humean thesis? I think no, though just how they reject it depends on how we interpret V.
Let’s start with a practical case of ignorance. The Giants are playing the Brewers, and our actualist agent’s credence in A, the proposition that the Giants win, is 0.6. That is, C(A) is 0.6. Since the agent is an actualist and an anti-Humean of Lewis’s preferred kind, then A*, the proposition that A is objectively good, is just A. So C(A*) = 0.6. What is V(A)? Two options stand out.
One possibility is that V should measure how much the agent _values_ A. In that case V(A) is either 1, if the Giants win, or 0, if they don’t, but the agent doesn’t know which. Objectivists about value tend _not_ to think that agents can always know what is valuable, or even what they value. So we shouldn’t assume that values (which are external) can link up with credences (which are in a sense internal). So our anti-Humean rejects Lewis’s statement of anti-Humeanism.
Another possibility is that V(A) just equals C(A*) which is just C(A). This looks more like we’re accepting Lewis’s statement of anti-Humeanism. Not so fast! Lewis’s thesis is meant to hold for _any_ credence function C. In particular it’s meant to hold still when Barry Bonds hits a first-inning grand slam and her credence that the Giants will win rises to 0.8. Now C(A) is 0.8, so V(A) should also be 0.8 by this method.
What the agent accepts is that _her_ valuation function V perfectly tracks her credence function. She doesn’t accept that her valuation function does (or even _could_) track _all_ possible credence functions. What she accepts is something more like the following
bq. There is a function * from propositions to propositions such that for any evidence E, the desirability of A _given E_ for her equals her credence in A* given E.
We know that theory is coherent because our actualist, who thinks the star function is the identity function, is coherent. (Or are they incoherent in a way I’m missing?) And the theory is anti-Humean, at least on some interpretations of star. I must be missing something here I guess. Suggestions as to just what I’m missing are welcomed.