Subjunctive credence and statistical chance
In her 2012 paper "Subjunctive Credences and Semantic Humility" (2012), Sarah Moss presents an interesting case due to John Hawthorne.
Suppose that it is unlikely that you perform a certain physical movement M tomorrow, though in the unlikely event that you contract a rare disease D, the chance of your performing M is high. Suppose also that the combination of contracting D and performing M causes death. Then many judge that the objective chance of 'if you were to perform M tomorrow, you would die' is low, but the conditional objective chance of this subjunctive given that you perform M is high.
The intuitive judgments Moss reports are
(1) Ch(M => Die) < 0.5
(2) Ch(M => Die/ M) > 0.5
where 'Ch' stands for objective chance and '=>' is the subjunctive conditional. Since the chances are known, it is also plausible that
(3) Cr(M => Die) < 0.5
where 'Cr' denotes credence. After all, you're unlikely to have the disease, and you're not going to get it by performing M. So it's unlikely that you would die if you were to perform M.
Moreover, by the centring principle for conditionals, (2) simplifies to
(4) Ch(Die/M) > 0.5
This also looks plausible, and it can be justified by simple probabilistic reasoning. The crucial point is that M is much more likely given D than given ~D. For concreteness, let's say that
(5) Ch(M/D) = 0.99
(6) Ch(M/~D) = 0.01
(7) Ch(D) = 0.1
By Bayes's Theorem, it follows that Ch(D/M) = 0.917. And since D & M entails death, Ch(Die/M) is at least 0.917. So (4) is correct.
Together, (3) and (4) provide a counterexample to Skyrms's Thesis, which says that one's credence in subjunctive conditionals should equal one's expectation of the corresponding conditional chance. Somewhat simplified:
(ST) Cr(A => C / Ch(C/A)=x) = x
There are well-known limitations to Skyrms's Thesis, but they typically involve agents with "inadmissible information". Nothing like this seems to be going on in Hawthorne's scenario.
I think what the scenario brings out is that Skyrms's Thesis relies on a special conception of chance on which (4) is actually false.
Here is the argument against (4). Suppose you do not have the disease D. In this case, what's the chance that you die if you perform M? Practically zero. M is evidence for D, but it doesn't cause you to have the disease if you don't already have it. So on the assumption that you don't have the disease, Ch(D/M)=0. Similarly, if you do have the disease, then M won't cure it: Ch(D/M)=1. Now we don't know whether you have D, and so we don't know whether the conditional chance is 1 or 0, but it's much more likely to be 0 than 1. The expectation of the chance is below 0.5, as predicted by (ST).
So what's wrong with the above derivation of (4)? The derivation relied on a statistical conception of chance. Statistically, there is a high probability of M given D, and a high inverse probability of D given M: most agents who perform M have D, and most patients with D perform M. But subjunctive conditionals and subjunctive credence does not track mere statistical correlation. So the chance Ch in Skyrms's Thesis shouldn't be interpreted in this statistical way.
Statistically, there's a high chance that you're a woman if you buy a certain magazine. That doesn't mean that a given male shopper would be a woman if he were to buy the magazine.
But the case does raise a worry. Philosophers with sympathies for Skyrms's Thesis (including Skyrms himself) often don't want to restrict chance to fundamental micro-physical propensities. They want to extend the principle to the higher-level probabilities of statistical mechanics. But aren't these statistical chances?
According to statistical mechanics, there's a high chance that an ice cube will melt if it is dropped in hot coffee. The chance is not 1 because there are unusual configurations of ice and coffee that would prevent the melting. Now suppose I have some ice and coffee in this unusual configuartion, so that the ice wouldn't melt if I were to drop it in the coffee. In a sense, there is still a high chance that the ice would melt, but this seems to be a merely statistical sense. Why does Skyrms's Thesis apply here, but not in Hawthorne's case or in the case of the shopper?