David Lewis thought the following is true:
The Principal Principle (PP): Ps(A|Po(A)=x)=x (where Ps is a rational subjective probability, and Po is some objective probability).
Here’s a gloss of this claim, more or less accurate: if you know that the objective probability of A is x, then you should assign degree of belief x to A given that information.
So, first, a confession: I know there’s been considerable discussion of this principle in the literature but I haven’t read as much of it as I should. So there may easy answers here to the things that concern me, and pointing them out in the comments would be useful.
Second, my problem: I think, to go straight to the bottom line, that I may not know how to interpret conditional probability claims.
Suppose I think of a conditional probability as implicitly referring to some background knowledge in the given condition, so that PP implicitly quantifies over every cognitive agent and their systems of information. Then the principle is false, regardless of what kind of probability is included here, since the background information might conflict with the stated condition.
One alternative here is to abstract from one’s actual system of information, ridding it of everything epistemically relevant to the truth or falsity of A before adding (just) the information contained in the condition. Then PP says something like this: if the only relevant information you had about A was that its objective probability is x, then your subjective probability for A should be x as well.
I’ve never been comfortable with abstractionist interpretations of conditional probability claims. What is the conditional probability that you exist given that I tell you so? (This is sloppy: I don’t want the report of testimony to guarantee your existence but just to be a sincere and honest assertion of it on my part.) The abstractionist interpretation requires you to consider a situation in which your only information about your existence is my testimony. I don’t see how such a situation is possible, or conceivable, or imaginable.
Perhaps one should put the claim in form of a conditional, as my gloss of PP above did, where I reported the principle as claiming that if one knows that the objective probability of a claim is x, then one’s subjective probability should be x as well. But that won’t work with probabilities that conditionalize on one’s own non-existence. What is the conditional probability of anything given that you do not exist? If we understanding probability in the subjectivist sense, this makes little sense. It’s hard to imagine your knowing of your non-existence. Of course, this might be an argument for a different understanding of probability, but we can’t take that route for the principle above, since it explicitly appeals to subjective probability.
So, assume that the condition in the principle is that the objective probability of my existence is zero. Note that this supposition is compatible with my existence, so we can’t rely on such a claim to generate an answer here concerning the application of the Principal Principle to such a case. What should my subjective opinion be about my existence?
Is there something I’m missing here? Maybe so, but if not, here’s what I think the idea is that makes the principle look plausible. It’s a claim about evidence or confirmation. It makes a claim about prima facie evidential support, to the effect that if you learn the objective probability of a claim, that is evidence that the total evidence supports the claim to the same degree. That allows one to say that the degree of confirmation provided for p by the evidence that the objective probability of p is x, is precisely the same (even though all such evidential claims are only prima facie).