A very plausible normative principle relating subjective degree of belief to objective chance is David Lewis’ “Principal Principle”. In a simplified version, this principle says that if you know the objective chance of some inherently chancy outcome, then your degree of belief in that outcome should equal the chance. Thus, if you know that the coin is fair, then you should have degree of belief 1/2 that it will come up heads.

This has some added bite because the chance information overrules a lot of other information – if you know the coin is fair, then it doesn’t matter how it happened to come up on the last 1000 flips, you should still believe in heads to degree 1/2. Even if the last 1000 flips were all tails – this is one idea of what’s fallacious about the gambler’s fallacy (or inverse gambler’s fallacy).

Of course, some sorts of information can overrule the chance information – if a very accurate fortuneteller has told you that the coin will come up heads, then maybe you should believe to a degree higher than 1/2, even though you still believe the coin is fair. This sort of information is what Lewis called “inadmissible” information. The question for the Principal Principle then is just what counts as inadmissible information?

To answer this, I think we need to consider just what chance really is. On one notion of chance, it requires that the world be objectively indeterministic, so that there is no fact of the matter about future chancy events. On this account, the idea of an accurate fortuneteller for chancy events doesn’t even make sense. This might be a natural view of chance that arises from the many-worlds interpretation of quantum mechanics. On this view, the chance of an event could potentially depend on anything for which there is a fact of the matter – but this only includes facts about the past and present. But since you’d need to know all this information (or the relevant parts anyway) to know the chances, there will trivially be no possibility of inadmissible evidence, so the Principal Principle stands (if at all) in a very simple form!

But there are other notions of chance I’ve heard people talk about. One is supposed to be compatible with strict determinism. I don’t know too many of the details, but I suspect that the idea is that there’s some natural class of “nearby worlds”, and chance is just some sort of probability measure on those worlds. This can definitely give rise to non-extreme values for chances, even though there is no possibility other than necessity. However, on this interpretation of chance, I don’t see why anything like the Principal Principle would have any normative force at all. I suppose if you can somehow narrow things down enough to know what the chances are, but can’t eliminate any of the worlds in the class that defines the chances, then it would make sense. But it’s far from clear to me why this situation would be at all common.

Then of course there’s Lewis’ own characterization of chance. I believe his idea is that one can read off the natural laws of a world by seeing what best systematizes the entire history of it. If there are certain types of events that have no interesting pattern to them at all except for a certain limiting frequency, then the best way to systematize these will be with chancy laws. In this setting it’s not clear how one would justify the Principal Principle, or how one would claim to have knowledge about the chances.

At any rate, the Principal Principle seems to say different things on these different interpretations of chance, and it gives rise to either different justifications or different accounts of what should count as “inadmissible evidence”.

Sridhar(14:17:12) :In terms of the first view, one needn’t restrict it in a temporally asymmetric way. Just as there may be no fact of the matter about future chancy events, the same could hold of past chancy events as well; the status of “Tomorrow’s coin flip will come up heads” could be entirely analogous to the status of “The bit which was written on this paper until being erased yesterday (with all traces of its particular value removed) was a 1”.

For that matter, there may be no fact of the matter about present chancy events either (say, “The bit which is written on the paper locked away right now in this box…”). The chanciness of an event could be entirely orthogonal to its temporal status, the attribution of “there is a fact of the matter” status to propositions occurring on some unrelated (or, at least, not necessarily related) basis.

rabriggs(08:09:40) :I don’t think the deterministic case is quite as bad as it seems. Suppose I believe (or suspect) that the chance of a roulette wheel’s landing on red is 1/2. I then learn that the wheel is being spun by an enthusiastic croupier. I’ve ruled out some worlds in which the chance of the roulette wheel’s landing on red is 1/2 (namely, those worlds in which its chance of landing on red is 1/2 and it is spun by an enthusiastic croupier). This does not mean that I have inadmissible information, because general propositions about the cropier’s level of enthusiasm don’t tell you anything about how the wheel will land, conditional on propositions about the (non-fundamental) chances. Your credence function screens the croupier-enthusiasm propositions off from the chance propositions.

Ideally, the believer in non-fundamental chances should have a story about why your credence function ought to screen croupier-enthusiasm propositions off from chance propositions. Michael Strevens seems to have such a story: very abstractly, the idea is that every set of worlds where the roulette wheel lands on black can be matched with a very similar set of worlds with the same measure where the roulette wheel lands on red, and that this still holds true if you restrict yourself to worlds with enthusiastic croupiers.