I do see the sense in what you’re saying. Here’s my thinking—maybe you could help me clarify it:
In the finite case, if we see one non-black raven, then the “all ravens are black” hypothesis is shot immediately[1]. If we see one white bear (or whatever), that doesn’t do much for “all ravens are black”, but it does a little; likewise if we see one black raven. But eventually, after we’ve examined every object in the universe, and found that none of them are nonblack ravens, then we can be as sure of “all ravens are black” as we would be of “not all ravens are black” in the “find one white raven” case[1].
[1] In either of these cases, the certainty would not be 100%, as we could be deceived by an evil demon, Descartes-style, or otherwise mistaken; but such possibilities exist in either scenario, and so the certainty is equally high, on the order of 1 minus epsilon.
In the infinite case, it is still true that if we see one non-black raven, then the “all ravens are black” hypothesis is shot immediately. However, there is no amount of objects we can examine, no amount of evidence we can gather, that would allow us to be as certain of “all ravens are black” as we would be of “not all ravens are black” in the “find one white raven” case. In the infinite case, therefore, the two situations are asymmetric.
To put it another way, in the finite case, the “all ravens are black” hypothesis may be disconfirmed[2] (by observing one white raven), and it may also be confirmed[2] (by observing all objects and finding no nonblack ravens). But in the infinite case, while the “all ravens are black” hypothesis may still be disconfirmed (by observing one white raven), it can never be confirmed—because however many objects we examine, there are still, always, an infinite number of objects remaining, of which one or more may be nonblack ravens.
[2] Modulo the epsilon chance of being deceived somehow.
This seems like a clear difference, to me. I do not see how the Bayesian approach resolves this asymmetry.
I agree that there’s an asymmetry here, and that it’s possible to confirm, in the usual sense, the all-ravens-are-black hypothesis in a finite universe but not an infinite universe. I think the Bayesian approach “resolves” this in the sense that it treats this confirmation as a special case of general evidence-gathering, with no particular special status.
An intuition pump: suppose that, instead of infinity ravens, there were merely a trillion of them for every atom in our visible universe (stored in the next universe over, of course). Clearly this is an important difference conceptually, with implications for physics at the very least. But what difference could it make for a human who has so far seen a mere thousand black ravens, and wants to predict the color of the next raven they see? Should they make different predictions, using different reasoning processes, in these cases?
If the ravens in question are in the next universe over, then we’ll never see them, regardless of their number or color.
That having been said, I think I get what you mean to say (sort of?), but it doesn’t seem to me to bear on the point. Consider these scenarios:
Scenario 1: There are a trillion ravens, and all are black.
Scenario 2: There are a trillion ravens, and all but one are black.
In both cases, if I’ve seen a mere thousand black ravens so far, I predict that the next raven I see will be black.
But in one case, “all ravens are black” is true, and in the other, it is false! So I am just not convinced that “what do you predict will be the color of the next raven you see” is even a relevant question, w.r.t. this paradox.
I do see the sense in what you’re saying. Here’s my thinking—maybe you could help me clarify it:
In the finite case, if we see one non-black raven, then the “all ravens are black” hypothesis is shot immediately[1]. If we see one white bear (or whatever), that doesn’t do much for “all ravens are black”, but it does a little; likewise if we see one black raven. But eventually, after we’ve examined every object in the universe, and found that none of them are nonblack ravens, then we can be as sure of “all ravens are black” as we would be of “not all ravens are black” in the “find one white raven” case[1].
[1] In either of these cases, the certainty would not be 100%, as we could be deceived by an evil demon, Descartes-style, or otherwise mistaken; but such possibilities exist in either scenario, and so the certainty is equally high, on the order of 1 minus epsilon.
In the infinite case, it is still true that if we see one non-black raven, then the “all ravens are black” hypothesis is shot immediately. However, there is no amount of objects we can examine, no amount of evidence we can gather, that would allow us to be as certain of “all ravens are black” as we would be of “not all ravens are black” in the “find one white raven” case. In the infinite case, therefore, the two situations are asymmetric.
To put it another way, in the finite case, the “all ravens are black” hypothesis may be disconfirmed[2] (by observing one white raven), and it may also be confirmed[2] (by observing all objects and finding no nonblack ravens). But in the infinite case, while the “all ravens are black” hypothesis may still be disconfirmed (by observing one white raven), it can never be confirmed—because however many objects we examine, there are still, always, an infinite number of objects remaining, of which one or more may be nonblack ravens.
[2] Modulo the epsilon chance of being deceived somehow.
This seems like a clear difference, to me. I do not see how the Bayesian approach resolves this asymmetry.
I agree that there’s an asymmetry here, and that it’s possible to confirm, in the usual sense, the all-ravens-are-black hypothesis in a finite universe but not an infinite universe. I think the Bayesian approach “resolves” this in the sense that it treats this confirmation as a special case of general evidence-gathering, with no particular special status.
An intuition pump: suppose that, instead of infinity ravens, there were merely a trillion of them for every atom in our visible universe (stored in the next universe over, of course). Clearly this is an important difference conceptually, with implications for physics at the very least. But what difference could it make for a human who has so far seen a mere thousand black ravens, and wants to predict the color of the next raven they see? Should they make different predictions, using different reasoning processes, in these cases?
If the ravens in question are in the next universe over, then we’ll never see them, regardless of their number or color.
That having been said, I think I get what you mean to say (sort of?), but it doesn’t seem to me to bear on the point. Consider these scenarios:
Scenario 1: There are a trillion ravens, and all are black.
Scenario 2: There are a trillion ravens, and all but one are black.
In both cases, if I’ve seen a mere thousand black ravens so far, I predict that the next raven I see will be black.
But in one case, “all ravens are black” is true, and in the other, it is false! So I am just not convinced that “what do you predict will be the color of the next raven you see” is even a relevant question, w.r.t. this paradox.