It has always seemed to me (and several commenters in the linked thread also note this, though no discussion seems to have ensued about it) that the critical issue in the black raven paradox is whether the number of objects in the universe is finite or infinite. All other considerations I’ve seen mentioned are, as far as I can tell, subordinate to this one.
I think the intuitive reason to think infinity is important is if you try to use the uniform distribution for everything. After all, there is no uniform distribution over an infinite extent—it becomes zero everywhere, which is an illegal distribution. So one might end up thinking something like “if there are infinite things, there might be evidence out there but I have zero probability of seeing it.” (Though apologies if this is crappy mindreading)
But if you just put your faith in a theorem called Bayes and say that you have some distribution over what you’re going to see, this solves the problem equally well for the finite and infinite case. In both cases, you’re allowed to have hypotheses involving unobservable yellow ravens, but they’re never mandated. Any distribution you choose has a quantitative answer for how much evidence you expect to see.
I think this is the wrong picture of how evidence works. The purpose of looking at things is not that it’s one step towards looking at all the things. The purpose of looking at things is that it helps you discriminate between hypotheses that make different predictions about what you’ll see.
Looking at all the things would be nice if we could do it, but it’s not necessary for knowledge. Like, we have a much better understanding of induction now (relative to Frege and Carnap’s day) - there’s no need to walk backwards from it, into some epistemology where you can’t know about stuff if you can never see all of it.
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.
Adding this stipulation would, however, largely remove the sense of paradoxicalness from the paradox. (Also, I don’t know if I believe that an object can provide an infinitesimal amount of evidence. I’m not sure what it would mean for this to be the case. How do infinitesimals apply to probability values? I have no idea! Certainly, adding infinitesimals into the mix quite severely damages any connection which might otherwise exist between numerical probability values and any intuitive sense of “how likely something is” that I might have—in other words, it severely undermines my confidence that probability values represent what they claim to represent.)
It has always seemed to me (and several commenters in the linked thread also note this, though no discussion seems to have ensued about it) that the critical issue in the black raven paradox is whether the number of objects in the universe is finite or infinite. All other considerations I’ve seen mentioned are, as far as I can tell, subordinate to this one.
I think the intuitive reason to think infinity is important is if you try to use the uniform distribution for everything. After all, there is no uniform distribution over an infinite extent—it becomes zero everywhere, which is an illegal distribution. So one might end up thinking something like “if there are infinite things, there might be evidence out there but I have zero probability of seeing it.” (Though apologies if this is crappy mindreading)
But if you just put your faith in a theorem called Bayes and say that you have some distribution over what you’re going to see, this solves the problem equally well for the finite and infinite case. In both cases, you’re allowed to have hypotheses involving unobservable yellow ravens, but they’re never mandated. Any distribution you choose has a quantitative answer for how much evidence you expect to see.
The reasoning I have in mind is basically that described in this comment (and also, to some extent, this comment).
I think this is the wrong picture of how evidence works. The purpose of looking at things is not that it’s one step towards looking at all the things. The purpose of looking at things is that it helps you discriminate between hypotheses that make different predictions about what you’ll see.
Looking at all the things would be nice if we could do it, but it’s not necessary for knowledge. Like, we have a much better understanding of induction now (relative to Frege and Carnap’s day) - there’s no need to walk backwards from it, into some epistemology where you can’t know about stuff if you can never see all of it.
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.
That’s not important if you believe that an object can provide an infinitesimal amount of evidence.
Adding this stipulation would, however, largely remove the sense of paradoxicalness from the paradox. (Also, I don’t know if I believe that an object can provide an infinitesimal amount of evidence. I’m not sure what it would mean for this to be the case. How do infinitesimals apply to probability values? I have no idea! Certainly, adding infinitesimals into the mix quite severely damages any connection which might otherwise exist between numerical probability values and any intuitive sense of “how likely something is” that I might have—in other words, it severely undermines my confidence that probability values represent what they claim to represent.)