Raven Paradox Revisited
The Raven Paradox Settled to My Satisfaction is a pretty good post, but there’s a few things that we can note to make this problem even clearer:
We can simplify this problem so that there are only two colors (black and white) and two kinds of objects (ravens and laptops). The post sort of did this, but not very consistently. I just thought this was worth mentioning as non-black and non-raven are slightly more abstract and so slightly harder to reason with.
Actually, we can make this even more legible. Make the objects medicine and food; and the properties fresh and expired. It’s then immediately clear that if we want to check that [All the medicine is fresh], we can either check each item of medicine to see if it is fresh or we can check all the expired objects and see that none of them is medicine. We’re used to this kind of practical reasoning, so it’s much easier for us than dealing with objects and colours.
If we have simplified the problem so there is only one shade of black, then A: [All ravens are black] is equivalent to B: [The first raven is black] plus C: [All ravens are the same color] apart from the degenerate case. This makes it rather clear why A is more likely to be true when there are less ravens. In particular, if there’s only one raven, then we only have to worry about B since C is trivially true.
We tend to confuse the following [An observation of a white laptop is independent of the color distribution of ravens] with [An observation of a white laptop is independent of how many ravens of a particular colour we will observe]. The first is true apart from the restrictions imposed by the number of black ravens having to be integral, but the second is only true if we knew it was a laptop before this observation. If it could have been a raven then it can influence the number of ravens of a particular color by influencing the number of ravens in total.
We tend to expect some correlation between the color of animals. For example, we are quite ready to guess that all ravens are black after only seeing quite a small random sample. On the other hand, suppose that all the non-black things keep being non-living creatures, whilst many of the black things are non-raven living creatures. We might guess that being black provides an evolutionary advantage in this world and so guess that all ravens will be black without ever having seen a single one. The point is that, given particular priors, you may have additional evidence beyond merely the numerical reduction.
Hempel’s resolution from Wikipedia is worth highlighting as it subtly reframes the problem to make the assumptions more obvious. Consider the statement [All sodium salts burn yellow], with the contrapositive [Whatever does not burn yellow is not a sodium salt]. Burning some ice and finding it does not tell yellow would be evidence towards the contrapositive and hence also the original statement. This seems paradoxical, but consider if the chemical makeup was unknown at the start. If something doesn’t burn yellow and then we analyse it and discover it is a sodium salt then we would have disproved the hypothesis. By conservation of evidence, if we discover it is ice we would gain evidence for the hypothesis. It’s easy enough to calculate the amount of evidence using Bayesian techniques. From here, it’s easy enough to see that this observation only provides no information if we implicitly assume that we know the chemical composition (or the type of object in the raven problem).
This is my go-to equivalent scenario that doesn’t sound so paradoxical:
Suppose you work at a bar frequented by young people, some of whom can legally drink alcohol and some of whom can’t. You’re trying to collect evidence for the statement “All underage people at this bar are drinking non-alcoholic drinks.” One way you could approach this is by going up to a sample of your patrons, carding them to learn their age, and checking the glasses of the ones who are underage. If you check the glasses of a lot of underage people, and none of them are drinking alcohol, that’s good evidence. But another thing you can do is check everyone’s glasses, and only card the people who are drinking alcohol. If you card a bunch of people who are drinking alcohol, and all of them are overage, then that’s also good evidence.
Similarly, there are two ways I could collect evidence for the statement “All ravens are black”. Every time I see a raven, I could remember to check that it’s black; or every time I see a non-black thing, I could remember to check that it’s also not a raven. But given that there’s an immense quantity of non-black things, and that my mind does not automatically categorize objects by color, the second approach doesn’t sound like something I could actually do. And so it doesn’t feel like observing a yellow banana is evidence for all ravens being black.
Yeah, this is better than my example of food & medicine.
I’m pretty happy with how this post has aged. Good work, 2014-self! Clearly, you’re much more reliable than 2011-self! I also endorse Richard Kenneway’s comments if you want more rigor, though they might be a bit confusing if you forget the central point of “how you should update one hypothesis depends on what other hypotheses are competing with it.”
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.)