Thursday, August 04, 2005

The Raven Paradox

Suppose you're wondering whether all ravens are black. Intuitively, observing black ravens would count as inductive evidence for the universal claim. However, the following three statements are all logically equivalent:
(S1) All ravens are black.
(S2) All non-black things are non-ravens.
(S3) Everything is either black or non-raven (or both).

Since the statements are all equivalent, they must be confirmed (or not) by the same evidence. (In particular, each statement is disconfirmed by finding a non-black raven.) But generalizing the sort of logic used in our intuitive claim above, we can say that observing non-black non-ravens is evidence for (S2), and therefore (S1). So observing a red herring is evidence that all ravens are black! But this seems absurd.

I think the "paradox" can be resolved once we recognize why it is that observing black ravens supports the claim that all ravens are black. Hempel claims:
R1: Whenever an object has two attributes C1, C2, it constitutes confirming evidence for the hypothesis that every object which has the attribute C1 also has the attribute C2.

But this, I claim, is mistaken. It makes a difference how you came by the observation. If you go searching solely for black things, and bring me back a sample of them, the observation of black ravens in this collection will not provide any evidence whatsoever that all ravens are black.

Why not? Well, recall that a non-black (henceforth, 'coloured') raven is what would falsify our statements S1-S3 (which from now on I will just call 'S'). The perfect test of S would be to exhaustively search the world for coloured ravens. If we found one, S would be shown to be false, but otherwise we would have established its truth. But we can also do imperfect tests, which merely sample the population of worldly objects, rather than doing an exhaustive census.

Suppose we took a random sample of objects from the world. Every sampled object that was not a coloured raven would count as (weak) evidence that there are no such creatures. For if there were coloured ravens then we should expect them to turn up occasionally in random samples. The fact that they don't, counts against them.

But this evidential justification requires that the sample be drawn from a population that would include coloured ravens were any to exist. For suppose that you go and collect a bucket of herrings for me. There is no chance that there are any coloured ravens populating this bucket. So their absence from my observations cannot count against them. My observations of nothing but herrings in the bucket is precisely what we should expect even if there are coloured ravens in the world. (Granted, each red herring we sample is evidence that there are no coloured ravens in our sample population - i.e. the bucket - but we already knew that!) Similarly for populations of black objects. If we purposefully sample only black things, then there is no chance of finding coloured ravens even if they exist. So, again, our failure to find any does not count against them. (It just tells us the trivial result that there are no coloured ravens among the black population.) So observing black ravens in such a sample does not at all count as evidence that all ravens are black! Hempel is simply mistaken about the nature of confirming evidence.

Now suppose instead that we took a random sampling from the population of ravens. If any coloured ravens exist, they would have some chance of turning up in the sample. So the failure to find any coloured ravens in such a sample would count as evidence against them, and in favour of S. Moreover, the same can be said of a random sample of coloured (non-black) objects -- or even a random sample of all the objects in the world. Every time our randomly chosen sample object turns out not to be a coloured raven (say it is a red herring instead), this offers some very slight evidence that there are no coloured ravens to be found in the broader population from which the sample was taken. So it helps confirm S.

There are pragmatic reasons why black ravens could count more heavily than red herrings in confirming S. This is because the population of ravens is much smaller than the population of coloured things. So, if any coloured ravens exist, they will form a much higher proportion of the raven population than they will the coloured population. Since they should be easier to find in the raven population, the failure to find them there will count more heavily against them. It's just a whole lot more practical to sample the raven population than it is the coloured (or even universal) population. But this difference is merely one of degree, not kind.

Indeed, if we are taking a random sample from the entire population of worldly objects, then there is no evidential difference between finding a black raven or a red herring in the sample. Each offers equally weak evidence for S, merely in virtue of not being a coloured raven.

I hope that's all relatively clear. I've tried to explain it informally, but one could probably formalize my arguments with Bayesian statistics. Anyway, the point is that "Hempel's paradox of confirmation" isn't really paradoxical at all. It simply serves to highlight some common misunderstandings about the nature of inductive evidence.


  1. I think this is a good argument, Richard; Hempel's claim, taken straightforwardly, can't distinguish evidentially between 'Some ravens are black' and 'All ravens are black, and obviously a theory of confirmation has to be able to do so. This would be a good post to work up more rigorously into a paper.

  2. Thanks! I might look into it for my philosophical logic paper. I imagine someone else will have made similar arguments before, though...

  3. The problem is that you already have to know an awful lot about ravens in order to be able to take a random sampling of them without also taking a random sampling of of everything.

    At minimum you have to know how to identify them and where to look.
    But you will presumably know these things on the basis of prior inductions, at least some of which would only seem to be possible by examining a random sampling of everything.

    Yet we are presumably in no position to ever do a random sampling of everything. Perhaps this can be avoided by limiting our inductions to claims with a limited geographic scope ('All ravens on earth are black'). Still it looks like inductions based on surveys of non-geographically defined subgroups will depend on prior surveys of the larger group.

    I don't know the literature on induction at all, so I don't know whether this is a standard problem and whether there are standard answers to it.

  4. Is he suggesting you search only for "non ravnens"? he cant be suggesting you search only for black things because he suggests the proof is finding a red herring...
    Anyway if everything was a red herring I guess there would be miniscule proof there but as long as the method of search has the potential to find disconfirming 9finding a coloured raven OR running out of things to find) or nessercary evidence (by nessercary I mean finding one black raven) it is ok I guess.

  5. I wouldn't be surprised if the arguments have been made before; but in philosophy, of course, that doesn't mean that the dispute has effectively been ended. And it's also possible that they haven't been made. I don't keep up with this sort of issue much, but I do browse a lot of phil. sci. articles. Your solution is to reject what's usually called Nicod's condition (which was the name eventually given to the principle in Hempel that you give) and to reject the claim that non-black things being non-ravens does not confirm that all ravens are black. There have been solutions before that have rejected Nicod's condition, but the most recent article I know of to discuss the matter, by Patrick Maher in Philosophy of Science (in 1999 or so), still claims that we have no good reason to reject Nicod's condition. And although I haven't looked closely at the debates at all, it seems to me that most of those who reject Nicod's condition try to do so by arguing that, given certain background evidence, it fails. But the original raven's paradox was formulated on the assumption that we have no such background evidence to make the probabilities go strange. And they all tend to assume that finding a non-black non-raven doesn't confirm (S1). Now, you do introduce background information (the random sampling of particular populations), but you go on to argue for a conclusion in which there is no background information (the random sampling of the whole world); and you allow non-black non-ravens to confirm (S1). So while there might be similar arguments, your solution is fairly distinctive. Given that it's not something I look at often, I could just be missing out on a large section of the discussion; but it seems to me to be fairly new. (And while it comes under similar qualifications, your general approach to the problem seems to me to be more promising than most approaches, which try either to accept Nicod's condition, or to deny that the observation of non-black non-ravens would confirm (S1).) Maher's paper gives a good brief overview of the major solutions proposed for the Raven Paradox, so if you're interested in that, you might look it up.

    You probably won't have difficulty finding a writing sample to submit when you get around to applying to grad school; but this sort of topic is ideal for such a sample, because it would give you an opportunity to show that you can work out a distinctive position, argue it analytically, do historical research to compare it to previous positions, etc. If you like how the paper turns out for your philosophical logic class, you might consider developing it even further.

  6. Very lucid, Richard. I had satisfied myself that the raven paradox wasn't really a paradox by supposing that you saw something red up in a tree and wondered if it was a raven. Once you look closer and see that it's a parrot, or a herring in a tree, or some other non-raven, then that provides some evidence for S.

    More theoretically, we can say that the way to confirm S is to try as hard as you can to falsify S. If you fail to find a colored raven despite your efforts, then that provides evidence in favor of S. Since searching for colored ravens among the set of ravens is more efficient than searching for them among the set of colored objects, the former kind of evidence should be given more weight.

  7. Yeah, that's a clear and helpful summary of the core idea behind by argument. Thanks :)

  8. This reminds me of Popper's theory of falsification as a necessary for proving a theory true. Since of course, we can never empirically verify that all ravens are black.

    You've givingan interesting idea of weak proof, so for any object O(1) with the property A(x), we find that any O(2) without the property A(x), it proves weakly that O(1) has the property A(x)?

  9. Not quite. Observing O(2) doesn't provide us with any positive information about other objects. It rather suggests that all other types of objects are less likely to exist. I explain this further here.

    As for Popper, he thought you could never confirm statements as true, or even likely to be true. By contrast, I hold that adopting his falsificationist methodology can provide confirming evidence for hypotheses (if the attempts to falsify them fail).

  10. I don't know if my logic is flawed but...

    The paradox works because intuitively it seems that a black raven is evidence of the claim "all ravens are black," when it is not. There are only two types of evidence for such a statement. All ravens are black, or all non-black things are non-ravens. No other evidence supports the claim because any number of ravens that are black but do not constitute the whole is not evidence.


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