Talk:Raven paradox

Quote:

The statement "all ravens are black" is logically equivalent to the statement "all non-black-things are non-ravens".

The argument above is simply the contrapositive which is logically equivalant. There should not be any discussion about this. -- Ram-Man

Ummm....please see: Tuxedos, night skies, crude oil, soot. The statement "all [objects] have [quality]" is not equivalent to "all [things not that kind of object] do not have [quality]". For example: All humans are mammals. All frogs are not mammals. So far, so good. But all cows are mammals! Danger, Will Robinson! Obviously "All non-humans are non-mammals" is ridiculous.

Should be rephrased as "All non-black-things are *not* ravens". This is obvious, like saying all non-mammals are not humans. There is no paradox, as far as I can see.

-Emmett

Not really. Your example wasn't done properly. Let me redo it for you

Following what the quote says, All humans are mammals is equivalent to All non-mammals are non-humans. Let's test that using your examples

All frogs are non-human -- looks good

All cows are non-human -- looks bad - cows are mammals, and we're supposed to be checking non-mammals according to the quote.

As you say Obviously "All non-humans are non-mammals" is ridiculous. but that's not what the quote is saying. It says that "All non-mammals are non-humans" which is obviously commonsense. -- Derek Ross

The argument is flawed. If you found a green raven, is it a raven? Is its greenness enough to make it a different bird? In that case all ravens are black is a tautology. Ravens are by definition black. Finding a green raven will either change your definition or won't. See No true Scotsman. EW

That's just stepping into biology. Of course, if you found a green raven, then that would alter your belief in the statement "all ravens are black". But this article is not about that. It's about observing a non-black thing which is not a raven (clearly not a raven). -- Tarquin 11:43 Oct 28, 2002 (UTC)

Tarquin is absolutely correct. In fact you do occasionally find white ravens in the wild. So ravens are not by definition black. They're actually the children of older ravens. But all that means to the argument is that we'd eventually work our way through checking the non-black objects until we identified one particular (white) non-black object as a raven. At which point we'd note that the statements "All non-black objects are non-ravens" and "All ravens are black" are both false but both still equivalent, and decide to pick a better example. -- Derek Ross 12:19 Oct 28, 2002 (UTC)

Let me rework my argument. The article assumes that the statement "All ravens are black." has an absolute meaning independent of context, but that is not so. It can mean either: All normal ravens are black. or It can mean: There are no creatures that are fundamentally similar to ravens that are not black.

This ambiguity exists in all statements of this type. It can only be resolved if there is some way of determining the true meaning of the statement. In other words, the argument is intentionally misleading because it assumes something that is not true. EW

"All ravens are black" is just an example. We're using it because it's traditionally used for this paradox, to the point that the paradox bears the name of this example. -- Tarquin 15:24 Oct 28, 2002 (UTC)

Perhaps this example will clear things up. Let's take the statement "All Klefs are Smodgy" where a "Klef" is a prime number between 0 and 2,000,000,000,000 and an object is "Smodgy" if its name has appeared in print somewhere prior to the 1st of January 2000.

The sort of ambiguity that you see in the Raven statement doesn't appear in this one. There is no doubt about which numbers we are talking about. Neither does the self-reference. Smodginess is not a part of the Klef definition in the way that blackness is (arguably) part of the Raven definition. Finally the statement isn't assumed to be true. The point is that it has to be checked. We can't say whether the statement is true or not because we don't know whether the Klefs are Smodgy for sure until we check. And that's what the "paradox" is really about: the checking process. It states that we can get the same results whether we:

Check all the objects which we know to be Klefs in order to find out whether they are also Smodgy.
Check all the objects which we know to be not Smodgy in order to find out whether they are also not Klefs

Logic states that these two methods are equivalent, as far as checking whether the statement is true or not is concerned, but we know that the latter method takes a lot more work before we can be sure whether or not "All Klefs are Smodgy" is a True statement or a False one, and so to many people it seems paradoxical that it can work at all. Hence the title of this article. -- Derek Ross 17:42 Nov 10, 2002 (UTC)

I do appreciate your clarification of the paradox, but I still run into a problem. The problem is with the word paradox. I understand the word paradox to mean a contradiction that cannot be satisfactorily resolved no matter how you work at it. After your clarification of the problem, it is so clear that there is nothing to bother you about it any longer, and it no longer fits my definition of paradox. <grin> Ezra Wax 01:39 Nov 11, 2002 (UTC)

All paradoxes can be resolved. If there were an actual contradiction in the world, the world wouldn't exist (at least mathematically speaking) :-) AxelBoldt 02:09 Nov 11, 2002 (UTC)

Further discussion of paradoxes in general should probably go in Talk:Paradox... on another topic, I'm concerned about the paragraph that starts, "This principle is known as "Bayes' theorem". It is foundational to the mathematics of probability and statistics..." Now, I am personally a Bayesian, and I agree with the content of the paragraph — but any devoted frequentist would likely find it pretty inflamatory. A rewrite is needed... Cyan 07:36 Apr 12, 2003 (UTC)

The claim that the paradox does not arise using Bayes’ theorem strikes me as somewhat contentious. In the first example, the criterion for selection is an apple, not a non-black thing or a raven, so the example is irrelevant, and it is not surprising that selecting the apple makes no difference to the belief. In the second example, in which the selection is relevant, the paradox arrises, albeit to a very small degree. Why should observing a red apple be supporting evidence for ‘all ravens are black’ at all, even to a very slight degree? Banno 02:02, 20 Dec 2003 (UTC)

please re-read what you have written above: "an apple, not a non-black thing or a raven". An apple IS a non-black thing, unless your apples are very different to mine. -- Tarquin 18:50, 20 Dec 2003 (UTC)

The article says: 'If you ask someone to select an apple at random and show it to you, then the probability of seeing a red apple is independent of the colors of ravens' - what is specifically requested is an apple. On the implicit assumption that no apples are black (except the one I pulled out of my daughter’s schoolbag after the vacation) the fact that the probability of the apple being red is independent of the colour of ravens is trivial. That is, the sampling method biases the result. If, instead, you were asked to select a non-black thing, then the probability of picking a non-black raven must indeed be included, and so the probability of seeing a red apple would not be independent of the colour of ravens.

Re-stating my point, the first example is not an example of the Raven paradox, and the second example (in the last paragraph) shows that the raven paradox holds; therefore the statement that the paradox does not arise is simply not true. I suggest replacing it with 'this principle shows that the influence of such examples is vanishing small' or some such. Banno 20:37, 20 Dec 2003 (UTC)

I believe the paradox arises because the induction principle ("If an instance X is observed that is consistent with theory T, then the probability that T is true increases") doesn't take background information (such as sampling conditions) into account, so it sometimes leads to improper inferences. Consider the case that you are an expert ornithologist (i.e. untrickable) and you find a white raven feather. While this object is indeed a non-black non-raven, it's pretty obvious that blind application of the induction principle leads one to a logically indefensible increase in the probability that all ravens are black.

Bayes' theorem (as I've formulated it in the article) explicitly forces one to consider the prior information that goes into the probability calculation, such as the sampling conditions and whatnot. Sometimes using Bayes' theorem leads one to the same conclusions as the induction principle; other times, it will not. (For instance, try applying it to my example above.) Bayes' theorem resolves the paradox because it takes care of the cases when the induction principle fails due to lack of consideration for background information. -- Cyan 02:23, 21 Dec 2003 (UTC)

I agree with your analysis. So in effect the Bayesian approach agrees with that of the red apple example given earlier in the article (from Quine isn’t it?)– observing a red apple really does increase the chance of all ravens being black, since it slightly increases the confirmation in the background information. OK, thanks for the reply. Banno 10:58, 21 Dec 2003 (UTC)

Actually, I'm inclined to say that given the implicit background information most people are using, the observation of a red apple (drawn at random from a world of objects) conveys no information about ravens at all: the color of the apple is irrelevant to (i.e. independent of) the color of any or all ravens, and vice versa. The reason I say this is because the red apple gives equal support to all theories of the form "All ravens are Z", where Z is any color. In this case, Pr(X|TI) = Pr(X|I) for any given T (in English, the probability of observing the apple is the same no matter what theory of raven color we assume), so by Bayes' Theorem, Pr(T|XI) = Pr(T|I) for any given T. -- Cyan 17:26, 21 Dec 2003 (UTC)