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Raven Paradox

dylancatlow
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7/9/2016 9:51:56 PM
Posted: 4 months ago
Take the following two hypotheses:

(1) All ravens are black.
(2) Everything that is not black is not a raven.

At first glance these two statements would seem to be saying essentially the same thing. If this is the case, then providing evidence for one of the statements is to provide evidence for the other. The paradox results from the fact that it is possible to provide evidence for the second statement (and thus the first) without ever referring to ravens at all. For example, if one observes a green apple then one has confirming evidence for the hypothesis that all non-black things are not ravens. This would seem to imply that it is possible to gain knowledge about ravens by looking at apples.

The notion that the above situation constitutes a paradox relies on two crucial assumptions, neither of which are true: that the two statements are in fact logically equivalent, and that observing a single black raven is evidence in favor of the hypothesis that all ravens are black. In other words, evidence in favor of the second statement is not evidence in favor of the first statement because they are not equivalent statements, and because observations do not constitute evidence in favor of either hypothesis anyway.

The statements are not logically equivalent because each statement contains a hidden assertion which the other doesn't. In order for all ravens to be black some ravens have to exist (unless it's tautologically true, in which case empirical evidence is beside the point and the paradox ceases to be meaningful). Thus, to claim that all ravens are black is also to claim that ravens exist. The statement can be reformulated as follows: "There are ravens and all of them are black". On the other hand, the second statement assumes the existence of non-black things, while the first statement doesn't. Their non-equivalence is illustrated by the fact that one statement can be true and the other false. Namely, if the world contains no ravens but contains non-black entities, then the statement "All ravens are black" is false while the statement "everything that is not black is not a raven" is true.

Even if the statements were equivalent, observing a green apple wouldn't be evidence in favor of the hypothesis that "all ravens are black" because observing a black raven isn't either. Imagine that you just observed a black raven and conclude on that basis that the statement "all ravens are black" has a higher than 50 percent chance of being true. You are then asked to consider how likely you think it is that there exists somewhere in the universe a planet populated entirely by white ravens. In the absence of any evidence you come up with an answer of 50 percent. You then realize that if you have no scientific grounds on which to rate the probability as being lower than 50 percent, you can hardly say that the probability of "all ravens are black" being true is higher than 50 percent. Since the two hypotheses are at odds with each other, you cannot claim that "all ravens are black" has a higher than 50 percent chance of being true unless you are also willing to claim that "there exists a planet filled with white ravens" has a lower than 50 percent chance of being true. Since the observation of a single black raven here on earth is completely irrelevant to the world-filled-with-white-ravens hypothesis, it's not evidence of anything on its own. It doesn't raise the probability of "all ravens are black" being true because it doesn't raise the probability of "there exists a world filled with white ravens" being false.
keithprosser
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7/10/2016 5:42:14 PM
Posted: 4 months ago
Essentally I see that as the problem of defining what 'All X are Y' means when there are no X's. (In the OP X is 'ravens' and Y is 'black').

Conventionally, 'All X are Y' is taken as true if there are no X. That has the advantage of making (1) All ravens are black and
(2) Everything that is not black is not a raven.
logically equivalent even if there is no suc thing as a raven.

It also means that (i)'All unicorns have a horn' is true but 'paradoxically' so is (ii)'All unicorns are hornless'! But (ii) is equivalent to 'Everything with a horn is a not a unicorn', which we readily can see is true - if you gathered everything with a horn into a (big!) field none of them would be a unicorn - QED!

As dylan points out the issue of whether some example of X exists is a factor. But if you adopt the convention described above then if you prove 'All X are Y' it would be invalid to infer X exists - you'd have to do a separte proof of existence.

So to to prove all ravens are black you can do it directly by checking all ravens for blackness or indirectly by checking all non-black things for non-ravenness and you will get the same answer(*) whether ravens exist or not. Which approach - direct or indirenct - makes a big difference in terms of efficiency and practicality but with the usual convention there is no difference in theory.

(*if ravens don't exist you can prove all ravens are black and you can prove all ravens are white, or green etc. I suppose if there no such things as ravens it's rather unimportant what colour they are, or rather would be if they did exist - rather as it doesn't matter if a non-existent god is 'omnipotent' or not.)
dylancatlow
Posts: 12,243
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7/12/2016 5:04:19 PM
Posted: 4 months ago
At 7/10/2016 5:42:14 PM, keithprosser wrote:
So to to prove all ravens are black you can do it directly by checking all ravens for blackness or indirectly by checking all non-black things for non-ravenness and you will get the same answer(*) whether ravens exist or not.

How are you supposed to know when all ravens have been checked?
ford_prefect
Posts: 4,138
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7/13/2016 1:41:57 AM
Posted: 4 months ago
The raven paradox is not a paradox, but not for the reasons described in OP. There was already a thread in this forum a while back where I and others explained why.
Adam_Godzilla
Posts: 2,487
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7/13/2016 2:14:38 PM
Posted: 4 months ago
I don't think observing an apple can be seen as evidence for "everything that is not black is not a raven". It serves as evidence for "some things that are not black, are not ravens."

Thus, if one observes everything in the universe that is not black, and sees that none of these things are ravens, then that's evidence for the conclusion that all ravens are black.
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keithprosser
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7/14/2016 4:46:30 PM
Posted: 4 months ago
At 7/13/2016 1:41:57 AM, ford_prefect wrote:
The raven paradox is not a paradox, but not for the reasons described in OP. There was already a thread in this forum a while back where I and others explained why.

Was that the thread about the red chair?

@dylan
In the real world it is impossible/impractical to check all ravens are black by examining all non-black things are non-ravens, bur there may be some restricted 'universe' (eg some set of numbers) where it would be easier to prove some theorem about those numbers using this trick than doing it directly. I can't think up an example off the top of my head but I hope Dylan get what I mean.