logical paradox

[ishmael]

Suppose you want to prove the statement "All crows are black".

You can't do it logically, it has to be done empirically. One way is to go out and look at every crow and note its color. Every black crow would be a confirming instance of the statement, and increases (however slightly) the likelihood of the statement being true. The more black crows you see, the more likely the statement is to be true. The statement could only be proven if you could examine every crow and find that each is black.

Now consider this: The statement "all crows are black" is the logical equivalent of "all non-black objects are not-crows". So if you go out and see a purple cow, that is a confirming instance of the statement, and increases the likelihood of it being true by an infinitesimal increment.

But... a purple cow is also a confirming instance of the statement "all crows are white".

How can one thing increase the likelihood of two opposite statements being true?

[Sisyphus]

"All crows are black" and "all crows are white" are not opposite statements.

[Mr Skeptic]

Well if you see a purple cow, it disproves both the statements "all cows are black" and "all cows are white".

 

A similar thing to what you were saying has been considered somewhat a problem with empiricism. Namely, finding non-cow non-black objects might count as evidence for "all cows are black" since it is evidence for the equivalent statement "all non-black things are non-cows". This would make evidence requirements rather lax.

Edited January 25, 2010 by Mr Skeptic

[phyti]

Suppose you want to prove the statement "All crows are black".

Every black crow would be a confirming instance of the statement, and increases (however slightly) the likelihood of the statement being true. The more black crows you see, the more likely the statement is to be true. The statement could only be proven if you could examine every crow and find that each is black.

 

The number of occurences does not increase the truthfulness of a statement. It only requires one sighting of an albino crow to falsify the 1st statement. Many species have been declared extinct only to be found later.

 

Now consider this: The statement "all crows are black" is the logical equivalent of "all non-black objects are not-crows".

 

The color of crows is not determined by the color of other objects, so there is no relationship of the 1st to the 2nd.

 

I don't see any paradox or logic here!

[the tree]

The Bayesian answer to the Raven's paradox is a confirmation of a contrapositive is gives only a minor confirmation of the original generality - so while it increases the probability of the generalisation being true, only slightly so.

 

And yeah, like Sisyphus said - the negate of 'black' is 'not-black', it isn't 'white'.

 

Beyond that you can play with ideas like natural kinds and stuff.

[D H]

This is the problem of induction as illustrated by Hempel's raven paradox. The problem is not one of opposites. Seeing a green apple or a purple cow does paradoxically (paradox: A seemingly contradictory statement that may nonetheless be true) lend credence to the statement "all crows are black" because that statement is logically equivalent to "all non-black objects are not ravens". Green apples and purple cows are not black and are not ravens. The extent to which the observation of a green apple or purple cow does lend credence to the thesis "all ravens are black" is very, very small, however.

[phyti]

Other considerations:

A life long study of all non-black objects will not allow you to state the color of crows, a study of crows is required.

But that study would not include all crows. There may have been a blue crow that lived and died before your study but was never recorded. These type of absolute 'true' statements are not possible due to lack of knowledge. The best you can do is a statement with a high degree of confidence. The history of science demonstrates that knowledge is always in a state of refinement.

[JN.]

It was because of the impossibility of proving something only with verification that Popper developed his theories of falsiability.

 

In fact, proving that all crows are black empirically is also impossible, because we can't prove there are no more crows in other part of our world. That's the old story: we cannot prove the non-existence of something.

 

 

[Guest mabil]

Its not co-relating the statement that all crows are black and all crows are white.. You can use any laws of thermodynamics like a=b, b=c so, a=c.. but even this thing fails here

[Mark Allan Barnes]

If the universe is objectively real then either all crows are black or some crows are not black.

 

Observations do not increase or decrease the likelihood of the statement "all crows are black" being true. There is not a probability, only a certainty. The probability is just a fiction in the mind of the observer.

The paradox seems to arise because ishmael, the person describing the situation, cannot tell the difference between what is true and what is thought to be true.

 

There are subtleties I have not included here (such as the colour of crows changing over time, the possibility of people deliberately painting a crow a different colour, or the colour black not being rigidly defined), but they are not relevant to my argument.

[LiquidMentality]

Cows and crows (the objects) are different subsets of the same unified set of all objects (S). Belonging to different sub-sets allows for calculation of odds (determinability) without the constraint of mutual exclusivity. Hence, announcing that a cow is purple doesn't falsify the conjecture that all crows are black.

 

Perhaps the 'paradox' could be resolved by instead postulating the following two llemas:

* All crows are birds and all crows are black.

*If a bird is not black, it is not a crow.

 

By fulfilling both of these postulates, we can logically determine that all crows are black.

 

 

[md65536]

Suppose you want to prove the statement "All crows are black".

You can't do it logically, it has to be done empirically. One way is to go out and look at every crow and note its color. Every black crow would be a confirming instance of the statement, and increases (however slightly) the likelihood of the statement being true. The more black crows you see, the more likely the statement is to be true. The statement could only be proven if you could examine every crow and find that each is black.

Now consider this: The statement "all crows are black" is the logical equivalent of "all non-black objects are not-crows". So if you go out and see a purple cow, that is a confirming instance of the statement, and increases the likelihood of it being true by an infinitesimal increment.

But... a purple cow is also a confirming instance of the statement "all crows are white".

How can one thing increase the likelihood of two opposite statements being true?

Keep in mind you've defined the problem as "not a logical problem", even though there is some intuitive logic in this example.

If you generalize this problem then you're basically talking about making statements about a set based on a sampling of the set.

 

When you make probabilistic statements based on samples (whether you take enough samples or, as the above example, not), you don't just have a probability that something is true, but you have a confidence interval or whatever. With your example, the "error bars" would be so big that the possibilities of the contradictory statements overlap or something.

 

 

Also... treating this as a general example and not a real-world example, you allow the case that there are no crows at all.

In this case, both statements about crows in the set are true.

If you sample objects and none of them are crows (as you did in your example), you increase your confidence in this case being a reality.

If you sampled all objects in the set and found that none of them were crows, you'd prove both statements with 100% certainty.

If you sampled all but one object, and none were crows, the error bars on the probability of both statements would still cover various contradictory statements about crows in the set.

Edited May 31, 2011 by md65536

[Dekan]

Surely this is just a matter of the definition of words.

 

If you choose to define the word "crow" as meaning:

 

A bird which includes among its defining characteristics, the possession of black feathers

 

then obviously any bird that doesn't possess black feathers, isn't a crow.

 

It's excluded from crow-hood by definition.

 

So where's the paradox?

[John Cuthber]

Not all crows are black

http://www.cbc.ca/news/canada/british-columbia/story/2008/06/19/bc-albino-crows-.html

the colour of cows has no influence on this.

 

I never saw a purple cow,

I never wish to see one.

But I can tell you anyhow,

I'd rather see, than be one.

[Dekan]

Not all crows are black

http://www.cbc.ca/ne...ino-crows-.html

the colour of cows has no influence on this.

 

I never saw a purple cow,

I never wish to see one.

But I can tell you anyhow,

I'd rather see, than be one.

 

Like I said, it's just the definition of words. If a crow with a genetic mutation that gives it all-white plumage is still a "crow", OK.

 

In that case, you extend the definition of crow.

 

If you extend the definition far enough, all birds could be crows, and we would resolve the paradox that way.