

The Raven Paradox (logic) - hhm
http://en.wikipedia.org/wiki/Raven_paradox

======
zoltz
I'd go with Dave Cole. From the Wikipedia article: _"when one finds a black
raven, it is a much more important discovery than finding a non-black object
simply because there are fewer black ravens than everything else"_

Finding a black raven yields evidence that all ravens are black.

Finding a green apple also yields evidence that all ravens are black, but
_very_ weak, much weaker than finding a black raven. It offers the same weak
(and in this case misleading) evidence that all ravens are pink.

~~~
icky
Non-paradox, since each non-black object that is not a raven will
infinitesimally increase your belief in the proposition, until you find a
counterexample or exhaust the set of all non-black objects.

The reaseon that finding a black raven is more important is because the set of
ravens is much smaller than the set of non-black objects, so it increases your
certainty more.

The green apple does indeed offer the same infinitesimal evidence that all
ravens are pink, but by going through the set of non-pink objects in the
universe and eventually finding a raven, the proposition will be disproven.

So part of the "paradox" here is a conflation of "evidence" and "proof". The
other part of the paradox lies in the fact that the evidence gained by seeing
one green apple is SO tiny that it won't even register on the human brain's
"is this evidence?" heuristic.

~~~
zoltz
_"The green apple does indeed offer the same infinitesimal evidence that all
ravens are pink, but by going through the set of non-pink objects in the
universe and eventually finding a raven, the proposition will be disproven."_

I had residual doubts about these pink ravens, but that's a great way to put
it. Now I really feel "on top" of the paradox.

------
jsnx
The Raven paradox is a good example of why we need Bayesian inference. The
paradox is not a real paradox -- it doesn't imply true and false at the same
time -- but it makes a strong case for a system that integrates degree of
confidence through and through, as opposed to merely assigning _true_ and
_false_.

------
fauigerzigerk
What came to my mind when I first read the paradox was that the class of
ravens is closed whereas the class of non ravens is open. We have a definition
of raven that's independent of its colour, otherwise we wouldn't be able to
tell whether a particular thing is a raven, and thus, whether its colour is
evidence for anything related to ravens at all. However, we don't have a
definition for all non raven things, so the number of non-ravens is not merely
large, as the article suggests, it is in fact infinite as counting needs
definition to tell one thing from another. Drawing conclusions by induction
from one of an infinite number of instances seems meaningless to me.

But there is another interesting question that arises from the Bayes
explanation at the bottom of the article. If we had all things in the universe
available and sufficiently defined to tell what is one thing and what is
another, and we found that there are indeed no non-black ravens, that would
still leave open the question of whether there could possibly be non-black
ravens in the future. They could be born right in this moment, so there would
be a race condition between making the statement and the statement being
falsified by the event of a non-black raven being born. And I think that's one
of the problematic things about purely extensional logic advocated by
philosophers like Quine.

~~~
jey
_"If we had all things in the universe available and sufficiently defined to
tell what is one thing and what is another, and we found that there are indeed
no non-black ravens, that would still leave open the question of whether there
could possibly be non-black ravens in the future."_

That's not a bug in Bayesian inference, that's a feature! It would be a
mistake to ever assign a probability of 0.0 or 1.0 to any statement. It means
you have infinite confidence in the statement, which is impossible unless you
have a screwy prior distribution.

You also don't have to evaluate _all_ of the data if you're giving a
probability. You can just report your degree of confidence in the proposition
based on the data you've evaluated so far, and this degree of confidence is
called a probability.

~~~
fauigerzigerk
I know the whole point of Bayesian statistics is that you actually need much
less data to get a good prediction. But the edge case I was talking about is
still interesting I think, if not as a criticism of Bayes.

------
immad
I don't get why this logic violates intuition. The proposed solution in the
article is fairly obvious and intuitive. Just because two logical statements
are equivalent doesn't mean evidence for one gives evidence for the other:

"The origin of the paradox lies in the fact that the statements "all Ravens
are black" and "all non-black things are non-ravens" are indeed logically
equivalent, while the act of finding a black raven is not at all equivalent to
finding a non-black non-raven. Confusion is common when these two notions are
thought to be identical."

~~~
jsnx
If two logical statements are equivalent, then yes indeed, evidence that
implies one of them (in classical logic) implies the other one. The solution
is indeed _intuitive_ but is not formalizable in classical logic.

Classical logic does not, in general, jibe with the intuition: consider the
case of the negative antecedent:

<http://www.earlham.edu/~peters/courses/log/mat-imp.htm>

For example, the statement, "If unicorns have one horn, then I am the Queen of
England." is logically true.

~~~
immad
hmm, I see the point. But isn't that just the limitations of classical logic?
Just seems like a question of semantics rather than a clash between logic and
intuition, but maybe I don't get it

~~~
jsnx
It is a clash between _classical logic_ and intuition, which is the point --
it indicates a difficulty with applying classical logic, and encourages us to
take up other avenues.

------
pius
The statement that "my pet raven is black" is not even weak evidence that "all
ravens are black." Rather, it's strong evidence that "there exist black
ravens." That's the flaw I see.

