
Raven paradox - ColinWright
http://en.wikipedia.org/wiki/Raven_paradox
======
gweinberg
There's no paradox here. Imagine that in the world of things, about 1 in 100
are black and about 1 in 1,000,000 are ravens. It's true as far as it goes
that observing a black raven and observing a nonblack nonraven are both
consistent with (and thus evidence for) the proposition that all ravens are
black, but the first observation is about 10,000 times as strong of evidence
as the second. Because Bayes.

~~~
andrewmutz
Also, if the world of things is infinite, then this doesn't hold up, right? if
the number of things is infinite then observations of nonblack nonravens is
not evidence of the proposition, right?

~~~
xxxyy
It still does hold up, but becomes more complicated. In general, there is no
uniform distribution over any infinite set. We might: a) think about a pipe
spitting out objects that have two possible features - being a raven and being
black, or b) use a nonuniform distribution over an infinite set of objects and
integrate (sum) to get a similar Bayesian result.

~~~
x0x0
What do you mean there is no uniform distribution over any infinite set? There
is the uniform distribution over [0,1] which is both infinite and not even
countable.

I'm confused.

~~~
xxxyy
I meant a distribution with "discrete" probability, i.e. a distribution where
the probabilities of singletons are all equal and nonzero, so that a simple
Bayesian argument could possibly be extended. My bad for not being precise
enough.

Perhaps I should have stuck to natural numbers in my previous comment,
otherwise yes, you can have uniform distributions with respect to some
additional structure of the probability space (like [0,1] with the Lebesgue
measure you suggest).

------
danbruc
Seems not really paradoxical to me, both formulations just suggest different
ways about acquiring evidence. The original formulation suggest to first pick
a raven from the set of all things and then verify that it is indeed black,
the inverted formulation suggest to first pick a non-black thing from the set
of all things and then verify that it is indeed not a raven.

Both ways work and provide evidence for the truth of the hypothesis. If there
are non-black ravens and the hypothesis is therefore wrong you can find that
out by first picking a raven and then realizing that it is non-black as well
as first picking a non-black thing and then realizing that it is a raven.

In our world the set of ravens and the set of non-black things vastly differ
in size and therefore the first way of doing it seems more intuitive but if
you for example think about sets of twenty triangles and squares colored
either black or white the paradox immediately disappears.

------
inglor
That's why an important criteria in research is falsification and not just
induction. Look at Popper
[http://en.wikipedia.org/wiki/Problem_of_induction#Karl_Poppe...](http://en.wikipedia.org/wiki/Problem_of_induction#Karl_Popper)

> ccording to Popper, the problem of induction as usually conceived is asking
> the wrong question: it is asking how to justify theories given they cannot
> be justified by induction. Popper argued that justification is not needed at
> all, and seeking justification "begs for an authoritarian answer". Instead,
> Popper said, what should be done is to look to find and correct errors.[27]
> Popper regarded theories that have survived criticism as better corroborated
> in proportion to the amount and stringency of the criticism, but, in sharp
> contrast to the inductivist theories of knowledge, emphatically as less
> likely to be true.

~~~
crimsonalucard
Not only is it "important," in science, falsification is the ONLY method
available, there is absolutely no other way. Proving any theory in reality is
fundamentally impossible. In fact, the concept of "proof" only exists in
mathematics and logic.

Karl Popper himself wrote: "In the empirical sciences, which alone can furnish
us with information about the world we live in, proofs do not occur, if we
mean by 'proof' an argument which establishes once and for ever the truth of a
theory,"

This flies in the face of our intuition and is very hard for people (even some
scientists) to understand. The reason this occurs is due to an asymmetry
between evidence needed to prove something and evidence needed to disprove
something. Evidence cannot prove a theory correct because other evidence, yet
to be discovered, may exist that is inconsistent with the theory. However to
falsify a theory, one only needs a single inconsistent fact.

------
learnstats2
You could check All Ravens Are Black by:

1\. seeking out all ravens and checking their colour, or

2\. seeking out all things that are not black and checking they are not
ravens.

Looking at an apple constitutes evidence, in that it contributes to the second
case, and that's not in doubt here. But you're really checking that the apple
is not a raven.

If you believe that apples are not ravens (perhaps by definition), then
checking an apple no longer contributes to the probability at all.

~~~
learnstats2
Alternative point of view, separated for clarity:

"seeking out all things that are not black and checking they are not ravens."

This statement seems a faintly ridiculous idea. Our mental model holds an
infinite number things that are "not black", so there are still an infinity to
check. Checking one thing should not change our internalised probability at
all.

Even if you disagree that there are an infinite number of things, checking all
apples contributes 0% towards this case - and is surely no easier than
checking every raven.

~~~
indrax
You can't be sure you're gotten all ravens unless you check everything in the
universe to see if it is a raven.

~~~
learnstats2
Well, not necessarily.

It depends on the definition of raven. If you define 'raven' as a current
resident of Earth, that simplifies things for you - it's meaningless to check
things off Earth.

------
dsjoerg
Very interesting, thanks. The idea of "supporting evidence" is murkier than it
first appears. Also the _scope_ of a proposition is unclear; "all ravens are
black" has a scope that is clearly constrained to ravens, but the logically
equivalent proposition "all non-black things are not ravens" has a scope of
all non-black things.

~~~
chaz72
The first proposition tests the blackness of ravens. The second proposition
tests the non-raven-ness of non-black things. When abbreviated, it sure looks
like a paradox.

------
mbillie1
One of my favorite paradoxes, I recall learning this in a formal logic class
in college. These sorts of mental exercises are fascinating and well worth
doing. The point of paradoxes is not to "solve" them, the point is to
understand more about how we reason, and about the boundaries of our reason.

------
dragonwriter
The issue, it seems, turns on whether either or both of "ravens" and "non-
black things" are infinite. If neither is, then both black ravens and non-
black non-ravens provide some evidence for the two equivalent propositions,
with evidence defined as any information which makes a conclusion more
strongly supported than it would be without that information.

That ravens are not infinite seems to be a well supported proposition. That
non-black things are not infinite is intuitively trickier.

(Of course, this all assumes ravens aren't defined by blackness, in which case
evidence isn't even an issue.)

~~~
SAI_Peregrinus
Ravens are defined by their species, since there are albino and leucistic
ravens which are still considered ravens (and can interbreed with other ravens
and produce viable offspring.)

Ornithologically, this is interesting, but other than for the definition it
doesn't have any bearing on the philosophical point.

------
kazinator
The reason that a paradox is perceived is that evidence-based inductive
reasoning is dressed up in logical language.

A logical proposition like "forall(x) : is_raven(x) -> black(x)" is not
something that is justified by _evidence_. It is a statement of logic, which
requires a _proof_.

Just because you've examined a hundred million ravens, and they have all been
black, doesn't mean you have reason to believe that the proposition is true. A
single counterexample destroys it. That is the case even if the counterexample
happens to be the last instance which is examined, of the entire set of ravens
in existence.

A counterexample is some X which satisfies is_raven(x), but fails black(x).

It is not the case that any other X is evidence; any other X just _fails to be
a counterexample_. A non-counterexample to a universal claim is not evidence
that it is true; evidence is simply not an applicable concept and misleading.
If you accept a logical statement on evidence, you're committing a logical
fallacy.

Now if we want to study the color of ravens _empirically_ , then we must throw
away the logical proposition. Our goal cannot be to prove that all ravens are
black. Rather, something else, like "based on our random sampling method, we
are 95% confident that between 99.996 and 99.998 percent of all ravens are
black".

If we strip away the misleading instances of over-reaching logical language
from an empirical investigation, then the paradox goes away.

------
jrapdx3
At least on an intuitive level, the propositions seem paradoxical because
there's a piece missing. After all, we _know_ that from a distance, it can be
hard to tell if it's a raven or a crow. I'd put it this way:

    
    
        1. All ravens are black.
        2. Non-black things are not ravens.
        3. Not every black thing is a raven.
    

With the addition of the third statement, it appears all cases are covered.
But maybe that's too simple, or could be I'm not understanding something...

------
Double_Cast
> _This conclusion seems paradoxical, because it implies that information has
> been gained about ravens by looking at an apple._

I remember how I used to be confused about ad-hominems. It's known as
fallacious because "an opponent's credibility has nothing to do with the
truth-value of their argument". Yet common sense dictates that if a
pathological liar makes a proposition, it would be unwise to take their
proposition at face value. At some point, I realized that my distrust was not
directed towards the proposal per se, it was directed towards what a bayesian
would call "the likelihood" [0]. I.e. how much the proposition changed my
beliefs. Therefore, the reputation for pathological lying did not imply that
the proposition was _false_ , but that I should hold my beliefs constant (set
likelihood to zero) as if the liar had never said anything at all. I think my
initial confusion is similar to the raven's paradox because we're confusing
the hypothesis itself (our prior) with the weight we assign the evidence (the
likelihood).

Let's consider the two proposals. The original proposition is "all raven are
black". Modus Ponens says "black, given a raven". Now if I were to see a raven
though a monochromatic lens, then I would expect to see a _black_ raven
without the lens. Now consider the contraposition "everything non-black cannot
be a raven". Modus Ponens says "not-raven, given non-blackness". So if I were
to see a green object, I would expect to not see a raven.

In the original proposition, the _raven_ was the signal while the _color_ was
the hypothesis. But in the contraposition, the _color_ was the signal while
the _raven_ was the hypothesis. The insight here is that when we contrapose
the claim, we also switch the hypothesis with the evidence. Therefore, the
sentence I quoted above is incorrect because a green (non-black) apple doesn't
provide information about ravens, a green (non-black) apple provides
information about non-blackness. Namely, whether non-blackness is a strong
signal of non-ravenness.

E.g. suppose we saw a raven and, upon closer inspection, the raven turned out
to be a green raven. The correct reaction is "huh, I guess raven-ness isn't
such a reliable signal of black-ness after all". And suppose we saw a blur of
green and, upon closer inspection, the green object was a green raven. The
correct reaction is "huh, I guess non-black-ness isn't such a reliable signal
of non-raven-ness after all."

[0] I didn't know the terminology back then, but I vaguely understood the
concept on an intuitive level.

------
orangecat
My solution, which I haven't come across elsewhere and is thus quite likely to
be wrong or incomplete: (Update: upon rereading the article this is
essentially the Carnap solution)

If you have no knowledge of the distribution of ravens and non-ravens, then
picking an object at random and observing that it is not a raven should
slightly decrease your estimate of the number of ravens in the population. In
turn that makes it more likely that all ravens are a single color. 100 ravens
are less likely to all be the same color than 5 ravens. (In the limit, if no
ravens exist than all ravens are black, and purple, and polka-dotted).

So under those conditions, it is true that observing a white cat should
slightly increase your belief that all ravens are black, but it should also
increase your belief that all ravens are purple, and even that all ravens are
white. That's entirely because the non-raven reduces the expected number of
ravens, thereby increasing the probability that all ravens are a single color.

On the other hand, if you already know how many ravens exist, then observing a
non-raven tells you nothing about the color of ravens, and your prior for "all
ravens are black" should be unchanged.

------
Booktrope
I don't understand how you can say that, even if you were to search all non-
black objects and none were ravens, this would prove that all ravens are
black. It also might prove, there aren't any ravens.

And, even if you were to find a black raven, it might just mean that you were
mistaken and it either wasn't black or wasn't a raven. Thousands of years ago,
the Chinese nominalist philosophers built an entire school out of saying, "A
white horse is not a horse", and if you read what they said, they were right.
You can say, "A white raven is not a raven" but I think you'd be wrong,
because it's easy to paint a white raven, especially if you use a dark
background. And please, don't tell me, it's the same with a white horse,
without reading a bit of Chinese nominalist philosophy.

And besides, what is black, actually?

------
miander
To me it seems like the intuitive resolution to this is that a non-black non-
raven does indeed provide some evidence that all ravens are black, but only an
incredibly small amount. The net effect on the argument is so small that it
can be ignored completely.

------
gweinberg
Another point: the idea that all ravens are black doesn't just come from
observing lots of ravens which are all black, and observing lots of non-black
things and noting that none of them are ravens. It's supported by a more
general world view. Ravens are animals and children tend to resemble their
parents, in particular birds of a given species tend to have the same color
scheme. If we occasionally observe a white crow, that will tend to make us
more inclined to believe there may be white ravens out there, even though
ravens are not crows.

------
neuigkeiten
Maybe we can solve this via a monte carlo simulation?

Create random worlds with random object categories that have random color
distribution.

Then let two agents experience one object after another and make assumptions
about the object categories.

One agent will only take into account objects of the same category. So he will
only learn about ravens when he sees a raven.

The other agent will take into account all objects to learn about every
category.

Then measure which agent makes the better assumptions.

~~~
kordless
Monte Carlo simulations are approximations, not complete knowledge of all
things. This is, for obvious reasons, why these 'paradoxes' are bullshit.
There's not enough energy in the universe to observe all black things. That's
the real paradox here.

------
crimsonalucard
Why is it called a paradox? The introduction on wikipedia states that it's two
different conclusions; one arrived through inductive reasoning and the other
arrived through intuition. The page seems to imply that that only the
conclusion arrived through logical induction is correct. Unless the definition
is stating that both conclusions are correct, I see no paradox.

------
nemo
There are leucistic ravens that are beige and albino ravens that are white.

------
hurin
To be clear: > _Nevermore, my pet raven, is black._

Is _not_ evidence (This is because you don't know anything about the
Distribution from which Nevermore is a sample):

> _supporting the hypothesis that all ravens are black._

It is evidence saying: there exists at least one Raven and that Raven is
Black.

and > _My Apple is Green_

Is evidence that there exists at least one thing which is not Black and Not a
Raven. That does _not_ tell you anything about the existence of Raven's or
Black things in the world.

------
astazangasta
"The correct answer is: who gives a shit?" \-- Triumph

Can someone explain what we gain by studying this question? It seems vastly
pointless to me. No one would seriously attempt to learn anything by finding
evidence for the contrapositive this way; if no one is doing it, why should we
study the oddness of this manner of inquiry?

~~~
bshimmin
(I suspect your comment may not be wildly popular...)

I must admit, while I was mildly interested by it, it did strike me as the
kind of "philosophising" that your regular bloke down the pub would find
rather baffling - "I know ravens are black. You know ravens are black. Why are
we even discussing this?"

~~~
mbillie1
It is a helpful context in which to view the seemingly "logical" arguments you
hear all the time from politicians and moralists of all persuasion. A reminder
that a good-sounding, intuitively appealing argument can have absurd or
utterly fallacious conclusions.

~~~
astazangasta
Except this is exactly the opposite of that: a logically sound argument that
is, intuitively, utter garbage.

------
Udik
\- All the white bears live in the arctic regions.

\- How do you know?

\- Well, have you ever seen a white bear roaming around?

\- True.

------
guard-of-terra
Isn't the whole topic stupid?

"All ravens are black" \- "Here is purple one, freshly painted."

You can rarely defend proposition that "All X are Y", because it's usually
pretty easy to forge X that barely passes and is not Y.

Except for elementary particles maybe. Everything else is forgeable.

