Raven paradox

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.

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.

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...

> 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.

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.

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.

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.

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.)

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.

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...

> 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.

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.

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?

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.

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.

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.

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.

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

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.

"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?

- All the white bears live in the arctic regions.

- How do you know?

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

- True.

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.