What's your explanation of the Raven Paradox?

The Raven Paradox starts with the following statement

(1) All ravens are black.

which is equivalent to the following statement

(2) Everything that is not black is not a raven.

In all the circumstances where statement (2) is true, (1) is also true. And, if (2) is false, i.e., if we find an evidence against it, then (1) will also be false.

Now, whenever we see a Black Raven, we see an evidence which supports the statement 'All Ravens are Black'. So, if we see more and more black Raven, then our belief gets stronger and stronger that all Ravens are black.

But since the statements (1) and (2) are equivalent, so collecting evidence supporting statement (2) is also an evidence that all Ravens are black. So, if we see , for example, a red apple, then it's an evidence supporting that 'All Ravens are Black'. It's because 'A red apple' is neither black (because it is red) nor is it a Raven (because it's an Apple. Apples can't be Ravens, can they?).

This conclusion seems paradoxical, because it implies that information has been gained about ravens by looking at an apple. Also, the evidence is completely unrelated. I attempted to explain it but I'm not completely convinced by my explanation.

How can we resolve this paradox?

EDIT: It can be used to collect evidence supporting completely false statements like: 'All dinosaurs are educated'. Because we've seen plenty of things until now which are neither educated nor they're dinosaurs.

EDIT2: I think that the paradox still remains. If we have a journey and took a look at every non-black thing in the universe and found it to be non-Raven, then from this argument it should be proved that all Ravens are black. But that's paradoxical because it would mean false statements can also be proved by taking a look at everything else.

Solution 1:

If you wanted to prove all ravens are black, you could try to find all the ravens in the world and check they are black. Or you could try to find all the non-black things in the world, and check that they're not ravens. I don't recommend either of these approaches, especially the second.

There are approximately 16 million ravens in the world (source: International Union for Conservation of Nature), so if you see a black raven, you've got a very tiny bit of evidence for your hypothesis: you're about $6.25\times 10^{-8}$ of the way to a full proof.

There are about 10 quintillion insects in the world (source: Smithsonian). There are a lot of things in the world that are not insects (source: I really should tidy my desk). Let's say, conservatively, that there are 20 quintillion things in total. About 79% of them aren't black (source: DuPont; they were only counting cars, but I'm going to assume that's a representative sample). So if you saw one non-black thing that wasn't a raven, that gives you a really really tiny bit of evidence for your hypothesis - you're about $2.5\times 10^{-16}$ of the way to a proof, i.e. 1 red apple $\approx$ 4 nanoRavens.

Solution 2:

An intuitive explanation of why evidence for the second statement carries less weight is that there are far more not black things than ravens.

Suppose that you are sampling marbles from a bag. Suppose that you draw 5 and they are all black; what is the probability that all in the bag are black? You need to know how many are in the bag. Try 10, 100, 1000, etc.

Solution 3:

First things first. From a mathematical point of view, statements (1) and (2) are clearly equivalent and that's all there is to it.

All the rest of the "argument" is completely heuristical. When I say "heuristical" I mean the words/concepts:

• "evidence"
• "supports a statement"
• "belief"
• "collecting evidence supporting statement (2) is also evidence supporting statement (1)"

which are used terribly imprecisely and happily lead to the disguised "paradox". If you care to define mathematically the words listed above, then a mathematical discussion is possible where we may or may not reach the conclusion that there is a paradox here. Such a chain of "arguments" leads to no paradox at all, at least not a mathematical one. A good example of a true mathematical paradox is Russel's famous paradox, whose conclusion lies well within mathematical definitions, and so finally forces us to abandon the notion of a universal set of all sets being a set. This however, is nowhere close, and hardly provides any interesting insights into mathematics.

Solution 4:

Here an answer a bit different. It is an "extreme" case, but it may convince you that finding something that is not a raven and it isn't black is evidence that all ravens are black:

Let's say there are only 7 things in the world. 4 of them are ravens, and 3 of them are not ravens. 2 of the non-ravens are also not black.

If you want to prove that all ravens are black you could check all of the 4 ravens, so each raven is 1/4 of the total evidence you need. Or you could also check if the 2 non-black things are not ravens, so each of the two things would be 1/2 of the total evidence you need. So checking that a non-black thing is not a raven is a stronger evidence than checking that a raven is black! After you have checked that all the non-black things are not ravens, you know that no raven will be able to be non-black anymore. So the ravens will be black.

This is not paradoxical and doesn't let you prove false things. If you want to prove that all dinosaurs are educate and you are able to prove that all the non-educate things are not dinosaur, then no non-educate thing will be left to be a dinosaur anymore! And so all the dinosaur are educate. But of course if some dinosaurs are not educate you will not be able to prove that all non-educate things are not dinosaurs.

Maybe the above sentences were a bit difficult to parse, so here some Venn diagrams to look at while reading:

These simple set theory considerations, together with the arguments about probability that other answers gave should give you a pretty clear overall picture.

Solution 5:

I'm inclined to be Bayesian about this, and for the basic approach, see e.g. the Wikipedia article.

But it isn't trivial to get the details right. If anyone really wants a careful account of Bayesian (yep, some maths!) treatments of the Ravens Paradox, then

How Bayesian Confirmation Theory Handles the Paradox of the Ravens

is helpful (Lemmas! Theorems!!) at bringing out some of the complications while aiming to give a "Best Buy" Bayesian response.

I'm not hereby fully endorsing that account by Branden Fitelson and James Hawthorne -- but it is salutary reading for anyone inclined to think that it is trivially easy to find a satisfactory response to the issues at stake.