What's your explanation of the Raven Paradox?

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.

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.

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.