I'm gonna take the definition of 'paradox' very literally here, that being:

A seemingly absurd or contradictory statement or proposition which when investigated may prove to be well founded or true.

And I'm gonna talk about the Raven Paradox.

This paradox is a favourite of mine, as it's a logical paradox, and I love logic. It shows the important different between logic and intuition.

It goes thusly:

Say you want to prove that all ravens are black. To do this you find a raven that is black. This is evidence towards your proposition. You also find a green apple, and claim that it helps prove all ravens are black. Rationally, this is ridiculous, how in the world does a green apple prove all ravens are black? But, logically, it does.

Here's how: by finding a raven that is black, you are showing that something that IS a raven IS black. And by finding a green apple, you are showing that something that IS NOT a raven IS NOT black, which suggests that something that is a raven, would be black, therefore logically a green apple helps prove all ravens are black.

Interesting, huh? Well, I think so at least; as I said, I adore stuff to do with logic. Also, as I said as well, this is an important paradox for showing the difference between logic and intuition (and it is a paradox, if you're in doubt, because it seems ridiculous at first, yet makes sense when you look at it). A green apple may prove all ravens are black *logically*, but rationally we know that's utterly ridiculous. It goes to show that logic isn't perfect, and (as much as I don't like it), isn't the answer to everything.