The paradox of the ravens was formulated by philosopher Carl Hempel. It is essentially a paradox of confirmation.

For any stated generalization, an instance of its truth confirms the generalization. For instance, if I made the generalization that all former and current U.S. Presidents are male, then any instance of a male president would confirm that generalization. Let us represent this principle, that a generalization is confirmed by any instance of its truth, as G1.

The other principle necessary to illustrate the paradox of the ravens is that if two hypotheses can be known to be logically equivalent, then any data that confirms one of them confirms the other. This can be known to be true, as if the statements have identical truth tables, any data that confirms one statement will confirm the other. This principle will be represented as E1.

Now let's imagine two equivalent hypotheses. First, "All ravens are black" (R1), and second, "All non-black things are non-ravens" (R2). These statements can be known to be logically equivalent by the Law of Contrapositives (p>q)≡(-q>-p). Since these statements are equivalent, any data that confirms one of them should confirm the other. This is where our paradox comes in. Statement R2 can be confirmed by the existence of anything that is non-black and also not a raven (e.g. almost anything). Therefore, by E1, we should expect statement R1 to also be confirmed by the same data. This is, on the face of it, an absurd notion. Why should the statement "All ravens are black" be confirmed by the existence of, say, a white piece of paper or a red apple?

There is no immediately obvious answer to the paradox. All of the premises seem to be true, and the reasoning valid. Therefore, the only way left to deal with the paradox must be to accept the conclusion. This is not necessarily as ridiculous as it seems. Simply because data that confirms one hypothesis confirms the other, it doesn't mean that the data confirms both of them to the same degree. In a way, the existence of a non-black thing that is not a raven does confirm the hypothesis that all ravens are black.

Think about it this way. If we actually managed to seek out and find every single non-black thing in the world, and were able to determine that none of them were ravens, then we would prove that all ravens are black. (This is probably not the most efficient way to prove this hypothesis.) Therefore, the existence of each individual non-black non-raven confirms R1, though not nearly to the same degree as the existence of a black raven does. This is simply because there are far more non-black things in the world than there are ravens.

Sources:
Buford, Chris. Critical Thinking. Santa Barbara, California. 2003 (Lecture presented at the University of California, Santa Barbara).