I read something today about this lovely little thing called Hempel's Paradox, and it's got to do with inductive logic. So first, a primer on that.

Induction is the type of logic used when we can't deduce answers from known facts. A prime example of this is the statement that gravity exists, or more commonly: "If I drop stuff they will fall to the ground." We know that this statement holds because we've observed many things fall to the ground after being dropped. Or do we really know for sure? Fact is, we can't logically come to the conclusion that all dropped objects will fall to the ground, or even that any particular object will always fall to the ground. It could well happen that the ball falls to the ground a thousand times, and on the thousand and first drop, it just hovers in the air.

Rather than say that a statement (like "If I drop an object it will fall to the ground") is necessarily true, inductive logic says that a statement is most likely true. And the thing about that is that you have to have a large observed base of cases to be able to say that something is most likely true. Dropping only one object wouldn't provide a very high probability, because even though that the statement held true 100% of the time, our base is very small. So, the more times we can observe something happening, the more likely it is to be universally true. The fact that we can assume that gravity works is derived from the fact that many millions of objects have been dropped in the history of man, and so far not a single one has mysteriously hovered in the air or shot off into space or anything like that.

So back to the paradox. The statement we will be observing is this: "All ravens are black" (1). This is a statement that has to be indicated as likely through inductive logic; we have to look at a lot of ravens and make sure none of them are white, or red, or any other color, and the more black ravens we observe, the more likely it is that our statement holds true. Now, the paradox, which isn't so much a logical contradiction as a statement contradicting our intuition. The statement above can be translated into another statement (Easily done with the aid of basic predicate logic): "All non-black things are non-ravens" (2). This is, according to traditional predicate logic, equivalent to the above statement.

Now imagine that we are attempting to prove that all ravens are black, and to do so we start looking at various objects, and come upon a red apple. The apple is red, which is non-black; and it is an apple, which is arguably not a raven; hence, (2) above is supported by us observing this red apple, because the apple is consistent with (2). However, since (2) is proven to be equivalent to (1), our observation of the red apple also supports (1). So in other words, by observing a red apple, we've increased the probability that all ravens are in fact black.

And that's Hempel's Paradox in a nutshell: Observing a red apple increases the probability of all ravens being black.

(Addendum: It might be worth pointing out that the reality of whether all ravens are black or not is entirely independent on our statement. The paradox is about reinforcing what we know about ravens, not about the actual truth about them.)