Does seeing a green apple help prove that all ravens are black?

If you are studying for the Law School Admission Test (LSAT) or the Bar Exam, you may be reviewing your formal logic skills. The “if-then” statements and contrapositives used in formal logic are a big part of those tests, and of the law.

Exploring formal logic sometimes leads to paradox, and Hempel’s Raven Paradox is a famous example.

In the 1940s, Carl Gustav Hempel proposed that seeing a green apple is evidence that all ravens are black. His logic proceeded like this:

Seeing a black raven provides some evidence, even if it is a small amount, for the hypothesis that all ravens are black. Certainly if we had seen 100,000 ravens and every one of them had been black, we would say that there was some evidence that all ravens are black, even if the hypothesis had not been proven. Therefore, each black raven that we see is one piece of evidence that all ravens are black.

Students of formal logic will easily translate the statement “All ravens are black” into “If raven then black,” also represented like this:

If raven → black

And we know that the contrapositive of such a statement is its logical equivalent. The contrapositive of the above statement is:

If not black → not raven

So, Hempel argued that if seeing a raven that is black is evidence that all ravens are black, then seeing a non-black thing that is not a raven is also evidence that all ravens are black. That is, it tends to prove that all non-black things are non-ravens, which is the logical equivalent of saying that all ravens are black.

So, seeing a green apple (or any other non-black non-raven) is a piece of evidence, though very tiny, that all ravens are black.

The paradox, of course, is that common sense tells us that seeing a green apple has nothing to do with whether all ravens are black.

Philosophers have attempted to resolve the paradox in different ways.

Some argue that neither seeing a black raven nor seeing a non-black non-raven actually constitutes evidence for the hypothesis that all ravens are black.

Others say that there is no real paradox: observing a green apple does provide some evidence that all ravens are black, just a very tiny amount.

Incidentally, in the real world, not all ravens are black. White ravens, though rare, do exist.