Hempel’s paradox works as follows: We notice first that the hypothesis 1) All Ravens are Black, is logically equivalent to 2) all non-black things are not ravens. Since these two things are logically equivalent (i.e., 1 if and only if 2), then any observation which raises the probability of 2 raises the probability of 1. If I observe a pink shoe, for example, I observe that a non-black thing is not a raven, which increases the probability of 2, and therefore of 1. Therefore, every observation of a non-black non-raven increases the probability that all ravens are black. It’s a cool paradox, and most thinkers like Hempel and Carnap [Edit: I should have said Maher, not Carnap; Maher gives a ‘Carnapian’ response, but Carnap doesn’t] just bite the bullet, while arguing that it doesn’t increase the probability very much.

Here’s another interesting point about this paradox that I hadn’t realized until a friend shared it with me today. It is possible not only for the observation of a pink shoe to increase the probability than all ravens are black, but also, even more paradoxically, for the observation of a black raven to decrease the probability that all ravens are black. Consider the following auxiliary assumption: a) either not all ravens are black and there are lots of ravens, or all ravens are black and there are very few ravens. Given such an auxiliary assumption, observing more black ravens decreases the probability that all ravens are black.