Black ravens and red herrings

The raven paradox is a dilemma in inductive logic posed by Carl Gustav Hempel. It starts by noting that the statement “All ravens are black” can be expressed in the form of an implication: “If something is a raven, then it is black.” This statement is logically equivalent to its contrapositive: “If something is not black, then it is not a raven.”

We then consider that observing a black raven would typically be considered evidence for the statement “All ravens are black.” The paradox comes from asking whether the same holds for the contrapositive; to put it another way, if we observe a non-black non-raven, such as a green apple, does that constitute evidence that all ravens are black?

The standard Bayesian solution, given by I. J. Good, goes as follows. Suppose there are N objects in the universe, of which r of them are ravens, and b of them are black, and we have a probability of 1/​N of seeing any given object. Let H_i be the hypothesis that there are i non-black ravens, and let us assume we have some sensible prior over our hypotheses. Then upon observing a non-black non-raven, our probability of H_0 increases, albeit only very slightly when N is large.

I. J. Good’s solution reveals a distinct way of thinking about evidence among Bayesians. Most logicians view logical fallacies as examples of poor reasoning, but the Bayesian interpretation is slightly different. Many fallacies, when stated in their appropriate inductive form, are actually valid in a Bayesian sense, though the evidence they provide is usually weak.

Which is to say that if you criticize a Bayesian for using a “red herring” they may not see the issue. Observing a red herring is indeed Bayesian evidence for the statement “All ravens are black.”