Black Ravens and Green Apples

The paradox

Consider the hypothesis H: "All ravens are black." Observing a black raven seems to confirm this. That is our basic intuition about how evidence works in science.

Now notice that H is logically equivalent to H*: "All non-black things are non-ravens." These two sentences say exactly the same thing; any world that makes one true makes the other true.

If observing a black raven confirms H, then observing a non-black non-raven should confirm H*. A green apple is non-black and a non-raven. Observing a green apple confirms H*.

But H and H* are equivalent. Confirming H* confirms H. Therefore, observing a green apple confirms that all ravens are black.

This strikes almost everyone as wrong. You cannot learn about raven color by looking at fruit. Yet the logical steps seem valid.

The logical equivalence that generates the problem

The paradox is not a trick. The equivalence between "All A are B" and "All non-B are non-A" is a theorem of classical logic, called contraposition. It holds for all universal conditionals. If accepting the logical equivalence and accepting the confirmation relation both seem reasonable, the paradox follows.

Hempel himself accepted the conclusion: in strict logical terms, observing a green apple does provide a tiny amount of evidence for the hypothesis about ravens. The problem is that our intuitions say it does not provide any evidence, and that intuition needs explaining.

What it shows about confirmation theory

The paradox reveals that confirmation, the relation between evidence and hypothesis, is not simply logical entailment and not as intuitive as it seems. Several responses have been developed.

One response is to relativize confirmation to background knowledge. If you are in a room full of ravens, observing another black raven provides much more evidence than observing a green apple, because the observation is more likely given the hypothesis than given its negation. A probabilistic treatment, using Bayes's theorem, can vindicate the intuition that green apples barely matter while preserving the technically correct result that they matter a little.

A second response questions whether H and H* really confirm the same things, even if they are logically equivalent. The hypothesis is about ravens, not about non-black things. Evidence is not just a logical relation between sentences; it is indexed to the domain of inquiry. This response requires giving up or qualifying the equivalence condition on confirmation, the principle that logically equivalent hypotheses must be confirmed by the same evidence.

Neither response is fully satisfactory, which is why the paradox remains a benchmark for any theory that tries to say what makes one observation relevant to a hypothesis and another irrelevant.