C. G. Hempel's ‘Raven Paradox’ involves derivation of the intuitively unpalatable conclusion that observation of things like a white shoe or a rainbow confirms the raven hypothesis ‘All ravens are black.’Footnote ¹ Here is how it goes. An earlier author, Jean Nicod, had put forward the following criteria for confirmation of hypotheses of the form ‘All A’s are B’s’:Footnote ²

1. (1) Observation of an object which has the property of being an A and also the property of being a B confirms ‘All A’s are B’s.’

2. (2) Observation of an object which has the property of being an A but not the property of being a B disconfirms ‘All A’s are B’s.’

3. (3) Observation of an object which does not have the property of being an A neither confirms nor disconfirms ‘All A’s are B’s.’Footnote ³

Add to these criteria the following highly plausible claim, which Hempel called ‘Equivalence Condition’: ‘Whatever confirms (disconfirms) one of two equivalent sentences, also confirms (disconfirms) the other.’Footnote ⁴ Let me write this condition in a more convenient form for our purposes:

The Equivalence Condition sounds perfectly intuitive, because to say that H ₁ and H ₂ are logically equivalent is to say that H ₁ and H ₂, having identical truth conditions, make exactly the same claims about the world. Thus if a piece of evidence confirms one of the logically equivalent hypotheses, it must equally confirm the other one.

Now, let H ₁ be the hypothesis ‘All non-black things are non-ravens’ and H ₂ be the hypothesis ‘All ravens are black.’ Since H ₁ is the contrapositive of H ₂, H ₁ and H ₂ are logically equivalent hypotheses. Suppose now that our observation O is the observation of a white shoe. A white shoe being a non-black non-raven, O confirms H ₁ by Nicod's criterion (1) above. And by the Equivalence Condition, O also confirms H ₂, the hypothesis ‘All ravens are black.’ This result, of course, is contrary to our intuitions and also contradicts Nicod's criterion (3).

I have just presented a classical, standard exposition of the Raven Paradox (or ‘paradox of confirmation’, in Hempel's wordsFootnote ⁵). Actually matters can be made to look worse. Not only does observation of things like white shoes or red tomatoes or rainbows turn out to confirm the raven hypothesis ‘All ravens are black’, we can also show that, given Nicod's criteria and the Equivalence Condition, observation of black ravens fails to confirm ‘All ravens are black.’ Let me first add the following condition which is logically equivalent to the Equivalence Condition given above:

This condition is obtained by an operation of contraposition on a portion of the original Equivalence Condition. Hence I called it ‘Contrap Equivalence Condition’. Next, let H ₁ be ‘All non-black things are non-ravens’ and H ₂ be ‘All ravens are black.’ H ₁ and H ₂ are, again, logically equivalent hypotheses. By Nicod's criterion (3) above, a black raven does not confirm H ₁ (nor does it disconfirm it, for that matter), since a black raven does not have the property of being a non-black thing. It follows by the Contrap Equivalence Condition that a black raven fails to confirm ‘All ravens are black.’ Combine this result with the standard, classical paradox: black ravens don't confirm the raven hypothesis, but white shoes do!

Our confirmatory intuitions are sure to revolt against this result. But the worst is yet to come. For it follows as a corollary from what we have said that:

Thus, the Equivalence Condition together with Nicod's criteria – plus the Contrap Equivalence Condition, which really expresses the same thing as the Equivalence Condition – not only lead to intuitively unacceptable consequences but to outright contradiction as well. Put otherwise, we don't have one raven paradox here but two, and the second paradox sounds more serious or devastating than the original one.