Fred Leavitt reveals how the whiteness of swans proves the blackness of ravens.

Many scientific theories and laws are of the form “All A is B.” Two examples are “Water at sea level boils at 100 degrees centigrade” and “Schizophrenia is associated with an excess of dopamine in the limbic system.” But philosopher and logician Carl Hempel pointed out a seeming paradox (Hempel, 1945). As virtually all logicians agree, the propositions “All ravens are black” and “All nonblack things are nonravens” are equivalent. To test the former, a scientist would look for ravens and check their colour. A black raven would provide supporting evidence. To test the latter, the scientist would look for nonblack things and check to see if they are nonravens. A white handkerchief would provide supporting evidence.

But if a white handkerchief supports the proposition that all nonblack things are nonravens; and if “All nonblack things are nonravens” is equivalent to “All ravens are black;” a white handkerchief would appear to support the proposition that all ravens are black. The paradox has two aspects: first, that a white handkerchief should be as informative as a black raven; second, that a white handkerchief should have any bearing at all on the proposition “All ravens are black” (or green or red). The paradox seems to have important implications for testing scientific theories.

What follows is my resolution.

There are two propositions (P1 and P2) and two pieces of evidence (E1 and E2).

P1: All ravens are black.
P2: All nonblack things are nonravens.
E1: This raven is black.
E2: This white thing is not a raven.

P1 and P2 are logically equivalent, i.e., any evidence that supports P1 supports P2 to the same extent. But E1 and E2 are not equivalent. E1 provides much stronger support for both propositions. To see why this is so, consider an aviary in which there are exactly two ravens among 100 birds.

P3: All ravens in the aviary are black.
P4: All nonblack birds in the aviary are nonravens.
E3: This raven is black.
E4: This white bird is not a raven.

E3 represents 50% of the evidence needed to prove P3. Less obviously but equally true, it represents 50% of the evidence needed to prove P4. (If the only other raven is found to be black, P4 must be true.)

By checking all 98 nonblack birds and verifying that they are not ravens, an investigator could prove P3. The proof would apply just as convincingly to P4. But a single nonblack nonraven would be less useful than a single black raven, because the former would represent 1/98th of the necessary evidence and the latter 50%.

E1 and E3 are more significant than E2 and E4 because they account for a greater proportion of the total number of cases under consideration. Nevertheless, E2 to a trivial extent and E4 to a much greater extent, account for something – a nonblack nonraven eliminates one potential falsifier of the proposition “All ravens are black.”