Our guiding purpose at Eureka is to encourage thinking as a form of play, and what better place to bring a novel thought to all our supporters than on the blog? Logic puzzles and board games abound in the store, but for those who know the rapture of reason, the comfort of contemplation and the simple solace of speculation, sometimes it’s enough not to solve the puzzle or win the game, but just to think about it.
Today we’re thinking about Hempel’s Ravens. This is a peculiar case of logic defying intuition put forward in the 1940s by the logician Carl Gustav Hempel. It proceeds as follows: Suppose you wanted to know whether all ravens are black. You might go outside and look at a few ravens. You couldn’t prove that every raven was black, but if all the ravens you observed were black you’d probably start to think you were onto something. And as you started touring parks and aviaries, states and countries, seeking out all the ravens you could find, and discovered only black ravens, you would only grow more and more confident that every raven was black. In fact, every single time you saw a black raven, the chances that every raven was black would increase.
It was Hempel’s insight, however, that every time you saw a red apple, the chances that every raven was black would also increase.
Hempel realized that, when we say “all ravens are black,” we are dividing the universe into those things that are ravens and those that are not, and checking the set of ravens for blackness. But there is another way to divide the universe in support of this proposition. The statement:
“All ravens are black.”
is functionally identical to the statement:
“All non-black objects are non-ravens.”
So we might instead separate all black objects from all non-black objects, and check the non-black set for non-ravenness. Red objects are non-black. If one of them is an apple, it must be a non-raven. Therefore, when we observe a red apple, the chances that every non-black object is a non-raven increase; which, we remember, is the same as saying that the chances that all ravens are black increase.
Think about it.