More Math

I said I would get around to another short post, so I'll do that. There should be a Christmas post coming, but I don't have pictures yet, so it wouldn't be as good as it could be.

Anyway, while reading about Bayesian inference on Wikipedia, I came across the Raven paradox, also known as Hempel's paradox. You can just read on there about how it works, or you can read my explanation which will be basically the same thing. So, here goes.

Consider the statement "all ravens are black." As a logical statement, this is the same as "if x is a raven, then x is black." Like all logical statements, it is logically equivalent to its contrapositive, which is "if x is not black, then x is not a raven." If you don't deal with logical statements like that, you should probably think about why statements and their contrapositives are logically equivalent.

Now, if we use the scientific (inductive) method, we can support or disprove this statement (or its contrapositive) with evidence. For example, if we see a raven that is black, we support our statement, and if we see a raven that isn't black, we have disproven the statement. Of course, since the statement and its contrapositive are equivalent, support one is the same as supporting the other and disproving one is the same as disproving the other.

So, what if we see a green apple. It is green, and since an apple is not a raven, it supports the contrapositive "if x is not black, then it is not a raven." So, this observation supports the original statement. But what happens if we start with the (obviously false) statement "if x is a raven, then x is white"? The contrapositive here is "if x is not white, then x is not a raven." Seeing the green apple again supports this contrapositive, so this observation simultaneously supports the (obviously contradictory) statements "all ravens are black" and "all ravens are white."

Weird.