Raven Paradox

While reading the less wrong blog I discovered a reference to the Raven paradox, an interesting problem posed in the 40s. I am aware that it is very much solved in every respect and solutions can be found on the Wikipedia page, it is however, still very interesting and quite difficult to get your head around (at least it was for me). Being a mathematician, I found it much easier to think about using ideas in maths. When converted to symbols and ever so useful Venn diagrams, it became very simple.

To say that all ravens are black is logically equivalent to saying all non-black objects are non-raven. This is a contrapositive and I've always struggled with this idea, but in the process of thinking about this problem, it became clearer. The paradox appears when we think that because of this, evidence that suggests ravens are black (ie a picture of a black raven) are equivalent to evidence that non-black things are non-raven (ie a picture of a red apple). It is intuitively and obviously incorrect. But why? The two statements that support are logically equivalent.

To start with, without evidence, we can only be sure that the top Venn diagram is correct, perhaps some ravens are black, perhaps some aren't. What we want is the second, where the whole set of ravens is contained within the set of black objects. Or, equivalently, there are no non-black ravens. The two piece of evidence have obvious uses now. The first tells us that the intersection of ravens and black objects is non trivial, and if the set of ravens is finite, that a non-zero percentage of ravens are black. Equivalently, the second piece of evidence suggests that a portion of non-black objects are non-ravens, and if the set of non-black objects is finite, it gives us a percentage. This is where the Probabilistic approach comes in, obviously the set of non-black objects is larger than the set of ravens, so the second piece of evidence tells us not nothing, but very little. To prove beyond doubt, we would have to check the whole set of non-black objects and find that none of them were ravens.

Our intuition was correct (nearly), the second piece of evidence is near useless, and that type of evidence nearly always is. On that note, while I've been not-working, I've been not-catching up with my work.