## Tuesday, July 8, 2008

I found a site about logical paradoxes. I love such paradoxes, so I was really happy to stumble on such a site. However, while some of the paradoxes mentioned on that site are classical examples, the explanations were of extremely low quality. The reason I am writing this is that one of the paradoxes I read there has an interesting connection with set theory. I am not going to link to this site, but I will write the "paradox" here:

The paradox arises from considering the following two statements:
1. All ravens are black.
2. Anything that isn't black isn't a raven.
Obviously, both of the statements are true. Also, they clearly say the exact some thing. Therefore if I can find somethings that shows that the second one is true, it must follow that so is the first one. Lets suppose that I found a green parrot. It is green and it is not a raven, so it supports the second statement. So, because I found a green parrot I conclude that all the ravens are black.

The problem is that the line of thought shown in this "paradox" is not correct. I never heard before about this paradox, so it might be an invention of the owner of the site I find it on. Or, it is not explained properly there (who ever wrote it managed to make the barber paradox look weird..).

If we will look on this paradox from set theory point of view, we will see that statements say:
1. If a is in A, then a is in B.
2. If a is not in B, then a is not in A.
The paradox tries then to sell us the idea that: If c is in C and not in B then all a are in B. In this form the fact that this paradox is just a game of words becomes clear. While there are paradoxes, the only paradox in this is that some people (at least the owner of the site I found it on) actually see this as a paradox...