Paraphrasing:
In Catch 22, you can get out of the military if you are crazy. If you are sane, then you have to stay. However, to fly the missions, you have to be crazy. Only thing is, if you know its crazy, that means you are sane. So if you tell the brass, they know you are sane, so you will have to keep doing crazy stuff. If you keep doing it, you are crazy, but you can't get out because you can't report you are crazy because then you would be sane.

That's the catch. And that's the paradox.
Two very logical things:
You can leave if you are crazy.
If you say you are crazy then you are sane.

Have a game where the students find two or more statements that sound logical but lead to contradicting or conflicting results. That's a paradox.

Catch 22 is a great book that teaches the concept of Paradox.