A koan is a question designed to have no logical answer, used in Zen training to help the mind escape the trap of words. As far as I know (which is not very far at all), Kabalah, the mystical tradition in Judaism, employs no parallel device.
However …
A young man asks a rabbi to teach him Talmud.
The rabbi replies, “I don’t think you’re ready to study Talmud, but I will give you a question as a test. Two men go down a chimney. One comes out dirty, and the other clean. Which one washes himself?â€
The young man says, “Rabbi, ask me a hard one! Clearly, the dirty man washes himself and the clean man does not.â€
The rabbi shakes his head and says, “As I suspected, you’re not ready to study Talmud. Your answer is wrong. Two men go down a chimney. One comes out dirty, and the other clean. Which one washes himself?â€
The young man thinks, and says, “Aha! I see the trick! The dirty man sees his clean companion, and thinks himself clean, so he doesn’t wash. But the clean man sees how filthy the dirty man is, and assumes that he himself must be equally dirty. So he washes. Therefore the clean man washes himself but the dirty man does not.â€
The rabbi shakes his head and says, “As I suspected, you’re not ready to study Talmud. Your answer is wrong. Two men go down a chimney. One comes out dirty, and the other clean. Which one washes himself?â€
The young man thinks again, and finally says, “Hmmm … so it must be that the clean man sees the dirty man and washes, and the dirty man, seeing the clean man wash, realizes that he himself is dirty, so he washes too. Therefore both wash. Rabbi, there’s much more to this Talmud stuff than I suspected.â€
The rabbi shakes his head and says, “As I suspected, you’re not ready to study Talmud. Your answer is wrong. Two men go down a chimney. One comes out dirty, and the other clean. Which one washes himself?â€
Now the young man is thoroughly puzzled, but not discouraged. He thinks for an even longer time, and says, “Oh my! This is really deep isn’t it? The dirty man sees the clean man, and never suspects that he himself might be dirty. The clean man sees the dirty man, goes to wash, but looks in the mirror and sees that he is in fact clean. So neither washes!â€
Once more the rabbi shakes his head and says, “As I suspected, you’re not ready to study Talmud. Your answer is wrong. Two men go down a chimney. One comes out dirty, and the other clean. Which one washes himself?â€
Now the young man, who hasn’t studied Torah but has studied logic, becomes upset. “But Rabbi! It must be one, or the other, or both, or neither. Those are the only possibilities. How can they all be wrong?â€
The rabbi says, “And how is it, my logical young friend, that two men go down a chimney and both don’t wind up covered in soot? When you learn how not to spend your time answering foolish questions, then you will be ready to study Talmud.â€
The parallel tradition in Jewish teaching is the parable.
I'm famiiar with a slightly different version of this, but thought the point was not, "Don't waste time answering foolish questions," but rather, "Consider the premises of a question, and be willing to challenge them if they seem wrong."
In the version I heard it was the rabbi that gave the incompatible answers, in order to explain to the man how dialectical materialism works. And there was no resolution about the impossibility of not getting dirty down a chimney. Your version is better.
The "Curse of the Liar" is one that has a liar believing that everyone else does, too, so that lying is normal. The liar then loses friends, until all that are left are the ones comfortable with lying and being lied to. Voila! The liar only knows liars, and his thought (everybody …) is confirmed.
We see conservatives like hannity, beck, all "reading our minds", telling people "what/how Liberals think." They are too crappy a people to even understand our motivations. Their minds are too full of lousy thoughts, distrust of themselves, too polluted to ever get us Liberals. They simply aren't good enough people.
I mean. look at the taxes on tobacco. Liberals see it as a punitive guidance, and a minimal revenue source that needs to mitigate the damage smoking does. Conservatives just scoff, saying that it will be a failure if we drive people to quit smoking. They simply can't measure the good. All they see is a tax, and possibly the demise of the (smoking) base of revenue.
Also, it seems that Republicans need laws to stoip THEMSELVES from having sex with dogs (santorum), or turles.
They can't even face up to ingrained morality; they need rules to be good! They probably do, but Liberals don't. It comes naturally to Liberals to do good; that's why we're Liberals.