Depending on the method, I've always loved the Liar Paradox. A lot of the work of Alfred Tarski examined the logical setup behind the Liar paradox, and resulted in an interesting arguement about the what "Truth" lies in language: Snow is white - but why? Tarski came to the conclusion that in order to confirm the whiteness of the snow (the object language) we make a meta-referential call to confirm its truthiness, but we eject truth in the process, and have to make a meta-reference in order to confirm its truth. However, this only applied to formalized languages and not natural language. Sort of paradoxical, but falls into the same line as Goedel's Incompleteness Theorem and other Theories of Truth.
