The encyclopedia of philosophy article did a decent job of motivating it, I thought. Understanding how to classify the Liar's sentence is linked to being able to use inconsistent information (like us humans do), without being able to prove absolutely everything from the inconsistency.
Also, it's interesting.
What we humans do is store our underlying representations as contingent networks and only "round them off" to categorical propositions when we reason explicitly about them.
That is, "This sentence is in English" is a categorical proposition, but if I trace it down to the cognitive structures that motivated me to generate it, I won't find any categorical representations, just contingent ones: spreading networks of activation. Ditto for "This sentence is true" and "This sentence is false" and everything else I might s...
Graham Priest discusses The Liar's Paradox for a NY Times blog. It seems that one way of solving the Liar's Paradox is defining dialethei, a true contradiction. Less Wrong, can you do what modern philosophers have failed to do and solve or successfully dissolve the Liar's Paradox? This doesn't seem nearly as hard as solving free will.
This post is a practice problem for what may become a sequence on unsolved problems in philosophy.