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.
Tordmor's first sentence below is correct, the system should be boolean arithmetic. (that's all that's correct in his post...)
Turing proved that any computational process (if we're being formalists and saying that our philosophical problems are computations) can be simulated in a universal turing machine, and you can write those in binary; so in some sense you really only have two values to deal with. Given a trinary table of truth values, you can run the same computation in a binary system, and then in that binary system write a liar's paradox and translate it.
I don't know what you'd get but it might be something along the lines of "this proposition is (true and false) xor (both)" as a wild guess.
The Liar's sentence is already uncomputable, so I've already abandoned Turning machines by attempting to give it a consistent classification. So his proposed desideratum 5 conflicts with what I consider to be the more important desideratum 1.