Tarski left out some of the fine print. That "if and only if" works only under the prior assumption that "snow" designates snow, "white" designates white, and "is" designates the appropriate infix binary relation.
In other words, "Snow is white" is true only if we know that "Snow is white" is a sentence in the English language.
Let me see if I understand you. "Snow is white" is true if and only if "snow" means snow, "is" means is, "white" means white, and snow is white? Because that still only makes sense if there's a fact of the matter about whether or not snow is white. And as ata pointed out, it's also false.
Edit: Maybe Tarski's undefinability theorem applies here. It says that no powerful formal language can define truth in that language. So if, as you say, truth is an attribute of linguistic objects, you have to invoke a metalanguage i...
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.