The Liar Paradox of antiquity goes something like this:
A man says, "Everything I say is a lie."
(I find the modern variation -- "This statement is false" -- to be less interesting. It seems to me to be nothing more than a simple self-contradiction.)
Define 3 logical predicates:
$S(x)$ means $x$ is a sentence
$T(x)$ means $x$ is true
$M(x)$ means the man says $x$
$S(x)\land M(x)\land \forall y(S(y)\rightarrow (M(y)\rightarrow \neg T(y)))$ (Premise)
$S(x)$ (Splitting premise, 1)
$\forall y(S(y)\rightarrow (M(y)\rightarrow \neg T(y)))$
$S(x)\rightarrow (M(x)\rightarrow\neg T(x))$ (Universal Specification, 4)
$M(x)\rightarrow\neg T(x)$ (Detachment, 2, 5)
$\neg T(x)$ (Detachment, 3, 6)
$\forall a (S(a)\land M(a)\land \forall y(S(y)\rightarrow (M(y)\rightarrow \neg T(y)))\rightarrow \neg T(a))$ (Conclusion, 1)
My question is, does this proof resolve the Liar Paradox?
By definition, everything a constant liar says is false. A contradiction arises only, it would seem, when he says something like, "Everything I say is a lie," that is, when he claims:
$$\forall x (S(x)\rightarrow (M(x) \rightarrow \neg T(x))$$
If, as required, this is false, then
$$\exists x (S(x) \land M(x) \land T(x))$$
This contradicts the requirement that everything he says is false. If the constant liar can refrain from making such an admission, no paradoxical situation should arise. No such contradiction would arise, for example, from his saying, "Everything I say is true."
To answer my own question then, the above theorem does not resolve the original Liar Paradox of antiquity. It doesn't "prove" much at all, I'm afraid, but I now feel I have indeed resolved the paradox: It arises from the liar himself claiming that everything he says is false, and my (not necessarily well-founded) assumption that everything he says is indeed false. In hindsight, I think the "prize" must go to Alex Becker for his insightful comment. See my formal proof (Corollary starting on line 19) at http://www.dcproof.com/LiarParadox.htm
FOLLOW-UP TWO YEARS LATER
In the time since I first posted this question, I have come to realize that "This sentence is false" is nothing more or less than meaningless nonsense. Somehow, it is easier to see that "This sentence is TRUE" is meaningless nonsense. Simply changing "true" to "false" should not somehow suddenly imbue this sentence with meaning. I don't really think you can formalize the notion of meaningless nonsense.