Member Rara Avis
A self-referential paradox is the same as a self-contradiction, and self-contradictions by their very nature are false.
This sentence is false.
Surely you see the problem with your logic? You CANNOT draw the conclusion that a paradox is false without creating the very paradox you seek to thwart. The Aristotelian Law of the Excluded Middle asserts that "every proposition is either true or false." That doesn't mean that we have to KNOW the truth value, just that it has to have one at any given point in time. A paradox doesn't, which is precisely why a paradox refuses to fit into our system of logic.
Here's a more comprhensive (and therefore more complicated) discussion of The Law of the Exluded Middle.
… but I would say that "this statement is false" doesn't exist.
You were, at least, on somewhat stronger ground with this attempt. One might argue, of course, that your very reference to the statement proves its existence, else you couldn't reference it. But we don't need to go that circuitous route, either. Essentially, you are denying the Law of the Excluded Middle and asserting a third possibility, one of non-existence (most who go this route try to argue ambiguity as a third alternative, but one's as good as another).
That doesn't work either. Let's assume a statement has three possible values, as you suggested:
a) True statements
b) False statements
c) Statements that don't exist
Now consider the statement:
This statement is false or does not exist.
If the statement is true, then it is false or does not exist, i.e., the statement is not true. Obviously, a contradiction.
If the statement is false, then the statement is true. Another contradiction.
If the statement doesn't exist, then the statement is true. Oops. Here we are again.
Any paradox that defies the Law of the Excluded Middle can be extended into Nth dimensions and will STILL be a paradox. It falls outside our system of logic and CANNOT be used within our system to prove anything (or, rather, it can prove anything you want it to prove).
I'm going to quote you on this the next time someone asks me why I don't believe in God.
All I've really demonstrated is that human constructs can't define God. That hardly seems surprising to me. On the other hand, if you COULD define God within human understanding, I would be willing to accept that as proof that such a defined God didn't exist as a God.