[sygtrmyz] Fun with the rule of the excluded middle

For all statements P, either P is true or P is false.  ("Either ... or" means exclusive or.)

Paradox: let P = "This statement is false."  (Or, "this sentence is false.")  This causes a problem because P is a statement that is neither true nor false, so violates the rule of the excluded middle, meaning the rule is not always valid.  If this axiom is not valid, then lots of mathematics built on top of it break down, for example, proof by contradiction.  This is a big problem!

I think the resolution of the paradox is that the rule does not apply to all statements, but only to some subset of "well formed" statements.  What is a good definition of well formed for the rule?  ("Good" mught be subjective.)  Forbidding self-reference is not enough, e.g., Quine's Paradox.