The axiom of excluded middle is that for every proposition, either that proposion is true or it is false. That's a universal quantification, so any counterexample makes it false. "This proposition is false." is such a counterexample (it can't be true, because than it's false and your logic is inconsistent, and it can't be false, because than it's true and ditto). This reasoning applies to pretty much any recognizable system of propositional logic (including, per Godel nineteen-thirty-something, systems that explicitly prohibit the basic self-referential counter example, as long as they're expressive enough to describe arithmetic), so it's at least colloquially correct to simply dismiss it as false, even if the pedantic version is that it's a axiom that cannot be part of a consistent set of axioms.
The theorem of included middle doesn't prevent specific classes of propositions from being exclusively divided into true and false, though. As a simple example, any proposition of the form "natural number SSS...SS0 is prime" will be either true or false, and there's nothing inconsistent about (say) "for every proposion P matching regex /(S)*0 is prime/, either P is true or P is false". Just like there's nothing weird about a linter that can reliably tell you that "while(1) {...}" doesn't terminate, halting problem or not.