Some statements are genuinely neither exclusively true nor false. It may be because the axiom set is insufficient to prove a statement, or it may be that the axiom set is too strong and can prove some statement to be both true and false.
Consider 3 different ways of defining true or false:
- True if provable from axioms, false if disprovable from axioms
- True if intuition (or a model) suggests it is true, false otherwise
- An expression is true or false if it is matched by some grammar $G$ (though not necessarily sure which)
The first way is how Hilbert (and others) would have described true/false in mathematical terms. The second way Godel (and others) used to analyze the first way. The third is the law of the excluded middle.
The third definition is what most people who study mathematics but not necessarily logic use. One may write $\forall x ~:~ x > y$ and say "it must be true or false, because it is a well written statement". And $\forall > x ~ y @$ is neither true nor false because it is garbage. Or maybe $x > 1$ isn't necessarily true or false because it isn't a complete statement, as the $x$ is unquantified. The problem with this intuitive approach is that it implicitly creating a grammar (an algorithmically defined set of of strings) and associating true/false with this grammar.
This is not necessarily compatible with the first definition of truth, given above. What if all statements that follow from axioms don't necessarily form a nice decidable grammar? In fact, Godel's incompleteness theorems say that that is the case.
So in very carefully defined logics (such as what would be used to prove correctness of avionics software, where incorrectness leads to people dying), whether a statement is true or false is left to be proved. Rather than assuming the excluded middle, that some set of grammatically defined strings are necessarily true or false, rather whether a statement is true or false is left to be proven. In casual usage, this is no big deal, but in formal logic, the excluded middle can't safely be assumed as an axiom.