This is the map territory relation problem.
We bring in the 'Trinity of thought' a priori and forgot about the advantages and costs.
You have the principal of excluded middle, principal of identity, and the principal of non-contradiction.
If your problems territory fits things are convenient.
Once you introduce known, unknown, or unknowable unknowns, the classic model falls down.
Unfortunately you have to choose what to sacrifice based on the context of the problem at hand.
This is exactly where Rice's theorm, the halting problem etc.. arise.
AC from ZF(C) and IID from statistics bring PEM, which gives or forces the classical behavior.
The 'non-trivial' properties of Rice's theorm are exactly those properties where PEM doesn't hold.
The frame problem in machine learning is also related. As is Gödels (in) completeness theories.
Sometimes you can use failure as negation, other times you can use methods like accepting that NULL is an unknown.
But you have to decide what you can sacrifice and still solve real problems.
I think thinking of a problem and it's complement is better.
E.G P is the truthy T and co-P is the truthy F. Obviously P=co-P means that the traditional method works, but jump to NP vs co-NP and ot doesn't.
NP chained ORs, co-NP is chained ANDs is another way of thinking about it.