Ok, so you can know if a proposition is false, or you can know if it is true, and these two things fundamentally behave differently if the proposition contains the word 'not' in it?
It's not just the word "not" it's an odd number of nots (double negatives and all that).
It is a statement about an abstraction, not existence. Like saying "the cup is above the table" is a statement about the existence of the "above-ness with respect to the table" of the cup. One needn't survey all of reality to know if it is possible for some abstraction to exist. It need only be consistent with the rest of your language, so that's all you need to survey.