[stakhanov]

...that would be more of a belief theory (like Dempster-Shafer) than a logic.

Or maybe what you are thinking of is simply the situation wherein a logical expression is neither valid nor unsatisfiable. Valid would mean it's true in all models (all possible worlds), and therefore provable. Unsatisfiable would mean it's false in all models (all possible worlds), so that its negation would be provable.

so "p or (not p)" would be valid "p and (not p)" would be unsatisfiable "p" is neither valid nor unsatisfiable. it may be true, or it may be false. it's a contingency. Maybe that's what you have in mind by "unknown".

There are various mathematical theories for navigating the space between unsatisfiable and valid. If you start thinking about the proportion of possible worlds wherein p is true, you are thinking of frequentist probability theory. If you start reasoning about whether p is satisfiable whereas not p is also satisfiable and distinguishing that from the case where p is either unsatisfiable or valid, then you are thinking of possibility theory.

[harekaze]

Yes, I use INVALID for scenarios where "p" is unsatisfiable, as well as scenarios where I encounter other types of nonsense sentences. For example, "the uppercase is having a daydream", or a more subtle one, "the country is proud of its own products" when considered literally (as a country is incapable of feeling proud). I should have called it NONSENSE instead.