It's not uncommon. Many people use the truth values {T}, {F}, {}, and {T, F} of one of the four-valued Belnap systems taken as partial logic. It is very adequate for linguistic modeling of truth conditional content and propositional attitudes, see for example Muskens's great little book Meaning and Partiality.

If I only need the "third case", I personally prefer a bivalent logic with nontraditional predication theory developed by A. Sinowjew (1970) and H. Wessel (1989). It's great fun to point out this system to philosophers who weren't trained very well in logic and are dogmatically convinced that it's impossible to express a third case in a bivalent logic. (Admittedly, that's a very petty motive. Anyway, NTPT will not convince any real intuitionist, because the quantifiers remain classical, too.)