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.)
It takes time for me to find and try understanding the materials you mentioned. I'm unlikely to have a meaningful reply to your comment anytime soon. Let me just thank you here :)
Of course not. They are both in German, though, and it didn't help that Wessel refused to change his name when officials of the GDR asked him to, because that would have contradicted his views about proper names. [The best formulation of FOL with NTPT is in the 1999 edition of Logik.]
Wessel, Horst (1989, 1999): Logik. Logos Berlin.
Sinowjew (Zinov'ev), Alexander Alexandrowitsch (1973): Foundations of the Logical Theory of Scientific Knowledge (Complex Logic). Springer.
Sinowjew (Zinov'ev), A. A. (1970): Komplexe Logik. Grundlagen einer logischen Theorie des Wissens. Vieweg.
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.)