Nakamura proposed two three-valued temporal logics. We present two extended Gentzen-type formulations of these logics. Then we prove the soundness as well as the completeness theorem.
"Extended Gentzen-type Formulations of Two Temporal Logics Based on Incomplete Knowledge Systems." Notre Dame J. Formal Logic 42 (1) 55 - 64, 2001. https://doi.org/10.1305/ndjfl/1054301355