  In this paper, we present an abstract framework of many-valued modal logic
with the interpretation of atomic propositions and modal operators as predicate
lifting over coalgebras for an endofunctor on the category of sets. It
generalizes Pattinson's stratification method for colagebraic modal logic to
the many-valued setting. In contrast to standard techniques of canonical model
construction and filtration, this method employs an induction principle to
prove the soundness, completeness, and finite model property of the logics. As
a consequence, we can lift a restriction on the previous approach~
\cite{Lin2022} that requires the underlying language must have the expressive
power to internalize the meta-level truth valuation operations.
