  We study elementary modal logics, i.e. modal logic considered over
first-order definable classes of frames. The classical semantics of modal logic
allows infinite structures, but often practical applications require to
restrict our attention to finite structures. Many decidability and
undecidability results for the elementary modal logics were proved separately
for general satisfiability and for finite satisfiability [11, 12, 16, 17]. In
this paper, we show that there is a reason why we must deal with both kinds of
satisfiability separately -- we prove that there is a universal first-order
formula that defines an elementary modal logic with decidable (global)
satisfiability problem, but undecidable finite satisfiability problem, and, the
other way round, that there is a universal formula that defines an elementary
modal logic with decidable finite satisfiability problem, but undecidable
general satisfiability problem.
