  We introduce a multi-sorted stratified syllogistic, called 4LQSR, admitting
variables of four sorts and a restricted form of quantification over variables
of the first three sorts, and prove that it has a solvable satisfiability
problem by showing that it enjoys a small model property. Then, we consider the
fragments (4LQSR)^h of 4LQSR, consisting of 4LQSR-formulae whose quantifier
prefixes have length bounded by h > 1 and satisfying certain syntactic
constraints, and prove that each of them has an NP-complete satisfiability
problem. Finally we show that the modal logic K45 can be expressed in 4LQSR^3.
