On the Satisfiability Problem for a 4-level Quantified Syllogistic and Some Applications to Modal Logic

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 ≥⃒ 2 and satisfying certain additional syntactical 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.