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

Abstract

We introduce a multi-sorted stratified syllogistic, called 4LQS^R, 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 (4LQS^R)^h of 4LQS^R, consisting of 4LQS^R-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 (4LQS^R)³.