On the satisfiability problem for a 3-level quantified syllogistic

We show that a collection of three-sorted set-theoretic formulae, denoted TLQSR and which admits a restricted form of quantification over individual and set variables, has a solvable satisfiability problem by proving that it enjoys a small model property, i.e., any satisfiable TLQSR-formula psi has a finite model whose size depends solely on the size of psi itself. We also introduce the sublanguages (TLQSR)^h of TLQSR, whose formulae are characterized by having quantifier prefixes of length bounded by h \geq 2 and some other syntactic constraints, and we prove that each of them has the satisfiability problem NP-complete. Then, we show that the modal logic S5 can be formalized in (TLQSR)^3.