On the satisfiability problem for a 4-level quantified syllogistic and some applications to modal logic (extended version)

TL;DR

This paper introduces a multi-sorted stratified syllogistic called 4LQSR, proves its satisfiability is decidable, and demonstrates its applications to modal logic K45, with specific fragments having NP-complete satisfiability.

Contribution

It defines the 4LQSR framework, establishes its small model property, and connects it to modal logic K45 through a specific fragment.

Findings

4LQSR has a small model property ensuring decidability.

Fragments (4LQSR)^h have NP-complete satisfiability.

Modal logic K45 can be expressed within 4LQSR^3.

Abstract

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.