Generalized quantifiers and modal logic (Q1314283)

The paper studies several modal systems designed to represent generalized quantifiers. The main system QUANT has infinitely many modal operators \(M_ n\), which are interpreted in a set \(W\) under a valuation \(V\) as follows: \(M_ n\varphi\) is true at a point \(x\in W\) if the number of points in \(W\) at which \(\varphi\) is true under \(V\) is greater than \(n\) [cf. \textit{K. Fine}, Notre Dame J. Formal Logic 13, 516-520 (1972; Zbl 0242.02025)]. It is proved that every first-order definable quantifier is definable in the language of QUANT. Subsystems of QUANT and a system for representing higher order quantifiers as modal operators are also studied. Completeness, complexity, normal forms and some other standard problems are investigated for the systems under consideration.