Generalized quantifiers using team semantics

TL;DR

This paper extends dependence logic with a new semantics to handle both monotone and non-monotone generalized quantifiers, matching the expressive power of existential second-order logic and maintaining conservativity over first-order logic.

Contribution

It introduces a modified semantical framework for dependence logic that accommodates all types of generalized quantifiers, including non-monotone ones, and proves its expressive equivalence to ESO.

Findings

Handles both monotone and non-monotone generalized quantifiers

Expressive power matches existential second-order logic

Maintains conservativity over first-order logic

Abstract

Dependence logic provides an elegant approach for introducing dependencies between variables into the object language of first-order logic. In [1] generalized quantifiers were introduced in this context. However, a satisfactory account was only achieved for monotone increasing generalized quantifiers. In this paper, we modify the fundamental semantical guideline of dependence logic to create a framework that adequately handles both monotone and non-monotone generalized quantifiers. We demonstrate that this new logic can interpret dependence logic and possesses the same expressive power as existential second-order logic (ESO) on the level of formulas. Additionally, we establish truth conditions for generalized quantifiers and prove that the extended logic remains conservative over first-order logic with generalized quantifiers and is able to express the branching of continuous generalized quantifiers.