Some observations about generalized quantifiers in logics of imperfect information

TL;DR

This paper compares different definitions of generalized quantifiers in logics of imperfect information, showing how they relate to team semantics and highlighting limitations in existing approaches.

Contribution

It provides a unifying analysis of Engström's definitions with higher-order team quantifiers and critiques their ability to fully capture team-theoretical generalized quantifiers.

Findings

Engström's quantifiers are special cases of team quantifiers.

Team quantifiers include both quantitative and qualitative components.

Engström's definitions embed first-order quantifiers but miss some team-theoretical aspects.

Abstract

We analyse the two definitions of generalized quantifiers for logics of dependence and independence that have been proposed by F. Engstr\"om, comparing them with a more general, higher-order definition of team quantifier. We show that Engstr\"om's definitions (and other quantifiers from the literature) can be identified, by means of appropriate lifts, with special classes of team quantifiers. We point out that the new team quantifiers express a quantitative and a qualitative component, while Engstr\"om's quantifiers only range over the latter. We further argue that Engstr\"om's definitions are just embeddings of the first-order generalized quantifiers into team semantics, and fail to capture an adequate notion of team-theoretical generalized quantifier, save for the special cases in which the quantifiers are applied to flat formulas. We also raise several doubts concerning the meaningfulness of the monotone/nonmonotone distinction in this context. In the appendix we develop some proof theory for Engstr\"om's quantifiers.