Generalized quantifiers in Dependence Logic

TL;DR

This paper extends Dependence Logic by incorporating generalized quantifiers from Tarskian semantics, proposing multivalued dependence atoms as a better tool for handling scope dependencies, and relating them to independence atoms.

Contribution

It introduces generalized quantifiers into Dependence Logic and advocates for multivalued dependence atoms over traditional dependence atoms for scope dependency management.

Findings

Multivalued dependence atoms are better suited for scope dependencies.

The multivalued dependence atom is equivalent to the independence atom.

Both monotone and non-monotone generalized quantifiers are considered.

Abstract

We introduce generalized quantifiers, as defined in Tarskian semantics by Mostowski and Lindstr\"om, in logics whose semantics is based on teams instead of assignments, e.g., IF-logic and Dependence logic. Both the monotone and the non-monotone case is considered. It is argued that to handle quantifier scope dependencies of generalized quantifiers in a satisfying way the dependence atom in Dependence logic is not well suited and that the multivalued dependence atom is a better choice. This atom is in fact definably equivalent to the \emph{independence atom} recently introduced by V\"a\"an\"anen and Gr\"adel.