Dependence Logic with Generalized Quantifiers: Axiomatizations

TL;DR

This paper establishes completeness results for dependence logic extended with generalized quantifiers, including monotone and uncountability quantifiers, providing axiomatizations that are sound and complete for FO(Q) consequences.

Contribution

It introduces two new axiomatizations for dependence logic extended with generalized quantifiers, covering monotone and uncountable interpretations, advancing the theoretical understanding of these logics.

Findings

Proves completeness for dependence logic with a monotone generalized quantifier Q.

Establishes completeness for dependence logic with the uncountability quantifier.

Shows axiomatizations are sound and complete for FO(Q) consequences.

Abstract

We prove two completeness results, one for the extension of dependence logic by a monotone generalized quantifier Q with weak interpretation, weak in the meaning that the interpretation of Q varies with the structures. The second result considers the extension of dependence logic where Q is interpreted as "there exists uncountable many." Both of the axiomatizations are shown to be sound and complete for FO(Q) consequences.