Dependence logic and its many variants are new logics that aim at establishing a unified logical theory of dependence and independence underlying seemingly unrelated subjects. The area of dependence logic has developed rapidly in the past few years. We will give a short introduction to dependence logic and review some of the recent developments in the area.

We study the extension of dependence logic \ by a majority quantifier \ over finite structures. We show that the resulting logic is equi-expressive with the extension of second-order logic by second-order majority quantifiers of all arities. Our results imply that, from the point of view of descriptive complexity theory, \\) captures the complexity class counting hierarchy. We also obtain characterizations of the individual levels of the counting hierarchy by fragments of \\).

We analyse the two definitions of generalized quantifiers for logics of dependence and independence that have been proposed by F. Engström, comparing them with a more general, higher order definition of team quantifier. We show that Engström’s definitions 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 Engström’s quantifiers only range over the latter. We further argue that Engström’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 Engström’s quantifiers. (shrink)