  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.
