Modern logics of dependence and independence are based on different variants of atomic dependency statements (such as dependence, exclusion, inclusion, or independence) and on team semantics: A formula is evaluated not with a single assignment of values to the free variables, but with a set of such assignments, called a team.

In this paper we explore logics of dependence and independence where the atomic dependency statements cannot distinguish elements up to equality, but only up to a given equivalence relation (which may model observational indistinguishabilities, for instance between states of a computational process or between values obtained in an experiment).

Our main goal is to analyse the power of such logics, by identifying equally expressive fragments of existential second-order logic or greatest fixed-point logic, with relations that are closed under the given equivalence. Using an adaptation of the Ehrenfeucht-Fraïssé method we further study conditions on the given equivalences under which these logics collapse to first-order logic, are equivalent to full existential second-order logic, or are strictly between first-order and existential second-order logic.