We introduce and study formal logics for reasoning about propositional determinacy and independence. These relate naturally with the philosophical concept of supervenience, which can also be regarded as a generalisation of logical consequence. Propositional Dependence Logic D, and Propositional Independence Logic I are recently developed logical systems, based on team semantics, that provide a framework for such reasoning tasks. We introduce two new logics L_D and L_I, based on Kripke semantics, and propose them as alternatives for D and I, respectively. We analyse and compare the relative expressive powers of these four logics and also discuss how they relate to the natural language use and meaning of the concepts of determinacy and independence. We argue that L_D and L_I naturally resolve a range of interpretational problems that arise in D and I. We also obtain sound and complete axiomatizations for L_D and L_I and relate them with the recently studied inquisitive logics and their semantics.