This paper investigates formal logics for reasoning about determinacy and independence. Propositional Dependence Logic ${\cal D}$ and Propositional Independence Logic ${\cal I}$ are recently developed logical systems, based on team semantics, that provide a framework for such reasoning tasks. We introduce two new logics ${{\cal L}_D}$ and ${{\cal L}_{\,I\,}}$ , based on Kripke semantics, and propose them as alternatives for ${\cal D}$ and ${\cal I}$ , respectively. We analyse the relative expressive powers of these four logics and discuss the way these systems relate to natural language. We argue that ${{\cal L}_D}$ and ${{\cal L}_{\,I\,}}$ naturally resolve a range of interpretational problems that arise in ${\cal D}$ and ${\cal I}$ . We also obtain sound and complete axiomatizations for ${{\cal L}_D}$ and ${{\cal L}_{\,I\,}}$ .