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
${{\cal L}_{\,I\,}}$.