This paper characterizes the powerdomain constructions which have been used in the semantics of programming languages in terms of formulas of first order logic under a preordering of provable implication. The goal is to reveal the basic logical significance of the powerdomain elements by casting them in the right setting. Such a treatment may contribute to a better understanding of their potential uses in areas which deal with concepts of sets and partial information such as databases and computational linguistics. This way of viewing powerdomain elements suggests a new form of powerdomain - called the mixed powerdomain - which expresses data in a different way from the well-known constructions from programming semantics. It is shown that the mixed powerdomain has many of the properties associated with the convex powerdomain such as the possiblity of solving recursive equations and a simple algebraic characterization.