This report 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. This provides an intuitive representation which 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 can be shown that the mixed powerdomain has many of the properties associated with the convex powerdomain such as the possibility of solving recursive equations and a simple algebraic characterization.