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 pre-ordering of provable implication. The goal is to reveal the basic logical significance of the powerdomains 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 artificial intelligence. Extended examples relating powerdomains to databases are provided. A new powerdomain is introduced and discussed in comparison with a similar operator from database theory. The new powerdomain is motivated by the logical characterizations of the three well-known powerdomains and is itself characterized by formulas of first order logic.