A second-order predicate is a predicate that takes a first-order predicate as an argument. Compare higher-order predicate.
The idea of second order predication was introduced by the German mathematician and philosopher Frege. It is based on his idea that a predicate such as "is a philosopher" designates a concept, rather than an object. Sometimes a concept can itself be the subject of a proposition, such as in "There are no Albanian philosophers". In this case, we are not saying anything of any Albanian philosophers, but of the concept "is an Albanian philosopher" that it is not satisfied. Thus the predicate "is not satisfied" attributes something to the concept concept "is an Albanian philosopher", and is thus a second-level predicate.
This idea is the basis of Frege's theory of number.