Summary: We propose an extension to logic programming called LPS (Logic Programming with Sets). The LPS language has two types of objects: individual objects, and sets whose elements are individual objects. We first consider only one level of set nesting in order to concentrate on the key problems that arise, in as simple a framework as possible. The rules in the LPS language are fairly similar to the Horn clauses of logic programming. The main difference between LPS rules and Horn clauses is that the right-hand side of a LPS rule may be preceded by restricted universal quantifiers. This means that a LPS rule has the form \[ A:- (\forall x_ 1\in X_ 1)...(\forall x_ n\in X_ n)(B_ 1\wedge...\wedge B_ m). \] This is a fairly conservative extension of Horn clause logic, since whenever the sets \(X_ 1,...,X_ n\) have known values, the body can be reduced to a normal Horn clause, i.e., the conjunction of the body (without the quantifiers) over all the elements of the sets. We shall see that our extension of Horn clause logic, unlike extensions that allow arbitrary quantification on the right-hand side, preserves the semantics of Horn clause logic.