Abstract. We develop a method for extending results about ultrafilters into a more general setting. In this paper we shall be mainly concerned with applications to cardinality logics. For example, assuming , Gdels Axiom of Constructibility, we prove that if then the logic with the quantifier there exist many is -compact if and only if either is weakly compact or is singular of cofinality . As a corollary, for every infinite cardinals and , there exists a -compact non- -compact logic if and only if either or or is weakly compact. Counterexamples are given showing that the above statements may fail, if is not assumed. However, without special assumptions, analogous results are obtained for the stronger notion of -compactness.