Ultrafilter translations .1. (lambda,lambda)-compactness of logics with a cardinality quantifier

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 V = L, Godel's Axiom of Constructibility, we prove that if lambda > omega(alpha) then the logic with the quantifier ''there exist omega(alpha) many'' is (lambda,lambda)-compact if and only if either lambda is weakly compact or lambda is singular of cofinality < omega(alpha-1). As a corollary, for every infinite cardinals lambda and mu, there exists a (lambda, lambda)-compact non-(mu, mu)-compact logic if and only if either lambda < mu or cf lambda < cf mu or lambda > mu is weakly compact. Counterexamples are given showing that the above statements may fail, if V = L is not assumed. However, without special assumptions, analogous results are obtained for the stronger notion of [lambda, lambda]-compactness.