  We present a logical framework for formalizing connections between finitary
combinatorics and measure theory or ergodic theory that have appeared various
places throughout the literature. We develop the basic syntax and semantics of
this logic and give applications, showing that the method can express the
classic Furstenberg correspondence and to give a short proof of the Szemer\'edi
Regularity Lemma. We also derive some connections between the model-theoretic
notion of stability and the Gowers uniformity norms from combinatorics.
