If Art is smart and Art is rich, then someone is both smart and rich – namely, Art. And if Art is smart and Bart is smart, then Art is something that Bart is, too – namely, smart. The first claim involves first-order quantification, a generalization concerning what kinds of things there are. The second involves second-order quantification, a generalization concerning what there is for things to be. Or so it appears. Following W.V.O. Quine, many philosophers have endorsed a thesis of Ontological Collapse about second-order quantification. They maintain that ultimately, second-order quantification reduces to first-order quantification over sets or properties, and therefore also carries the latter’s distinctive ontological commitments. In this revised version of his doctoral dissertation, awarded the Wolfgang-Stegmüller-Prize in 2012, Stephan Krämer examines the major arguments for Ontological Collapse in detail and finds all of them wanting. Quantifications, he argues, fall into at least two irreducible kinds: those on what things there are, and those on what there is for things to be