In Iceberg semantics the denotation of an NP is an i(ceberg)-set, a pair of sets <body, base>, where the base generates the body under sum. You can think of the body as what was the full interpretation of the NP in Mountain semantics; for count NPs, the base is the distribution set, a disjoint set in terms of which the body is counted. Sections 5.1, 5.2, 5.3, 5.4 and 5.5 develop Iceberg semantics for count NPs and DPs. The interpretations of singular NPs, plural NPs and DPs are specified (Sect. 5.1); the notion of a distribution set, presupposing disjointness, is introduced, and cardinality is redefined in terms of this notion (Sect. 5.2). Section 5.3. introduces the compositional theory. The Head principle specifies that the base of the denotation of a complex NP is determined by the part set of the body of the denotation of that complex NP, intersected with the base of the grammatical head of the NP. This principle plays a central role throughout this book, since, as argued in Sect 6.1 and later sections, it has the consequence that the interpretations of complex NPs and DPs inherit their mass-count characteristics from the interpretation of their nominal head. Section 5.4 illustrates the compositional theory by giving a detailed step-by-step derivation of the Iceberg interpretation of the three white cats; Section 5.5 shows how the theory deals with the problems that were discussed in Sect. 4.7 of counting and distribution in a non-sorted theory.