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An important motivation for Montague’s work on quantification (Montague 1974) was to achieve uniformity with respect to referential and quantificational subjects. This was attained by type raising all NPs to denote sets of sets (indeed there are claims that such a move is theoretically necessary) and by giving up a subject–predicate semantics where the verbal predicate predicates of the nominal argument. In this paper we argue for essentially the opposite move whereby all predication is genuine predication and involves arguments -- witnesses of type individual or set of individuals (for plurals). We argue that such an approach is crucial if one is to capture a variety of fundamentally important phenomena involving anaphora, clarification interaction, and speech-gesture cross-references associated with the use of quantificational noun phrases in dialogue, and to explicate several recent key psycholinguistic results on quantifier processing -- all features of an NP semantics which give rise to what we call “Referential Transparency”. The discussion is couched in a new set-denotational framework for plural count nouns, namely sets of ordered set bipartitions. We argue that quantification happens entirely within the noun phrase and involves ref(erence)sets, comp(lement)sets, and max(imal)sets. As a corollary of this denotational foundation, the semantic conservativity universal is an immediate consequence and the range of quantifier denotations is significantly reduced. In addition to collecting empirical motivation for quantification from Referential Transparency Theory and to developing a count noun semantics, a theoretically grounded explanation for complement set anaphora is given.