Jacobs (1980,1991) and Rullmann (1995) claim that lexical decomposition of the German determiner kein "no" and its Dutch counterpart geen "no" is necessary to account for readings in which a scope-bearing operator such as an intensional verb or a universal quantifier intervenes between the negation and the existential quantifier part of the determiner. In this paper, I argue that lexical decomposition is not only undesirable, but empirically incorrect. As an alternative, I develop a higher-order interpretation of negative quantifiers in terms of quantification over properties. The analysis is built on the observation that split readings are restricted to monotone decreasing NPs in predicative positions.