According to the standard view of scope underspecification, an underspecified representation denotes a set of interpretations (readings), while the (partial or full) specification of an underspecified denotation amounts to narrowing down the set of interpretations. This perspective on underspecification has, arguably, two shortcomings. First, since the denotation of the underspecified representation is a set of fully specified readings, it does not capture what the different readings have in common. And secondly, the denotation of the underspecified representation is not in any sense part of the denotation of each specified representation. In this paper I will propose an alternative theory of quantification and scope underspecification, where (i) the denotation of the underspecified representation does indeed capture what the possible readings all have in common, and (ii) the specification of an underspecified denotation amounts to adding information, so that there is a clear sense in which the underspecified denotation is part of every specified reading.