A Remark on Collective Quanti cation

We consider collective quanti cation in natural language. For many years the common strategy in formalizing collective quanti cation has been to de ne the meanings of collective determiners, quantifying over collections, using certain type-shifting operations. These type-shifting operations, i.e., lifts, de ne the collective interpretations of collective determiners systematically from the standard meanings of quanti ers. All the lifts considered in the literature turn out to be de nable in second-order logic. We argue that second-order de nable quanti ers are probably not expressive enough to formalize all collective quanti cation in natural language.