This paper presents a unified analysis of the licensing conditions for NPs headed by the particle de in argument position in French (e.g., J'ai beaucoup lu de livres; Je (n')ai pas lu de livres). These expressions are only possible in direct object position when c-commanded by an appropriate operator: either a degree adverb (e.g., beaucoup, tellement, trop, etc.) or an anti-additive (negative) quantifier (e.g., pas, personne, rien, etc.). This paper argues that the unifying generalization that groups these two groups of operators together and explains their licensing abilities is semantic: they are polyadic operators, that is, elements that can bind more than one variable at once. It is argued that the polyadic nature of the degree adverbs can be seen in the interpretations of 'Quantification at a distance' sentences, and that the polyadic nature of the anti-additive quantifiers can be seen in their participation in negative concord phenomena.