Potts et seq. presents an analysis of gap-containing supplements where the gap is modelled as a variable over the semantic type of the constituent that the as-clause adjoins to. This much allows the meaning of the gap to be resolved purely compositionally, by defining as as a function that allows the anchor to bind the gap variable. This article presents a class of as-clauses where Potts’s analysis seems to break down, in that the gap cannot be modelled as a variable (...) over the semantic type of the anchor. I propose that these cases can be unified with those in Potts’s work, as well as a larger class of ellipsis phenomena, by assuming that, under certain conditions, surface gaps are composite entities, containing a bound variable and a free variable that are resolved independently of each other. The bound variable is bound by the anchor, and the free variable is resolved by anaphora to a salient discourse object. (shrink)