It is well known that vanilla LFG+Glue overgenerates when it comes to scope ambiguities. This has both theoretical and practical implications: it generates unattested readings, and also produces many spurious ambiguities which do not correspond to linguistically relevant differences. We propose a solution in keeping with the LFG philosophy: a new level of the parallel projection architecture, called proof-structure, which can be used to limit scopal interactions. Unlike earlier attempts to solve this problem, the modularity of our approach means that we do not need to alter the underlying linear logic or proof algorithm of Glue Semantics, with the effect that existing analyses and tools continue to be relevant and usable.
Findlay et al. (Fri,) studied this question.