The semantics of human languages involves complex representations and operations over those representations. The goal of my research is to design logics in which one can define representations and operations over those representations that closely mimic what is observed in human languages. This allows us to then model how people interpret the meanings of arbitrary expressions in the languages they speak. To understand how meaning in human languages works in general, we need evidence from a large variety of languages. For this reason my research has a strong empirical component. I have done primary fieldwork on a number of Mayan languages spoken in Guatemala. I work to describe, document, and analyze these languages in their own right, while also discovering what they can tell us about how human languages work in general.