Monotonicity phenomena are well-studied in the descriptive semantics of natural language, and to a lesser extent in formal work. This talk begins by reviewing some basic examples of monotonicity/antitonicity and of the associated concept of polarity. The first proposal for a formal calculus in the area is due to Johan van Benthem, and to Victor Sanchez-Valencia. Our work builds on the ideas and implements them in the context of typed lambda calculus interpreted over hierarchies of preorders rather than sets. This leads to an enrichment of the simply-typed lambda calculus which could be of independent interest. The talk will present many of the known basic results on the monotonicity calculus. For example, there is a version of Friedman's Theorem. The talk will also touch on how the overall topic of monotonicity gets used in natural language processing. But I'm reporting on work at an early stage, and as of now it has not been applied.
Joint work with Thomas Icard.