Monotonicity Reasoning in Formal Semantics Based on Modern Type Theories

Modern type theories (MTTs) have been developed as a powerful foundation for formal semantics. In particular, it provides a useful platform for natural language inference (NLI) where proof assistants can be used for inference on computers. In this paper, we consider how monotonicity reasoning can be dealt in MTTs, so that it can contribute to NLI based on the MTT-semantics. We show that subtyping is crucial in monotonicity reasoning in MTTs because CNs are interpreted as types and therefore the monotonicity relations between CNs should be represented by the subtyping relations. In the past, monotonicity reasoning has only been considered for the arrow-types in the Montagovian setting of simple type theory. In MTT-semantics, richer type constructors involving dependent types and inductive types are employed in semantic representations. We show how to consider monotonicity reasoning that involve such type constructors and how this offers new useful mechanisms in monotonicity reasoning in MTTs.