Types, Sets and Probabilistic Semantics

TTR, a theory of types with records, which has been used in work on natural language for overfifteen years, is perhaps unique in that it combines a large number of features of Martin-Löf type theory with a foundation in standard Zermelo-Fraenkel set theory with urelements.

This immediately raises the questions of how much of the set theory is crucially used by TTR and whether TTR or a fragment of it could be accommodated within type theory.

To answer these questions we use the successful method from logic of defining relative interpretations betweentheories, to relate TTR to previously well-studies foundational theories.This is a unique opportunity to study a system that combines set theory and type theory, whichwould cast light on the relation between classical and intuitionistic foundational systems in thecontext of natural language semantics, and pinpoint the amount of mathematical abstractionrequired for formal natural language semantics.

The results of this project will serve also as aformally rigorous foundation for the large body of contemporary linguistic work within the TTRframework.We will also develop a probabilistic version of TTR, which can work as an interface between formal semantics and current machine learning approaches to probabilistic aspects of meaning and reasoning.

This would contribute significantly to the Natural Language Inference task, and in the long run also to the general endeavour of explainable artificial intelligence.