Henkin Semantics for Reasoning with Natural Language

Frank Richter and Michael Hahn

The frequency of intensional and non-first-order definable operators in natural languages constitutes a formidable challenge for automated reasoning with the kind of logical translations that are deemed adequate by formal semanticists. Whereas linguists standardly employ expressive higher-order logics in their theories of meaning, the most successful logical reasoning strategies with natural language to date rely on sophisticated first-order theorem provers and model builders. In order to bridge the fundamental mathematical gap between linguistic theory and computational practice in grammar implementations, we present a general translation from a higher-order logic frequently employed in the linguistics literature, two-sorted Type Theory, to first-order logic under Henkin semantics. We investigate alternative formulations of the translation, discuss their properties, and evaluate the availability of linguistically relevant inferences with standard theorem provers in a test suite of inference problems stated in English. The results of the experiment indicate that translation from higher-order logic to first-order logic under Henkin semantics is a promising strategy for automated reasoning with natural languages.