A single-type logic for natural language

Abstract

In this article, we develop a single-type logic for natural language along the lines of [Partee (2009, Snippets, vol. 20)]). This logic, called ‘${\text{TY}}_0^3$’ takes objects of different syntactic categories and model-theoretic domains to be structured by the same logical type. Its language, a variant of the simply typed lambda calculus, is interpreted in partial Henkin models. We give a Gentzen-style sequent calculus for ${\text{TY}}_0^3$ and prove its soundness and completeness with respect to the class of models. To show the logic's application adequacy, we provide a ${\text{TY}}_0^3$ semantics for a standard fragment of English. Partial possible worlds, which are identified with elements in the logic's base domain, enable us to obtain the standard modal operators.