Logical types and linguistic types

One of the primary aims of linguistic semantics is to translate the expressions of natural language into formulas of some logical calculus. These formulas, in turn, can be interpreted in the appropriate models, and semantic notions like truth, entailment, etc. can be formally defined in the usual manner. In addition to the well-known theoretical advantages of such an intermediate logical form, there is a practical advantage as well: given a system of rules for translation from formulas to natural language, it will be possible to translate from one natural language to another (via the interlingua) without actually evaluating the expressions of the source language. Although the calculi used as intermediate language in machine translation range from first-order predicate calculus (e.g. Schubert – Pelletier 1982) to the higher order intensional calculus of Montague Grammar (e.g. Landsbergen 1977), so far no type-free calculus has been employed for this purpose. The aim of this paper is to outline an intermediate language for machine translation which is based on combinatory logic. Section 1 sketches the background assumptions of type theory. Section 2 discusses some of the problems with the traditional (typed) approach in Montague Grammar, and Section 3 outlines some general problems with type theory. In Section 4 a type-free intermediate language is defined and exemplified: its interpretation is discussed in the last section.