Title:

Constraining Montague grammar for computational applications

This work develops efficient methods for the implementation of Montague Grammar on a computer. It covers both the syntactic and the semantic aspects of that task. Using a simplified but adequate version of Montague Grammar it is shown how to translate from an English fragment to a purely extensional firstorder language which can then be made amenable to standard automatic theoremproving techniques. Translating a sentence of Montague English into the firstorder predicate calculus usually proceeds via an intermediate translation in the typed lambda calculus which is then simplified by lambdareduction to obtain a firstorder equivalent. If sufficient sortal structure underlies the type theory for the reduced translation to always be a firstorder one then perhaps it should be directly constructed during the syntactic analysis of the sentence so that the lambdaexpressions never come into existence and no further processing is necessary. A method is proposed to achieve this involving the unification of metalogical expressions which flesh out the type symbols of Montague's type theory with firstorder schemas. It is then shown how to implement Montague Semantics without using a theorem prover for type theory. Nothing more than a theorem prover for the firstorder predicate calculus is required. The firstorder system can be used directly without encoding the whole of type theory. It is only necessary to encode a part of secondorder logic and this can be done in an efficient, succinct, and readable manner. Furthermore the pseudosecondorder terms need never appear in any translations provided by the parser. They are vital just when higherorder reasoning must be simulated. The foundation of this approach is its fivesorted theory of Montague Semantics. The objects in this theory are entitites, indices, propositions, properties, and quantities. It is a theory which can be expressed in the language of firstorder logic by means of axiom schemas and there is a finite secondorder axiomatisation which is the basis for the theoremproving arrangement. It can be viewed as a very constrained set theory.
