Overview of the Approach

In a series of papers culminating in [Mon 73], henceforth PTQ, Richard Montague embarked upon a program of providing a formal syntax coupled with a model-theoretic semantics for increasingly sophisticated fragments of English. The semantics for the PTQ fragment is given indirectly, as follows. The set of sentences of the fragment is defined inductively from a set of Basic Expressions (words). Direct translations into an Intensional Logic (IL) are provided for each of the Basic Terms. Each of the formation rules in the inductive definition is coupled with a translation rule which specifies the translation into IL of the output of the rule as a function of the translations of the input(s) to the rule. Thus the interpretation of any English sentence in the fragment is given by means of the model-theoretic interpretation given by the semantics of the logic to its corresponding representation in IL.

Subsequent to the presentation of the PTQ fragment, a number of researchers (e.g. [Tho72], [Ben74], [Par75], [Tho76], [Kar77], [Ben79], [Dow79], [Hir83], and [Hin88]) have explored various extensions to the PTQ fragment. These extensions have been motivated by the desire to provide a formal syntax and semantics to a larger set of English syntactic constructs; occasionally they have necessitated changes or extensions to the underlying logic IL. [DWP81] provides an excellent introduction to the area known as Montague Semantics, while [vB88] surveys the entire field of intensional logic.