Phrase structure grammar and model theory

The idea that one might do for the expressions of natural language what model theoretic structures do for a string of symbols in a notation was first entertained by Alfred Tarski in his pioneering work on model theory (Tarski 1935, 164) and was mentioned subsequently by other prominent logicians such as Paul Rosenbloom (1950, 153) and Alonzo Church (1956, 3)—all of whom thought the undertaking infeasible, since they thought that the grammatical expressions of natural language do not form a well defined set. However, in the early 1970’s a number of people undertook to carry out a least part of such a project. They included Richard Montague (a student of Tarski’s), Renate Bartsch and Theo Vennemann (1972), David Lewis (1972) and Max Cressell (1973). They all used some form of categorial grammar, a form of grammar with which only a few linguists are conversant. These days most semanticists use a form of an enriched phrase structure grammar instead, following the examples found in each of the four well-known textbooks of Chierchia and McGonnell-Ginet (1990), of Kamp and Teyle (1993), of Larson and Segal (1995) and of Heim and Kratzer (1998). However, these textbooks, and the work based on them, overlook a crucial question about the application of model theory to the study of natural language semantics using phrase structure grammars. The aim of this paper is to raise the question and to provide an answer to it. As we shall see, the answer provided also provides a solution to a fundamental problem posed by optional complements for the application of model theory to natural language expressions, noted as early as the 1960’s (Cf. Kenny 1963, ch. 7, for example).