Computational semantics in type theory

This paper aims to show how Montague-style grammars can be completely formalized and thereby declaratively implemented by using the Grammatical Framework GF. The implementation covers the fundamental operations of Montague’s PTQ model: the construction of analysis trees, the linearization of trees into strings, and the interpretation of trees as logical formulas. Moreover, a parsing algorithm is derived from the grammar. Given that GF is a constructive type theory with dependent types, the technique extends from classical Montague grammars to ones in which the Curry-Howard isomorphism is used to explain anaphoric reference. On the other hand, GF has a built-in compositionality requirement that is stronger than in PTQ and prevents us from formulating quantifying-in rules of Montague style. This leads us to alternative formulations of such rules in terms of combinators and discontinuous constituents. The PTQ fragment will moreover be presented as an exmple of how a GF grammar is modified by replacing English with another target language, French. The paper concludes by a discussion of the complementary rôles of logically and linguistically oriented syntax.