Type-theoretical grammar
Syntactic Categories in the Language of Mathematics
TYPES '94 Selected papers from the International Workshop on Types for Proofs and Programs
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The aim of this study is to look at the the syntactic calculus of Bar-Hillel and Lambek, including semanticinterpretation, from the point of view of constructive type theory. The syntactic calculus is given a formalization that makes it possible to implement it in a type-theoretical proof editor. Such an implementation combines formal syntax and formal semantics, and makes the type-theoretical tools of automatic and interactive reasoning available in grammar.In the formalization, the use of the dependent types of constructive type theory is essential. Dependent types are already needed in the semantics of ordinary Lambekcalculus. But they also suggest some natural extensions of the calculus,which are applied to the treatment of morphosyntactic dependencies and to an analysis of selectional restrictions. Finally, directed dependent function types are introduced, corresponding to the Π types ofconstructive type theory.Two alternative formalizations are given: one using syntax trees, like Montague grammar, and one dispensing with them, like thetheory called minimalistic by Morrill. The syntax tree approach ispresented as the main alternative, because it makes it possible to embed the calculus in a more extensive Montague-style grammar.