Journal of Logic, Language and Information

, Volume 7, Issue 4, pp 413-431

First online:

Syntactic Calculus with Dependent Types

  • Aarne RantaAffiliated withDepartment of Philosophy, University of Helsinki

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access


The aim of this study is to look at the the syntactic calculus of Bar-Hillel and Lambek, including semantic interpretation, 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 Lambek calculus. 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 of constructive type theory.

Two alternative formalizations are given: one using syntax trees, like Montague grammar, and one dispensing with them, like the theory called minimalistic by Morrill. The syntax tree approach is presented as the main alternative, because it makes it possible to embed the calculus in a more extensive Montague-style grammar.

Constructive type theory Lambek calculus proof editors