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**1 - 1**of**1**### A Lambek Calculus with Dependent Types

"... In this note, we discuss how to introduce dependent types into the Lambek calculus [7] (or, in general, an ordered calculus [17] and extensions such as those found in the studies of cate-gorial grammars [12, 13]). One of the motivations to introduce dependent types in syntactical analysis is to faci ..."

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In this note, we discuss how to introduce dependent types into the Lambek calculus [7] (or, in general, an ordered calculus [17] and extensions such as those found in the studies of cate-gorial grammars [12, 13]). One of the motivations to introduce dependent types in syntactical analysis is to facilitate a closer correspondence between syntax and semantics, especially when modern type theories are used in formal semantics [14, 9, 10, 2]. We then hope to establish a uniform basis for NL analysis: from automated syntactical analysis to logical reasoning in proof assistants based on type theories and formal semantics. Dependent types have been used in various contexts in computational linguistics (see, for example, [14, 3, 11] among others). In [15] that formalises the Lambek calculus in (a proof assistant that implements) Martin-Lof's type theory, Ranta discussed the idea of introducing type dependency into directed types and gave an inspiring example to represent directed types of quantiers, although the paper did not study a formal treatment of such an extension. De Groote et al [4] studied how to extend the underlying type system for ACGs by (intuitionistic) dependent product types so that type families can be used to represent syntactic categories indexed by linguistic features.