Polymorphism in English Logical Grammar

The quantifiers and connectives of English present difficulties to standard (Church-style) type theory in linguistics in that they can apply systematically to objects of different form. This paper sketches solutions to this problem with some extensions to the simply typed lambda calculus, including Damas-Milner types and System F or the polymorphic lambda calculus. The goal is to maintain the “proofs-as-readings” spirit of the logical grammar tradition while allowing flexible types for the logical terms. 1 Background and Outline Type theory in linguistics grew out of the categorial investigations of BarHillel [1], Lambek [15, 16, 17] and Curry [5]. Later, Montague’s PTQ semantics [18] utilized a simple Church-style type system to accompany the simple applicative categorial grammar used to define the fragment of English under investigation. This unification and the rediscovery of Lambek’s work led van Bentham [26, 27] and many others to develop the tradition of type-logical grammar. In this tradition (substructural) fragments of propositional intuitionistic logic are used to characterize both natural language (nl) syntax and as the type-theory accompanying lambda terms used to describe nl denotations. The easily defined (injective) functors from the associative syntactic calculus (often Lambek’s L) to the type theory (often Lambek’s associative syntactic calculus with permutations, LP, or the full positive propositional intuitionistic calculus J[→]) became the generalized format of Montague’s rule-to-rule notion of compositionality. Also, implicitly or explicitly, type-logical grammar was inspired by the Curry-Howard-de Bruijn isomorphism between typed lambda terms and formulae of intuitionistic logic—the tradition characterized by the slogan “formulae as types, proofs as terms”. Applied in the setting of nl semantics, and the above theories of LPand J[→], the isomorphism is relaxed to a correspondence [19, p.116]. Within the Montagovian tradition, Partee and Rooth [20] addressed the problem that the boolean connectives “and” and “or” in English induce the same basic intuition or reading irrespective of the types of arguments, e.g. Serge and Jane sing ≈ Serge sings and Jane sings (1) Serge sings and smokes ≈ Serge sings and Serge smokes (2)