On Pregroups, Freedom, and (Virtual) Conceptual Necessity

Pregroups were introduced in (Lambek, 1999), and provide a foundation for a particularly simple syntactic calculus. Buszkowski (2001) showed that free pregroup grammars generate exactly the -free context-free languages. Here we characterize the class of languages generable by all pregroups, which will be shown to be the entire class of recursively enumerable languages. To show this result, we rely on the well-known representation of recursively enumerable languages as the homomorphic image of the intersection of two context-free languages (Ginsburg et al., 1967). We define an operation of cross-product over grammars (so-called because of its behaviour on the types), and show that the cross-product of any two free-pregroup grammars generates exactly the intersection of their respective languages. The representation theorem applies once we show that allowing ‘empty categories’ (i.e. lexical items without overt phonological content) allows us to mimic the effects of any string homomorphism.