Higher-Order DisCoCat (Peirce-Lambek-Montague semantics)
Alexis Toumi Giovanni de Felice
We propose a new definition of higher-order DisCoCat (categorical
compositional distributional) models where the meaning of a word is not a
diagram, but a diagram-valued higher-order function. Our models can be seen as
a variant of Montague semantics based on a lambda calculus where the primitives
act on string diagrams rather than logical formulae. As a special case, we show
how to translate from the Lambek calculus into Peirce's system beta for
first-order logic. This allows us to give a purely diagrammatic treatment of
higher-order and non-linear processes in natural language semantics: adverbs,
prepositions, negation and quantifiers. The theoretical definition presented in
this article comes with a proof-of-concept implementation in DisCoPy, the
Python library for string diagrams.