  This article contains a proposal to add coinduction to the computational
apparatus of natural language understanding. This, we argue, will provide a
basis for more realistic, computationally sound, and scalable models of natural
language dialogue, syntax and semantics. Given that the bottom up, inductively
constructed, semantic and syntactic structures are brittle, and seemingly
incapable of adequately representing the meaning of longer sentences or
realistic dialogues, natural language understanding is in need of a new
foundation. Coinduction, which uses top down constraints, has been successfully
used in the design of operating systems and programming languages. Moreover,
implicitly it has been present in text mining, machine translation, and in some
attempts to model intensionality and modalities, which provides evidence that
it works. This article shows high level formalizations of some of such uses.
  Since coinduction and induction can coexist, they can provide a common
language and a conceptual model for research in natural language understanding.
In particular, such an opportunity seems to be emerging in research on
compositionality. This article shows several examples of the joint appearance
of induction and coinduction in natural language processing. We argue that the
known individual limitations of induction and coinduction can be overcome in
empirical settings by a combination of the the two methods. We see an open
problem in providing a theory of their joint use.
