# Nabla: A Linguistic System Based on Type Theory

## Jørgen Villadsen

### Monograph

More than three decades ago Richard Montague presented a formal description
of the syntax and semantics of a fragment of English that included
verbs, nouns, negation, coordination and quantification as well as
propositional attitudes like assertion, knowledge and belief.
From an artificial intelligence point of view the propositional attitudes
are of particular interest, since agents act on the basis of their beliefs
(and other attitudes) and constantly try to determine the attitudes of
other agents.
But beliefs can be false and are not necessarily closed under logical
consequence; furthermore intensionality is required since (extensional)
equivalent expressions can not be substituted for each other in beliefs.

Montague grammar uses a collection of so-called syntactic rules
and translation rules to define a relation between natural language
expressions of different categories and terms of an intensional logic.
The intensional logic is a special higher order logic with a typed
lambda calculus that makes a treatment of the propositional attitudes
possible.
Based on recent work on generalized categorial grammar and type logic we
present a linguistic system called Nabla which uses lexical and logical
combinators in a linguistic theory that covers about the same fragment
as Montague grammar, but without any rules and without the intensional logic.
We use a multi-dimensional type theory as model theory and a sequent calculus
as proof theory.
A paraconsistent logic is used in the analyses of the propositional attitudes,
but classical logic is available as a special case.

The book starts from classical Montague grammar but improves the
syntactical part of this approach by following modern "lexicalist"
trends and, most importantly, enriches the type system underlying the
semantics by several innovations. By this it meets the current on-going
debate about extensions of type theory recently discussed in computer
science, logic, and linguistics.

*Prof. Dr. Klaus Robering*

Sponsored by a Danish Research Council for the Humanities Grant

Foundations of Communication and Cognition (New Series)

LIT Verlag Münster-Hamburg-Berlin-Wien-London

Volume 3 - 224 pages - 2010

*
Jørgen Villadsen
2014-09-20
www2.compute.dtu.dk/~jv/nabla
*