Nabla: A Linguistic System Based on Type Theory

Jørgen Villadsen


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