Mathematical Structures in Natural Language

Edward L. Keenan
Section: Language and Logic
Level: Advanced


The course focuses on characterizing certain properties of natural language that we can represent in a mathematically enlightening way. Indeed in several cases the properties were not noticed until they were formulated in a mathematical way. Below is an outline of the topics to be covered together with some readings on those topics. A course reader with the indicated articles will be provided. It will contain in addition an introduction to the mathematics used. The introductory material will be assumed in the lectures and not covered explicitly. The lectures are lighter than the readings and are intended as guides enabling the readings to be more accessible.

Course Outline

Lectures 1 & 2: Linguistic applications of generalized quantifier theory.

We characterize various classes of Noun Phrases linguists have concerned themselves with. E.g. Which NPs occur naturally in Existential There contexts? (There are X in the garden: ok for X = exactly five cats, no student's cats, more cats than dogs, ...; not ok for X = most cats, the ten cats, seven out of ten cats). Which NPs occur naturally in the post 'of' position of partitives? (two of X: ok for X = the cats, these cats, John's cats; not ok for X = no cats, most cats, more cats than dogs). Which NPs in subject position license negative polarity items (X have ever been to Pinsk: ok for X = no students here, fewer than ten students here, at most ten students here; not ok for X = all students, most students, more than ten students).

We provide a mathematical classification of the major classes of Determiners that occur in natural language and motivate one universal denotational constraint.


  1. The Semantics of Determiners. 1996. E.L. Keenan. in The Handbook of Contemporary Semantic Theory. S. Lappin (ed)
  2. Generalized Quantifiers in Linguistics and Logic. 1997. E.L. Keenan & D. Westerstahl. in The Handbook of Language and Logic J. van Benthem and A. ter Meulen (eds) Elsevier.

Lecture 3: Beyond the Frege Boundary

We show that English presents some types of quantified NPs which are properly polyadic: they bind more than one argument position of predicates and are provably not representable by iterated (Fregean) application of quantifiers of lesser arity. Core examples are ones like Different students answered different questions on the exam.


Further Beyond the Frege Boundary. 1996. E.L. Keenan. in Quuantifiers, Logic, and Language J. van der Does and J. van Eijck (eds). CSLI

Lecture 4: Sortal Reducibility

We show that natural languages present a range of quantifiers that make essential use of the restriction on the domain of quantification imposed by the noun the quantifier combines with. Classical existential and universal quantifiers, and their intersective and co-intersective generalizations, do not. E.g. Some As are Bs is logically equivalent to Some Entities are both As and Bs, where we quantify over all the objects in the universe of the model, compensating with a new predicate consisting of a boolean compound of the original predicate and the noun property. Similarly All As are Bs is logically equivalent to All Entities are either non-As or Bs. We exhibit simple quantifiers (e.g. most) in English which provably have no such logical paraphrase.


  1. Natural Language, Sortal Reducibility and Generalized Quantifiers. E.L. Keenan. in the Journal of Symbolic Logic 58: No. 1:314-325.
  2. Quantification in English is Inherently Sortal. in The History of Philosophy and Logic 20:251-265.

Lecture 5: Logical Objects and Grammatical Invariants

We present a characterization of logical constants and grammatical constants, which we generalize by increasing type to logical and grammatical invariants. The latter notion is, to my knowledge, the only intrinsic characterization of the expressions, properties, relations, ... determined by a grammar. Semantically we show how to distinguish in terms of denotations the "logical" Determiners like the ten, all, all but ten, infinitely many, ... from "non-logical" ones like John's ten, every...but John (as in John's ten cats are black; Every student but John left early).


  1. Syntactic Invariants. E.L. Keenan and E.P. Stabler. to appear with CSLI.
  2. Logical Objects. E.L. Keenan. to appear in Logic, Language, and Computation: Essays in Honor of Alonzo Church. C.A. Anderson and M. Zeleny (eds). Kluwer


Edward L. Keenan
Dept. of Linguistics, UCLA
ph: (310) 825-0634
fax: (310) 206 5743