Introduction to Vector Space Semantics

Yoad Winter and Joost Zwarts
Section: Language and Logic
Level: Introductory


Vector Space Semantics (VSS) is a branch of model-theoretic semantics that aims to apply logical approaches to meaning to the domain of spatial expressions by adopting vectors as the basic elements of spatial ontology. In this course we will review current work in VSS, giving sufficient introductory background on model-theoretic tools and previous work on spatial expressions.

We will show how meanings of various types of spatial expressions can be described uniformly using an ontology of space that employs vectors. This use of vectors reveals that certain basic semantic properties of natural language like conservativity, monotonicity, and denotational constraints on grammaticality, which are familiar from Generalized Quantifier Theory, also hold in the spatial domain.

Part 1 (1 lecture) - formal and linguistic preliminaries on model-theoretic semantics and prepositions.

Part 2 (2.5 lectures) - motivation of VSS and applications to locative prepositions.

Part 3 (1.5 lectures) - recent extensions of VSS to gradeable adjectives and other expressions for size, orientation and shape. Most of the material on VSS will be made available electronically.

Course Outline

Part 1 - Background

In this part we will briefly review some relevant topics from model-theoretic semantics and generalized quantifier theory. Three main issues will be discussed:

  1. Mathematical properties of natural language determiners that correspond to simple inferences with sentences (e.g. Monotonicity, Intersectivity).
  2. Properties that restrict the possible denotations of natural language determiners (e.g. Conservativity, Continuity).
  3. Relations between the denotation of the determiner and its syntactic distribution (e.g. negative polarity items and monotonicity, existential 'there' sentences and intersectivity).


Keenan, E. (1996). The semantics of determiners, in S. Lappin (ed.),
    The Handbook of Contemporary Semantic Theory, Blackwell.
Keenan E. and D. Westerstahl (1996), Generalized Quantifiers in
    Linguistics and  Logic, in J. van Benthem and A. ter Meulen (eds.),
    Handbook of Logic and Language, Elsevier, Amsterdam.

Part 2

Motivation for VSS and application to locative prepositions. This part will consist of four sections:


Jackendoff R. and B. Landau (1991), Spatial language and spatial cognition,
    in D. J. Napoli and J. A. Kegl (eds.), Bridges between Psychology
    and Linguistics, Lawrence Erlbaum Associates, Hillsdale, NJ.
O'Keefe J. (1996), The spatial prepositions in English, vector
    grammar, and the cognitive map theory. In P. Bloom et al. (eds.),
    Language and Space. MIT Press, Cambridge, Mass.
Zwarts J. (1997). Vectors as relative positions: a compositional
    semantics of modified PPs. Journal of Semantics 14:57-86.
Zwarts, J. and Y. Winter (2000), Vector Space Semantics: a model-theoretic
    analysis of locative prepositions. Journal of Logic, Language and
    Information 9:169-211.

Part 3

Recent extensions of VSS In this part we will review recent extensions of VSS to the following domains:

  1. vectors as axes in the description of adjectives
  2. modification of gradeable adjectives and comparatives
  3. parallels between spatial and temporal domains, and between space terms and kinship terms (e.g. 'father', 'uncle' etc.)


Faller, M. (2000). Dimensional adjectives and measure phrases in vector space
    semantics. In M. Faller et al. (eds.), Formalizing the Dynamics of
    Information. CSLI Publications, Stanford.
Winter, Y. (2000). Measure phrase modification in vector space semantics.
    In preparation.
Zwarts, J. (2000). Vectors across spatial domains: from place to size,
    orientation, shape and parts (1). Unpublished ms.


The course is planned as a terse but self-contained introduction. Some sophistication in elementary set theory and logic, as well as familiarity with the basic objectives and methodologies of linguistic research will be presupposed, but not much specific background in formal semantics of natural language or linear algebra will be needed in order to follow the discussion.



Yoad Winter
Computer Science
Technion - Israel Institute of Technology
Haifa 32000, Israel

Joost Zwarts
P.O.Box 6645
Eldoret, Kenya