Peter Gärdenfors

Why must language be vague?

Why must language be vague?• Philosophers since Leibniz have dreamt of a

precise language• Vagueness is a design feature of natural

language• Brief answer: Because of cognitive economy• Vagueness has been analysed in terms of the

utility of language in a game theoretic setting

What is language for?

• Signalling systems: About what is here and now

• Symbolic communication: About what is not present

• Hockett’s central criterion for language: displacement

We communicate about our inner worlds

• Required for colloboration about non-present goals

• Requires coordination of absent referents

Mental structures (different for different individuals)

Action

Semanticsas the meeting of minds

Language

Conceptual structure

Meaning W orld

Action

Meeting of minds

Joint attention as a meeting of minds

• The pointer indicates the direction of the focal object (this can by pointing or by gaze directing).

• The attendant looks at the angle of the pointer’s indicated direction.• The attendant follows the direction until his own gaze locates the first salient

object.• The pointer looks at the angle of the attendant’s indicated direction.• The pointer follows the direction until his own gaze locates the first salient

object and checks that it is the same objects as he has indicated.• Joint attention is achieved • Can be described as a fixpoint in

product of two visual spaces• Words point to regions of mental spaces

Conceptual spaces

• Consists of a number of quality dimensions (colour, size, shape, weight, position …)

• Dimensions have topological or geometric structures

• Concepts are represented as convex regions of conceptual spaces

The color spindle

Intensity

Hue

Brightness

Green

Red

Yellow

Blue

Why convexity?

• Handles fuzzy concepts• Makes learning more efficient• Connects to prototype theory

Voronoi tessellation from prototypes

Cognitive economy: Once the space is given, you need only remember the prototypes – the borders can be calculated

Modelling the evolution of colour concepts

• Communication game studied by Jäger and van Rooij• Signaller and receiver have a common space for colours

(compact and convex)• Signaller can choose between n messages

Convex tessellation in a computer simulation of a language game

Modelling the evolution of colour concepts

• Communication game studied by Jäger and van Rooij• Signaller and receiver have a common space for colours

(compact and convex)• Signaller can choose between n messages• Signaller and receiver are rewarded for maximizing the

similarity of the colours represented• There exists a Nash equilibrium of the game that is a

Voronoi tessellation

Voronoi tessellation as a fixpoint

Illustrates how a continuous function mapping the agents meaning space upon itself is compatible with the discreteness of the sign system.

The model• States of mind of agents are points x in the product space

of their individual mental representations Ci

• Similarity provides a metric structure to each Ci

• Additional assumptions about Ci: convexity and compactness

• If Ci are compact and convex, so is C=Ci

• An interpretation function f: CC• It is assumed that f is continuous• “Close enough” is “similar enough”. Hence continuity of f

means that language can preserve similarity relations!

The central fixpoint result• Given a map f:CC, a fixpoint is a point

x* C such that f(x*) = x*• Theorem (Brouwer 1910): Every

continuous map of a convex compact set on itself has at least one fixpoint

• Semantic interpretation: If individual meaning representations are “well-shaped” and language is plastic enough to preserve the spatial structure of concepts, there will be at least one equilibrium point representing a “meeting of minds”

Language preserving neighbourhoods

This spaceis discrete, but combinatorial

1 2C CL

Language does not preserve neighbourhoods perfectly

Why do we use vague terms when we refer?

• Why can’t everything have a name?

• Memory limitations

” … words are only names for Things … ”

What has names?

• People (often not unique)• and some domestic animals• Places• regions, towns, villages, streets,

some prominent buildings (mainly part of local language)

• place names are often vague • Some events: New Year, WW2,

9/11

Hierarchy of categories• Rosch’s theory of basic, subordinate

and superordinate levels• Several criteria for identifying the basic

level• Based on cognitive economy

Why is the basic level special?

• Most informative for shared properties• Most informative for shared interactions with

objects• Response times• Priming: When primed with the super-ordinate

category, subjects are faster in identifying if two words are the same

• When asked to name a few exemplars, the more prototypical items come up more frequently

Experimental coordination games

• PP

“Looks like a motor from a motorboat. It has a thing hanging down with two teeth”

”The bird””The black bird””The black one”

Pechman 1984Kraus and Glucksberg 1977

The pragmatics of vagueness

• ”Better safe than sorry”• Does not fit directly with maximizing

expected utility• Politeness and diplomacy• Doctors’ reports• Politicians’ promises

Compositionality• Linguistic (and other communicative) elements can be

composed to create new meanings• Modelled by composition of continuous functions• Products of convex and compact sets are again convex

and compact• Products and compositions of continuous functions are

again continuous• So to a large extent compositionality comes for free• Simple example: the meaning of “blue rectangle” is

defined as the region which is the Cartesian product of the “blue” region of color space and the “rectangle” region of shape space

Products of regions

Concepts are sensitive to context

degrees Celcius0 30

tap water

60

bath waterhot

hot

x

Hot bath water is not a subcategory of ”hot water”

The effect of contrast classes

red: of the colour of fresh blood, rubies,human lips, the tongue, maple leaves in theautumn, post-office pillar boxes in Gt. Brit.Advanced Learner's Dictionary of Current English.

• Red book• Red wine• Red hair• Red skin• Red snapper• Redwood

The embedded skin color

space

The mechanism of metaphor

”We have had a bumpy relationship”

Time

Problem level

Why must language be vague?

• Language is finite because of evoutionary pressures on production, comprehension and memory

• The meaning of an expression is a product of the common ground of the speakers and the context

• Meanings can be made sufficiently precise by composition

Peter Gärdenfors

Why must language be vague?