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 nonpresent goals
• Requires coordination of absent referents
Semantics
as the meeting of minds
Mental structures (different for different individuals)
Action
Meeting of minds
Action
Lang uage
M e a n in g
W
Co n c e p t u a l s t r u c t u r e
o r ld
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
Brightness
Yellow
Green
Intensity
Blue
Red
Hue
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: CC
• 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:CC, 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
C1
L
This space
is discrete,
but combinatorial
C2
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 superordinate 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
Pechman 1984
”The bird”
”The black bird”
”The black one”
Kraus and Glucksberg 1977
“Looks like a motor from a
motorboat. It has a thing hanging
down with two teeth”
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
Hot bath water is not a subcategory
of ”hot water”
hot
bath water
hot
tap water
0
30
x
60
degrees Celcius
The effect of contrast classes
red: of the colour of fresh blood, rubies,
human lips, the tongue, maple leaves in the
autumn, pos t-office pillar boxe s in Gt. Brit.
Advanced Learne r'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”
Problem
level
Time
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?