Peter Gärdenfors&
Massimo Warglien
Semantics as a meeting of minds
What is a semantics?
Extensional semantics
TruthLanguage World
Intensional semantics
Truth
Language
Possible worlds
Situation semantics
Pola ri ty
Language
W orld
Si tua tion
Language
Truth (partial)
Situation
World
Language
Conc ep tua l s truc tu re
Mean ingW orld
Language
Mental structure
association WorldAction
Meaning
Cognitive semantics
”Meanings ain’t in the head”
Putnam:Suppose you are like me and cannot tell an elm from a beech tree. We still say that the extension of 'elm' in my idiolect is the same as the extension of 'elm' in anyone else's, viz., the set of all elm trees, and that the set of all beech trees is the extension of 'beech' in both of our idiolects. Thus 'elm' in my idiolect has a different extension from 'beech' in your idiolect (as it should). Is it really credible that this difference in extension is brought about by some difference in our concepts? My concept of an elm tree is exactly the same as my concept of a beech tree (I blush to confess). (This shows that the identification of meaning 'in the sense of intension' with concept cannot be correct, by the way). ... Cut the pie any way you like, meanings just ain't in the head!
Sharing mental representations results in an emergent semantics• Image schemas in cognitive semantics provide a clue to
the mental structures• But, if everybody has their own mental space, how can
we then talk about a representation being the meaning of an expression?
• Semantics is also a product of communication – meanings arise as a result of communicative interactions
• Sharing of meaning puts constraints on individual meanings
• Socio-cognitive approach
Language
Conc eptual s truc ture
MeaningW orld
Language
Mental structures (different for different individuals)
association
World
Actionassociation
Semanticsas the meeting of minds
Language
Conc eptual s truc ture
MeaningW orld
Language
Meaning
Meaning
Action
World
Meeting of minds
Fixpoint semantics• ”Meeting of minds” ≈ reaching agreement on a contract• A semantics is a function that maps communicative expressions
on mental states (conceptual spaces), and vice versa • Minds meet when the representation-interpretation function
mapping states of mind on states of mind via gestures or language finds a resting point – a fixpoint (or an approximation of it)
• Related to equilibria in communication games• Topological and geometric properties of mental states help
generating fixpoints in communication activities • Same mechanisms in speaking and pointing
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
Why convexity?
• Handles fuzzy concepts• Connects to prototype theory• Makes learning more efficient• Makes it possible for minds to meet via
communication• Just as wheels are round to make
transport smooth, concepts are convex to make communication efficient
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
The mathematical 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
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.
Pointing
Imperative Interrogative Declarative
Evaluative Informative Goal-directed
Steps in the development of pointing• Grasping
• Imperative pointing
• Interrogative pointing
• Declarative pointing
• Deixis at Phantasma (Bühler)
• Analysis in terms of expanding conceptual space (product spaces)
• Visual space + emotional space + goal space + category space
• Involves several forms of intersubjectivity
Emotional space
Goal space
• Locations in visual space transformed into goal space
• Extended by metaphorical mappings to more abstract goal spaces
• Cf General Problem Solver
Category space
• Domains for properties of objects
• Size, shape, weight, color, taste …
• Properties are convex regions of domains
• Categories are sets of properties (+ correlations)
An example of a category: ”Apple”
Domain Region
Fruit Values for skin, flesh and seed type
Color Red-green-yellow
Taste Values for sweetness, sourness etc
Shape "Round" region of shape space
Nutrition Values for sugar, vitamin C, fibres etc
Pragmatics of pointing• Grasping• Imperative pointing• Evaluative pointing
• Informative pointing
• Goal-directed pointing
• Deixis at Phantasma
• Possession of object• Help to obtain object• Vicarious learning
about value of object
• Vicarious learning about object
• Helping attendant to achieve goal
• Visual support for linguistic communication
Imperative pointing• Grasping is moving Object to Subject
• S can move to O in other ways
• S can get O to S by imperative pointing
• Attendant is used as an instrument
• No joint attention
• No intersubjectivity in the pointer, but the attendant must understand the desire of the pointer
• Need not involve intentional communication
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
Evaluative pointing
• The pointer does not desire O but desires an evaluation of it
• Goal is also to achieve joint emotion• Attendant reacts emotionally and
pointer can assign emotional coordinates to O
• Involves meeting of minds in emotional space (in addition to visual space)
Informative pointing
• Pointer wants to achieve information about O
• Goal is to achieve joint attention• Attendant must understand the
informative goal of pointer, e.g. by linguistic description of O
• Involves meeting of minds also in category space
• Scaffolds language learning
Goal-directed pointing
• Pointer helps attendant locate a goal object O
• Joint attention is achieved• Pointer must understand goal of attender• Involves meeting of minds in goal space
(in addition to visual space)• Joint intention is achieved
Product spaces used in pointing• Grasping• Imperative pointing• Evaluative
declarative pointing• Informative pointing
• Goal-directed
declarative pointing• (Detached language)• Deixis at phantasma
• Visual space• Visual space• Visual space x
emotional space• Visual space x
category space• Visual space x goal
space• Category space• Category space x
visual space
Peter Gärdenfors&
Massimo Warglien
Semantics as a meeting of minds
Compositionality• Linguistic (and other communicative) elements can be
composed to create new meanings• 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
• However, there are other modifier-head compositions requiring more elaborate mappings