Peter Gärdenfors & Massimo Warglien Semantics as a meeting of minds

What is a semantics?

Extensional semantics

Intensional semantics

Situation semantics Language Truth (partial) Situation World

Language Mental structure association World Action 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 Mental structures (different for different individuals) association World Actionassociation Semantics as the meeting of minds Language 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 C i Similarity provides a metric structure to each C i Additional assumptions about C i : convexity and compactness If C i are compact and convex, so is C=  C i 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 This space is discrete, but combinatorial 1 2 CC L

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 InterrogativeDeclarative EvaluativeInformativeGoal-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” DomainRegion FruitValues for skin, flesh and seed type ColorRed-green-yellow TasteValues for sweetness, sourness etc Shape"Round" region of shape space NutritionValues 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 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