Peter Gärdenfors & Massimo Warglien Semantics as meeting of minds: A fixpoint approach based on conceptual spaces

Extensional semantics

Intensional semantics

Situation semantics

Mental structure Action Cognitive semantics

Who determines the meanings of words? "When I use a word," Humpty Dumpty said, in rather a scornful tone, "it means just what I choose it to mean – neither more nor less." "The question is," said Alice, "whether you can make words mean so many different things." "The question is," said Humpty Dumpty, "which is to be master – that's all." Lewis Carroll: Through the Looking-Glass, 1871.

”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 – vague meanings arise as a result of communicative interactions Sharing of meaning puts constraints on individual meanings Socio-cognitive approach

Mental structures (different for different individuals) association Action Semantics as the meeting of minds Action Meeting of minds

Meanings are in the heads Meanings emerge through the interaction between the members of a linguistic communicty. Language is a game with speech act moves where we try to coordinate our meanings. A semantics for a language is ideally an equilibirum point for the coordination game. Linguistic power structures determine how meanings are settled (cf. Putnam’s ”division of linguistic labor”). Topological and geometric properties of mental states help generating fixpoints in communication activities La parole est moitié à celui qui parle, moitié à celui qui écoute - Michel de Montaigne

Declarative pointing as a meeting of minds The “signaller” points to an object or spatial location and at the same time checks that the “receiver” focuses his or her attention on the same object or location The receiver in turn must check that the signaller notices that the receiver attends to the right entity Joint attention is achieved (can be described as a fixpoint)

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 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

Convex tessellation in a computer simulation of a language game Illustrates how a continuous function mapping the agents meaning space upon itself is compatible with the discreteness of the sign system.

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!

Language preserving neighbourhoods This space is discrete, but combinatorial 1 2

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 does not preserve neighbourhoods perfectly

Relaxing the continuity assumptions simplicial approximation decompose the meaning space in simplexes (convex, compact sets); map the vertexes of the decomposition on corresponding vertexes “fill” the rest by linear composition of the vertexes

Compositionality Linguistic (and other communicative) elements can be composed to create new meanings Cognitive economy: We express ourselves in a sufficiently precise way by combinations of a finite set of vague concepts 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

Concepts are sensitive to context Hot bath water is not a subcategory of ”hot water”

The effect of contrast classes Red book Red wine Red hair Red skin Red snapper Redwood

The embedded skin color space

The mechanism of metaphor The peak of a mountainThe peak of a career

Meanings are in the heads Meanings emerge through the interaction between the members of a linguistic community Language is a game with speech act moves where we try to coordinate (or negotiate) our meanings A semantics for a language is ideally a fixpoint for the game Reality enters via the payoffs of communication. If meaning is not aligned with reality, then the communicators will suffer costs

Peter Gärdenfors & Massimo Warglien Semantics as meeting of minds: A fixpoint approach based on conceptual spaces

The fixpoint theorem in one dimension

Relaxing the continuity assumptions Multivalued maps: If a point maps on multiple points, a fixpoint is defined as x*  F(x*). Theorem (Kakutani 1941): Let M  R n be a compact convex set. Let F: M  M be an upper- hemi-continuous convex valued correspondence. Then there is some x*  M such that x*  F(x*).

Communication games Stalnaker (1979):“One may think a nondefective conversation as a game where the common context set is the playing field and the moves are either attempts to reduce the size of the set in certain ways or rejections of such moves by others. The participants have a common interest in reducing the size of the set, but their interest may diverge when it comes to the question of how it should be reduced. The overall point of the game will of course depend on what kind of conversation it is – for example, whether it is an exchange of information, an argument, or a briefing” If the conversation is successful, the result is a fixpoint