Peter Gärdenfors

1. Peter Gärdenfors Why must language be vague?

2. 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

3. 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

4. We communicate about our inner worlds • Required for colloboration about non-present goals • Requires coordination of absent referents

5. Semanticsas the meeting of minds Mental structures (different for different individuals) Action Meeting of minds Action

6. 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

7. 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

8. The color spindle Brightness Yellow Green Intensity Red Blue Hue

9. Why convexity? • Handles fuzzy concepts • Makes learning more efficient • Connects to prototype theory

10. Voronoi tessellation from prototypes Cognitive economy: Once the space is given, you need only remember the prototypes – the borders can be calculated

11. 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

12. Convex tessellation in a computer simulation of a language game

13. 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

14. 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.

15. 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!

16. 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”

17. Language preserving neighbourhoods This space is discrete, but combinatorial L C C 2 1

18. Language does not preserve neighbourhoods perfectly

19. Why do we use vague terms when we refer? • Why can’t everything have a name? • Memory limitations ” … words are only names for Things … ”

20. 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

21. Hierarchy of categories • Rosch’s theory of basic, subordinate and superordinate levels • Several criteria for identifying the basic level • Based on cognitive economy

22. 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

23. Experimental coordination games • PP Kraus and Glucksberg 1977 Pechman 1984 “Looks like a motor from a motorboat. It has a thing hanging down with two teeth” ”The bird” ”The black bird” ”The black one”

24. The pragmatics of vagueness • ”Better safe than sorry” • Does not fit directly with maximizing expected utility • Politeness and diplomacy • Doctors’ reports • Politicians’ promises

25. 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

26. Products of regions

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

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

29. The embedded skin color space

30. The mechanism of metaphor ”We have had a bumpy relationship” Problem level Time

31. 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

32. Peter Gärdenfors Why must language be vague?