Philosophy Lexicon of Arguments

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Fixed point: a point that satisfies the equation f (x) = x is a fixed point i.e. x is mapped to itself. S.A. Kripke based his alternative theory of truth from 1975 on fixed points in order to resolve the problem of paradoxes when dealing with self-reference. (Kripke, S., 1975. Outline of a Theory of Truth, The Journal of Philosophy, 72 690-716.). See also self-reference, paradoxes, liar paradox, truth theory.

Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments.

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Peter Gärdenfors

I 97
Fixed point/communication/Gärdenfors: a fixed point in a communication is reached when, for example, a person sees that the other person looks in the direction of the object they are referring to.
I 99
There should also be a consistency between the mental representations for the consistency of word meanings. Communication is also possible without this: e.g. children often have fewer domains in the representation of their terms or the domains are differently weighted.
Equilibrium: Communication can work restrictedly before the equilibrium of the partners (the same level of information) is reached.
I 100
Signal game/Jäger/Rooij/Gärdenfors: (Jäger & van Rooij, 2007)(1): randomly selected color samples are ordered by a second person. The goal of the game is to achieve an equal division of the color space in regions. (Nash-equilibrium or fixed point).
Gärdenfors: thesis: this is achieved if the conceptual spaces are convex and compact.
I 101
Equilibrium/Fixed point/Gärdenfors: further experiments have shown that repeated interactions lead to a stable communication system. (E.g. Hurford, 1999,(2) Kirby, 1999,(3) Steels, 1999,(4) Kaplan, 2000,(5) Steels & Belpaeme, 2005 (6)).
I 102
Meanings: do not necessarily have to change when the composition of the communicators involved changes or new parties join or disappear.
Fixed point/Dewey/Gärdenfors: (Dewey 1929, p. 178) (7): in order for V to understand A's moves, he must react to the thing from A's standpoint of view. So not I-centered and vice versa. Thus, something is literally made into a common.
I 104
Fixed point theorem/Gärdenfors: in order to achieve fixed points, it is not necessary for the conceptual spaces of the participants to be identical, nor that they divide the spaces equally.
I 105
We assume that the rooms are convex and compact. The following theorem from Warglien & Gärdenfors (2013)(8) is a consequence of Brouwer's fixed point theorem (Brouwer 1910)(9):
Theorem: every semantic reaction function, which is a continuous mapping of a convex compact set on itself, has at least one fixed point.
That is, there will always be a fixed point representing a Meeting of Minds.
Conceptual spaces: that they are assumed to be convex makes the communication flowing and memory performance efficient.
I 106
Gärdenfors: I do not mean that convex spaces are a reliable representation of our world, but that, because they are effective, they will be widespread.
Fixed points: the fixed point approach allows to consider a variety of types of communication such as color determinations and negotiations. The fixed-point theorem guarantees that the consciousness of the participants together (> Meeting of Minds) but it does not show how the semantic reaction function emerges from the communicative interaction.
I 109
Fixed Point/Communication/Gärdenfors: how do we know if a fixed point (balance, agreement) has been reached?
I 110
If the listener believes to understand, this is not a guarantee for a meeting of minds.

(1) Jäger, G., & van Rooij, R. (2007). Language structure: Psychological and social constraints. Synthese, 159, 99–130.

(2) Hurford, J. (1999). The evolution of language and languages. In R. Dunbar, C. Knight, & C. Power (Eds.), The evolution of culture (pp. 173–193). Edinburgh: Edinburgh University Press.

(3) Kirby, S. (1999). Function, selection, and innateness: The emergence of language universals. Oxford: Oxford University Press.

(4) Steels, L. (1999). The talking heads experiment. Antwerp: Laboratorium.

(5) Kaplan, F. (2000). L’émergence d’un lexique dans une population d’agents autonomes. Paris: Laboratoire d’Informatique de Paris 6.

(6) Steels, L., & Belpaeme, T. (2005). Coordinating perceptually grounded categories through language: A case study for colour. Behavioral and Brain Sciences, 28, 469–489.

(7) Dewey, J. (1929). Experience and nature. New York: Dover.

(8) Gärdenfors, P., & Warglien, M. (2013). The development of semantic space for pointing and verbal communication. In J. Hudson, U. Magnusson, & C. Paradis (Eds.), Conceptual spaces and the construal of spatial meaning: Empirical evidence from human communication (pp. 29–42). Cambridge: Cambridge University Press.

(9) Brouwer, L. E. J. (1910). Über ein eindeutige, stetige Transformation von Flächen in sich. Mathematische Annalen, 69, 176–180.

Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution.

Gä I
P. Gärdenfors
The Geometry of Meaning Cambridge 2014

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Ed. Martin Schulz, access date 2018-04-21