Gärdenfors’s work on conceptual spaces hinges on a principle for natural concepts, which he has dubbed Criterion P in (2004, Minds and Matter), among other places.
Criterion P: A natural concept is a convex region of conceptual space.
He mentions a conjecture from Berlin and Kay (1969) that, although different languages divide the color circle in different ways, all divisions seem to divide basic color terms into convex regions. So, if point x in the color wheel is counted as red and point y is counted as red, then all points of the color wheel on the line segment between x and y are counted as red, too. On the other hand, the term ‘grue’, Gärdenfors argues, is artificial because it does not correspond to a convex region in the conceptual space of color. (Indeed, it fails to be so by design.) Thus, ‘grue’ does not denote a natural concept by Criterion P. Thus, the projectability of ‘blue’ and ‘green’ but non-projectability of ‘grue’ and ‘bleen’, on Gärdenfors proposal, can be explained by the non-logical Criterion P for natural concepts.
Questions: (1) What is the status of Berlin and Kay’s conjecture? Is convexity an accepted feature of basic color concepts? (2) Assuming that theirs is a reasonable account of color concepts, do you think Gärdenfors’s strategy to extend this account to natural concepts in general is persuasive?
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Hi Greg: I think that Gärdenfors’ proposal is very interesting. His book offers one of the best accounts of cognitive semantics in the psychological and philosophical literature. One of the problems of the account is how to select the relevant dimensions. For example, in the case of color, should one work with three dimensions necessarily? There are accounts of color concepts that only use two dimensions. And why a color spindle? Why not a cylinder for example?
The idea of using convexity to give an account of natural concepts has some pedigree in psychology, although Gärdenfors’ account seems more sophisticated mathematically and conceptually than the predecessors.
In any case, the main problem seems to be how to provide a sensible account of the following question: where cognitive dimensions come from? Familiar philosophical issues arise here: conventionalism, evolutionary arguments, etc. None of these options seems theoretically satisfying, but I still think that Gärdenfors’ proposal is, as usual, suggestive and fruitful.
Under a semantical point of view the approach offers a good account of lexical semantics but it is very poor when the problem is interpreting entire sentences or pieces of discourse. It seems that it would be desirable to complement Gärdenfors’ proposal with a richer account of discourse processing (along the lines of DRT for example).
So far Berlin and Kay’s conjecture seems to be confirmed in all the existing accounts of color (even when the account is in terms of two dimensions).
Hi Horacio – I also like his proposal very much. I take your point that he is primarily interested in lexical meaning and to give an account of similarity in terms of (some) distance measure within a conceptual space. (I forgot to mention for readers: A conceptual space is defined by a class of quality dimensions,
. The rough idea is that concepts are ‘regions’ of this 
-dimensional space. One of Horacio’s points, then, is that the proposal depends upon the selection of the 
‘s, since these are the parameters of your conceptual space.)
I am interested in pressing a bit on two fronts, both concerning concept similarity. The first concerns lexical meaning of open compound nouns. The second concerns the notion of similarity itself.
Open compound nouns: Consider the noun phrase
(1) Water meter adjustment screw
I assume that (1) can be constructed recursively, which entails that the concept for (1) is some region of conceptual space that is the intersection of the concepts [water], [meter], [adjustment], and [screw]. Now consider,
(2) Gas meter adjustment screw
(3) Water meter adjustment screwdriver
I take it that (1) and (2) are more similar than (1) and (3). But one would think that the concepts [screw] and [screwdriver] would be more similar than [(flammable) gas] and [(liquid) water] would be. Granted, neither (1), (2), nor (3) are natural concepts. But they are familiar concepts and, unlike ‘grue’, I don’t think the question hangs on Criterion P.
Similarity: Similarity in this framework is symmetric: Iff concept A is similar to concept B, then concept B is similar to concept A. But psychological judgments of similarity seems to fail symmetry. Children, when asked to compare dogs and cats, will report that cats are similar to dogs, but deny that dogs are similar to cats. Irrational? On the contrary: it appears that children are responding to the different variations in the populations. Domesticated dogs are a varied lot: big and small; slobbering and neat; fickle and friendly. Domesticated cats, by comparison, are less so. So, with respect to dogs, a cat is similar: a cat could fit into that population. But with respect to the population of cats, no dog would fit. So, the second point is that psychological similarity is antisymmetric but similarity within Gärdenfors’s framework is symmetric.
Hi Greg: The points you raise are legitimate and interesting. I do not have the book with me here in NYC so my answer relies on memory and common sense. Regarding the first point I think that the intuition that (1) is more similar to (2) than to (3) is based on the fact that (1) and (2) are two types of screws, while (3) is a kind of screwdriver, which is an entirely different thing. If I am remembering correctly I think that Gärdenfors would analyze (1) and (2) in such a way that the concept under analysis is `screw’ and the previous words modify it. So, for (1) and (2) we get two regions of conceptual space that correspond to `screw’. The regions are probably adjacent and overlap. In the third case we get the modification of an entirely different area of conceptual space focusing on `screwdriver’. For example, brick red and red brick would pick up different areas of conceptual space. The first is a kind of color and the second is a type of brick. If we just take the intersections we can lose valuable information. I think that Gärdenfors discusses these issues when he talks about contrast classes.
The second point is crucial of course. Tversky raised various objections to the geometrical account of concepts, including your point. He proposed an alternative account where identity, symmetry and the triangular inequality fail. Gärdenfors analyzes this rival account and finds it problematic (this is the account in terms of feature similarity). In addition he proposes a way of qualifying his account that seems to deal with the usual counterexamples (by adding saliency parameters in his definition of similarity). So, North Coreas is similar to Cuba if we privilege the political dimension. And Cuba is similar to Jamaica if we privilege the geographical dimension. So, we block that North Corea is similar to Jamaica because we obtained the previous similarities by using different saliency parameters. All things considered is not a bad response, especially considering that Tversky’s model seems to be more problematic.
Hi Horacio – The reason I picked a compound noun rather than a more typical adjective-noun construction was because, if you were to diagram (1), (2), or (3), all the words would be on the same line rather than playing different grammatical roles as ‘brick’ and ‘red’ do in ‘brick red’ versus ‘red brick’. The pileup of nouns in (1), (2), and (3) are intended to be inseparable, even though we know that the first two compound nouns talk about screws and the last talks about a tool.
Since a conceptual space S is convex, S is a type of Moore collection; therefore, S is closed under intersection. So, the idea is that since open compound nouns are “flat”, the construction must be (I conjecture) through intersections of the constituent noun concepts. If you buy that much, then the rest of the example just tries to wrench the geometry in enough different directions to make it hard to realize the semantic similarity relations we would intuitively judge to hold.
I should get Gärdenfors book to see how he deals with the symmetry question! It is neat stuff!
Yes, you are right, but I think that Gärdenfors considers similar cases as well in the book. Take for example `stone lion’. The pileup of nouns is also inseparable here, with the additional feature that the resulting compound does not refer to a kind of lion but a decorative artifact. Obviously one cannot take the intersection here (it would be empty). In these cases Gärdenfors proposes an operation of revision of the concept `lion’ rather than an intersection. Perhaps one can use the same strategy with your examples. We know little about revision of concepts. The topic seems completely open.
Nice example! Consider too ‘paper tiger’, where here the empty intersection is the point rather than a problem. Better still: ‘lead balloon’. Speaking of lead balloons….