Igor Douven
email: igor.douven@univ-paris1.fr

The Role of Naturalness in Concept Learning: A Computational Study *

Keywords: concepts, design, learning, naturalness, optimality, similarity spaces, simulations

This paper studies the learnability of natural concepts in the context of the conceptual spaces framework. Previous work proposed that natural concepts are represented by the cells of optimally partitioned similarity spaces, where optimality was defined in terms of a number of constraints. Among these is the constraint that optimally partitioned similarity spaces result in easily learnable concepts. While there is evidence that systems of concepts generally regarded as natural satisfy a number of the proposed optimality constraints, the connection between naturalness and learnability has been less well studied. To fill this gap, we conduct a computational study employing two standard models of concept learning. Applying these models to the learning of color concepts, we examine whether natural color concepts are more readily learned than nonnatural ones. Our findings warrant a positive answer to this question for both models employed, thus lending empirical support to the notion that learnability is a distinctive characteristic of natural concepts.

Introduction

In the conceptual spaces framework, as developed by [START_REF] Gärdenfors | Conceptual spaces: The geometry of thought[END_REF][START_REF] Gärdenfors | The geometry of meaning: Semantics based on conceptual spaces[END_REF] and others, concepts can be represented geometrically, specifically as regions, or sets of regions, in so-called similarity spaces. For example, the concept of redness can be identified with a certain region in perceptual color space (see below), and the concept of tartness, with a certain region in taste space (e.g., [START_REF] Churchland | Plato's camera[END_REF]. However, not every region in a similarity space can represent a concept, at least not a natural one, where natural concepts are concepts like blue, or green, or tiger, or gold, which carve nature at its joints (in the words of Plato's Phaedrus) and which have or can have a place in our thinking and theorizing. The question then arises what distinguishes those regions that do or can represent natural concepts from those that cannot. Gärdenfors (2000, p. 71 ff ) suggests that natural concepts are represented by convex regions in a similarity space,1 where a region is convex precisely if for any two points lying in the region, every point between them lies in the region as well. Although both empirical evidence and considerations of cognitive economy support this idea, it is clear that convexity alone is not enough to define naturalness. For instance, it is easy to pick a convex region in perceptual color space that groups together color shades which we would not naturally associate.

In response to this problem, [START_REF] Douven | What are natural concepts? A design perspective[END_REF] argue that natural concepts are ones that are represented by the cells of an optimally partitioned similarity space. As they show, this proposal encompasses, but goes beyond, the aforementioned convexity criterion. At the core of their proposal is a set of constraints that a partitioning must satisfy to be considered optimal. While, as [START_REF] Douven | What are natural concepts? A design perspective[END_REF] show, there is already support-both empirical and from computational studiesfor most of the constraints they propose, this cannot be said for what they call Learnability, according to which an optimal partitioning creates concepts that are easy to learn and allow for reliable generalization from a few samples.

Previous studies have shown that people generally learn new concepts quickly. However, most of this research focuses on the role of prototypes in concept learning. It does not specifically address whether learnability is a distinctive characteristic of natural concepts, given that much of it concerns artificial concepts-like ones about triangular patterns of dots distorted to varying degrees [START_REF] Posner | Perceived distance and the classification of distorted patterns[END_REF][START_REF] Posner | On the genesis of abstract ideas[END_REF][START_REF] Posner | Retention of abstract ideas[END_REF] or about artificial faces [START_REF] Reed | Pattern recognition and categorization[END_REF]-or involves both natural and artificial concepts [START_REF] Rosch | Family resemblances: Studies in the internal structure of categories[END_REF], without however examining whether naturalness differentially affects learnability.

We present evidence for Learnability in the form of the outcomes of a computational study using two distinct models of concept learning. We use these models to simulate the learning of color concepts, with an eye toward investigating whether natural color concepts (i.e., the mental correlates of basic color terms like "blue," "red," "green," etc.; see [START_REF] Berlin | Basic color terms[END_REF] are learned more readily than systems with equal numbers of nonnatural color concepts. It will be seen that, given either model, the answer to this question is positive. Before we delve into our study, we outline the theoretical background that informs our work.

Theoretical background

We start by stating the basic tenets of the conceptual spaces framework, then summarize recent work on the problem of how to define naturalness of concepts in that framework, and lastly describe the two methods of concept learning our study will rely on.

From similarities to concepts

At the core of the conceptual spaces framework is the thought that concepts can be represented as regions in similarity spaces, where a similarity space is a one-or multi-dimensional metric space whose dimensions represent measurable qualities that objects can have to varying degrees. Objects are mapped onto points in such spaces depending on the degree to which they possess these qualities. In principle, there is a great choice of metrics, but the ones most commonly encountered in practice are the Manhattan and the Euclidean metric, which, given an n-dimensional space S, are the instances of the schema below with p = 1 and p = 2, respectively:

δ S (x, y) = ( n ∑ i=1 | x i -y i | p ) 1/p ,
where x = ⟨x 1 , . . . , x n ⟩ and y = ⟨y 1 , . . . , y n ⟩. The measure in which two objects representable in S are similar to each other, in the respect corresponding to S (e.g., similar in taste, if S is taste space), is then some inverse function of the distance between them in S, meaning that they are the more similar in the said respect the closer together they are represented within S. We find a great variety of similarity spaces discussed in the cognitive science literature, including the already mentioned taste space, auditory spaces [START_REF] Petitot | Morphodynamics and the categorical perception of phonological units[END_REF], olfactory space [START_REF] Castro | Categorical dimensions of human odor descriptor space revealed by non-negative matrix factorization[END_REF], various shape spaces, such as a space for shells (Gärdenfors, 2000, pp. 142-150), for human faces [START_REF] Valentine | Face-space: A unifying concept in face recognition research[END_REF], and for container objects [START_REF] Douven | Vagueness, graded membership, and conceptual spaces[END_REF], various animal spaces (e.g., [START_REF] Henley | A psychological study of the semantics of animal terms[END_REF], as well as various action spaces (e.g., [START_REF] Gärdenfors | Using concept spaces to model actions and events[END_REF], moral spaces (e.g., [START_REF] Peterson | The ethics of technology: A geometric analysis of five moral principles[END_REF][START_REF] Verheyen | Can we use conceptual spaces to model moral principles?[END_REF], and social spaces (e.g., [START_REF] Deauvieau | Les catégorisations ordinaires de l'espace social français[END_REF][START_REF] Bendifallah | Conceptual spaces for conceptual engineering? Feminism as a case study[END_REF]. Perhaps the best known similarity space, and certainly the one most easily accessible for experimentation, is perceptual color space. Because it is also the space to be used in our study, we will describe this in more detail.

There are in effect two perceptual color spaces, CIELUV space and CIELAB space, where the former is assumed to best represent similarity judgments concerning colors perceived on screen while the latter is assumed to best represent such judgments when colors are shown on cloth or paper (Malacara, 2002, pp. 86-90;Fairchild, 2013, Ch. 10). The spaces look very similar: both are three-dimensional, spindle-like Euclidean spaces. The left panel of Figure 1 visualizes CIELUV space by placing in it all 1,625 chips from what is commonly known as "the Munsell book of colors" [START_REF] Munsell | A color notation: An illustrated system defining all colors and their relations[END_REF]. 2 The right panel of this figure highlights the 320 chromatic Munsell chips that have been widely used in color naming studies, most famously those reported in [START_REF] Berlin | Basic color terms[END_REF] and the World Color Survey [START_REF] Cook | The World Color Survey database: History and use[END_REF]. These chips will also serve as the main materials in our computational study.

There are different ways to obtain a conceptual space from a similarity space, but the currently dominant approach combines prototype theory with the mathematical technique of Voronoi tessellations [START_REF] Gärdenfors | Conceptual spaces: The geometry of thought[END_REF][START_REF] Gärdenfors | The geometry of meaning: Semantics based on conceptual spaces[END_REF]. Locating the points representing prototypes in a given space S, we can let them generate a Voronoi tessellation of S by associating each point in S with the prototype or prototypes closest to it. This will result in a partitioning of S, the cells of which can be considered to rep-resent concepts, while the points which are equally close to two or more prototypes form the concept boundaries.

We briefly mention also a different approach to representing concepts in similarity spaces, which is advocated by [START_REF] Nosofsky | Attention, similarity, and the identification-categorization relationship[END_REF][START_REF] Nosofsky | Attention and learning processes in the identification and categorization of integral stimuli[END_REF]. In his Generalized Context Model, concepts are conceived not as regions of points but as sets of exemplars, or individual instances of a concept, represented in a similarity space. Our focus will be on Gärdenfors' model, which offers a richer mathematical structure, thereby facilitating the formalization of more complex cognitive operations and relations. 3 Nosofsky's GCM is still relevant to our study, inasmuch as it may provide an important complementary viewpoint concerning concept learning, specifically, by framing this as a comparative process of assimilating new items based on their similarity to items stored in memory, in a way to be detailed shortly.

Naturalness and optimality

The study we are to report is directly related to the question of what distinguishes natural from nonnatural concepts. What, for instance, distinguishes the natural color concepts from ones obtained by picking a number of points in CIELUV space randomly and letting those generate a Voronoi tessellation of that space? Given that any Voronoi tessellation of a Euclidean space partitions that space into convex regions (Okabe et al., 2000, p. 58), the convexity criterion is not going to help answering this question. As mentioned in the introduction, [START_REF] Douven | What are natural concepts? A design perspective[END_REF] propose to analyze the notion of a natural concept in terms of optimally partitioned similarity spaces. More exactly, natural concepts are those that are represented by a cell in an optimally partitioned similarity space.

In doing so, these authors took their cue from work in cognitive science relating categorization to principles of rationality and optimality (e.g., [START_REF] Rosch | Principles of categorization[END_REF][START_REF] Marr | Vision: A computational investigation into the human representation and processing of visual information[END_REF][START_REF] Anderson | The adaptive character of thought[END_REF][START_REF] Anderson | Is human cognition adaptive?[END_REF][START_REF] Oaksford | A rational analysis of the selection task as optimal data selection[END_REF][START_REF] Chater | Ten years of the rational analysis of cognition[END_REF][START_REF] Goodman | A rational analysis of rulebased concept learning[END_REF][START_REF] Frank | Predicting pragmatic reasoning in language games[END_REF][START_REF] Griffiths | Rational use of cognitive resources: Levels of analysis between the computational and the algorithmic[END_REF] as well as from recent work in biology, which explains the occurrence of certain biological traits and processes by reference to what they would look like had they been designed according to principles of good engineering; for instance, [START_REF] Alon | Biological networks: The tinkerer as an engineer[END_REF] argues that many biological networks look as if they had been constructed by someone who obeyed the same principles of modularity, use of recurring circuit elements, and robustness of component tolerances that guide the design of engineered networks.

The main part of [START_REF] Douven | What are natural concepts? A design perspective[END_REF] proposal consists of a set of design principles that a good engineer would want to follow if tasked with designing one or more conceptual spaces for creatures with our perceptual and cognitive make-up. More exactly, these authors formulate a number of design principles which they believe to define optimal design for representational systems generally, regardless of which format one prefers for concepts (so regardless of whether one commits to the conceptual spaces framework). But then their main concern is to show what these principles imply for the design of conceptual spaces, specifically, for when such spaces are optimally designed-optimally designed for creatures whose memories have limited storage capacity, whose discriminatory capacities are limited as well, and who are to survive in a world with scarce commodities where competition for those commodities can be fierce.

One of the principles is what Douven and Gärdenfors call Informativeness, according to which a partitioning should offer fine distinctions across the similarity space. But the authors also stress the need for Parsimony, cautioning against partitioning a space so finely that it overloads memory. Two further principles-Contrast and Representation-have to do with the placement of prototypes in the space.

According to the former, the partitioning should allow a placement of the prototypes such that they can easily be told apart from one another (i.e., prototypes of different concepts should be at a sufficiently large distance from each other in the space). According to the latter, the placement should at the same time be such that the prototypes are good representatives of the items falling under the concept they are the prototype of (i.e., they should not be too distant from any of the items falling under the concept). The final principle on Douven and Gärdenfors' list is Learnability, which, as already mentioned in the introduction, demands that the concepts resulting from the partitioning should be easy to learn, in particular, that concept learners should be able to safely (even if not infallibly) generalize to unlabeled instances from a relatively small number of labeled instances.

Douven and Gärdenfors explicitly leave open the possibilities that their list of principles is incomplete as well as that it contains redundancies. They further recognize that different ones of their principles can pull in different directions. Therefore, they define an optimal partitioning of a similarity space to be one that strikes the or a best balance between the different desiderata.

In their paper, Douven and Gärdenfors muster a range of experimental and computational results reported in the scientific literature which support various of their principles. For example, Informativeness and Parsimony receive strong support from [START_REF] Regier | Color naming reflects optimal partitions of color space[END_REF]. Following an idea put forward by [START_REF] Jameson | It's not really red, green, yellow, blue: An inquiry into perceptual color space[END_REF], these authors apply a computational clustering algorithm implementing principles that, at bottom, amount to Informativeness and Parsimony to the Munsell chips used in the WCS, finding that the algorithm produces clusterings that closely match how those same chips are carved up into categories by various languages spoken across the globe.4 Douven (2019) provided evidence specifically for Contrast and Representation by first experimentally determining the locations in CIELUV space of the basic color prototypes and then showing via simulations that the constellation of those prototypes is a near-to-optimal trade-off between the two designated constraints. As said, however, evidence supporting Learnability is still sparse, which motivated the present paper.

Concept learning

Laboratory studies involving human participants might seem ideal to examine whether natural concepts are acquired more readily than nonnatural ones. However, such studies are likely to be confounded by the fact that the participants are already familiar with the natural concepts. Whichever difficulties they might have acquiring nonnatural concepts, at least as compared to natural concepts, could be due to the fact that the concepts to be acquired (as part of the experimental task) clash with the ones the participants have been using for most of their lives. A way around this might be to conduct an experiment in which toddlers in the earliest stages of language development are trained on nonnatural concepts (say, nonnatural color concepts) prior to their having acquired the natural ones. But it would probably be difficult to get such an experiment approved by the ethics committee of one's university.

At any rate, in this paper we take a computational rather than an experimental approach to investigating concept learning. More exactly, we consider two algorithms that are suited for modeling concept learning, supposing concepts are represented as regions in similarity spaces. One is a computational implementation of Gärdenfors' (2001) Approximate Prototype Model (APM), which is his account of concept learning set within the conceptual spaces framework. The other is the k-Nearest Neighbors (KNN) algorithm, which is a popular machine learning algorithm and which, as Nosofsky (2011, p. 22) notes, is closely connected, or even identical (depending on the exact functional relationship between similarity and distance), to his GCM.

According to the APM, we learn a family of concepts (e.g., color concepts) by going through the following steps:

1. we are shown examples of each of the concepts in the family, where these examples are labeled, meaning that it is given, for each, under which concept it falls; 2. we estimate the prototype of each concept by averaging the examples of the concept, meaning that, where these examples are given as points in an n-dimensional similarity space, we first group the examples according to their labels and then calculate, for each group and each dimension i (1 ⩽ i ⩽ n), the average i-th coordinate of the members of the group (this yields the group's so-called center of gravity in the space); 3. we use these estimates to generate a Voronoi tessellation of the relevant similarity space; 4. when new labeled examples come in, we repeat steps 2 and 3.

The hope is that, if all goes well, the Voronoi tessellation generated by our best guesses of prototypes will converge toward the actual conceptual structure, as we come to learn more and more labeled examples. How accurately you will be able to classify unlabeled examples at a later point in time will depend on how close, at that point in time, the best-guess structure is to the actual one, and is a measure of the extent to which you can be said to possess the concepts represented in the relevant space. Figure 2 illustrates the process in the abstract.

The GCM-based model of learning is simpler. In this model, you learn a family of concepts by storing in memory more and more exemplars (i.e., labeled examples) of each of the concepts. Upon encountering an unlabeled example, you find the exemplars most similar to it, and you label it on the basis of those exemplars' labels. This leaves some details to be filled in. For instance, if all the most similar exemplars have the same label, then the decision how to label the new example will be straightforwardbut what if they have different labels? And how many most-similar exemplars are we to consider in order to determine the label of the new example?

To start with the latter question, the k-Nearest Neighbors algorithm can be regarded as implementing the GCM-based model of concept learning, and as its name suggests, this algorithm considers the k most similar exemplars. It allows the number k to be set by the user, though a common recommendation is to set k equal to the integer nearest to the square root of the total number of exemplars. With k fixed, the k most similar exemplars vote on the label to be given to the new example, where this vote can be weighted depending on the distances in the relevant similarity space between the exemplars and the new example. For instance, suppose k = 5 and three of the exemplars most similar to a new example are labeled l 1 while two are labeled l 2 . Then the unweighted majority vote would yield l 1 as the label for the new example. If, however, we take weights into account, this could be different. For example, if the l 2 -exemplars are much closer to the new example than any of the l 1 -exemplars, then the new example could end up being labeled l 2 . Most packages providing KNN allow users to set a great variety of weighting functions. 5The two learning methods to be used in our study have in common that they are both forms of supervised learning, meaning that they both require labeled data to learn from. But they are different in the sense that the APM is a form of what in machine learning is called greedy learning, while KNN is a form of lazy learning. That is to say that, in learning, the former creates a map-in our case a conceptual structure-and then labels new examples according to their place on the map. KNN, on the other hand, runs one and the same routine each time a new example is received in order to determine the label of that example, without ever creating a map. Of course, if we want, we can let it create a map by approximation, by asking for a fine grid of points in a given space how the algorithm would classify them. To show how this would work, we can use the same space and sample of examples that was used to illustrate the APM and let KNN predict the label for each point in a grid of 500 × 500 points, spaced uniformly in each dimension. The result is shown in Figure 3. Just glancing at the results from the two illustrations, it would seem that KNN offers a somewhat better approximation of the real conceptual structure than the APM does. Needless to say, though, this is a toy example of a space and so we should be cautious to infer anything in general from the relative performance of the two learning methods.

The main focus of this paper is on the question of how to distinguish natural from nonnatural concepts, not on the different learning methods (though we will briefly compare their performance in our study). It is still worth commenting on what could appear as problematic aspects of the methodsproblematic, at least, from the standpoint of the conceptual spaces framework.

The first concerns the APM and is illustrated in Figure 4. As can be seen in that figure, some of the examples that underlie the estimate of the conceptual structure end up, not in the concept they were supposed to be an example of, but in a neighboring one. The problem was already mentioned, though only in passing, in Gärdenfors (2001, p. 176 n), where it is speculated that these problem cases might actually give rise to the introduction of new concepts. A different response, which we want to propose here, points to the fact that the issue of vagueness is typically "idealized away" in presentations of the conceptual spaces framework. The framework can model vagueness, as was shown in [START_REF] Douven | Vagueness: A conceptual spaces approach[END_REF], [START_REF] Decock | What is graded membership?[END_REF], [START_REF] Douven | Measuring graded membership: The case of color[END_REF], [START_REF] Verheyen | Typicality and graded membership in dimensional adjectives[END_REF], and Douven (2020), but "fuzzifying" concept boundaries in a realistic manner can be computationally costly. A quick and dirty way of representing vagueness uses the notion of silhouette width, as developed by [START_REF] Kaufman | Finding groups in data[END_REF], which helps to distinguish between instances that are more central to a concept and those that are more peripheral to it. Given a family of concepts C = {C 1 , . . . , C k } representable in some similarity space S, with x an instance of concept C i , its silhouette width is defined to be

( b(x) -a(x) ) / max { a(x), b(x)
} , with a(x) the average distance in S of x to the other instances of C i and b(x) the smallest average distance in S of x to the instances falling under C j ∈ C, for j ≠ i. Sizing the point markers according to the silhouette width of the exemplars they represent, as is done in Figure 5, suggests that the problem cases might all turn out to be borderline cases, once vagueness is taken into account.

To see why this is relevant, note that a parent trying to teach a child the concept of redness would probably not want to pick red-orange, or red-pink, or red-blue borderline cases. That does not mean she will only pick examples that are typically red, as this might fail to give a good impression of the extension or width (in [START_REF] Carnap | A basic system of inductive logic, part II[END_REF], sense) of the concept. But she will either want to leave out borderline instances completely or explicitly single them out as being, for instance, reddish-orangish. Note also that, in other spaces, the boundary regions between concepts may not even have instances. For instance, mammal space, as represented in [START_REF] Henley | A psychological study of the semantics of animal terms[END_REF], can plausibly be thought of as having continuous dimensions, but for biological reasons, there may be gaps in this space. Thus, one diagnosis of the problem cases as highlighted in Figure 4 is that we have been assuming that examples are sampled uniformly from a space, which in reality may often be false. For pedagogical purposes, we may want to select examples that are rather central to a concept, or we may be bound to select such examples, because there are no borderline cases, for biological or more broadly scientific reasons.

The second problem concerns KNN, or rather, KNN as a method for concept learning, where concepts are understood as per the conceptual spaces program. After all, it was said that even if convexity is not a sufficient condition for naturalness, it is generally agreed among proponents of the framework that it is a necessary one. And from Figure 3 it is evident that if we learn concepts via KNN, we cannot expect them to be convex. Note, though, that the deviations from convexity might in their entirety fall within the boundary regions of the concepts and thus could, in a realistic setting, be nothing but artefacts of the idealizing assumption that concepts have sharp borders. This is particularly plausible in view of the results reported in Douven (2016), [START_REF] Douven | Measuring graded membership: The case of color[END_REF][START_REF] Jraissati | Delving deeper into color space[END_REF], which show for different real similarity spaces that boundary regions can have substantial volume, relative to the total volume of the space.

Computational study

As previously mentioned, perceptual color space is readily available to experimenters. CIELUV and CIELAB space can both be accessed via dedicated packages in a great number of popular computer languages, which is true for none of the other similarity spaces known from the psychological literature (Sect. 2.1). For our study, we have used the Colors.jl package for the scientific computing language Julia (Bezanson et al., 2017) to work with these spaces. The results to be reported in the following were obtained in CIELUV space, but interested readers are invited to use the code provided in the Supplementary Materials to verify that rerunning the procedures to be stated below in CIELAB space yields qualitatively identical outcomes. Given our focus on color space, the specific research question we aim to answer becomes this: Is the system of color concepts we use, which is widely regarded as a prime example of a system of natural concepts (e.g., [START_REF] Rosch | Natural categories[END_REF][START_REF] Kripke | Naming and necessity[END_REF][START_REF] Hardin | Color for philosophers: Unweaving the rainbow[END_REF], more easily (more quickly, more reliably) learnable than systems of color concepts that we would regard as nonnatural? Given a number of exemplars of all color concepts, can we more reliably generalize from those exemplars to the conceptual structure if that structure is the natural one (i.e., the carving-up of perceptual color space into natural concepts) than if it is a nonnatural structure? To further clarify, consider a paradigmatic situation of concept learning in which a parent teaches her child color names by showing the child shades of blue, shades of red, shades of green, and so on, while naming those shades. We can then ask how accurately the child will be able to predict the color names of shades it has not yet seen or in any case has not been taught to name. We can further ask whether, if our system of color concepts consisted of nonnatural concepts, the learning process would be slower in that the child would have to see more exemplars of each color to achieve a similar level of accuracy in naming so-far-unseen color shades. To repeat, we are not going to use human participants to test the hypothesis that the answer will turn out in favor of the natural color concepts, as Douven and Gärdenfors (2020) predict it will, but rather try to model the learning process as best we can by computational means.

The materials for the computational study consisted in the 320 chromatic chips from the WCS, which were already highlighted in the right panel of Figure 1 and which Figure 6 shows in a chart, in the way they are usually presented in publications related to the WCS. 6 The study is in two parts, each part centering on one of the learning methods described in the previous section. The procedure is the same in both parts and involves simulating color concept learning for 10,000 randomly generated systems of nonnatural color concepts and comparing the accuracy achieved in the learning process with that achieved in a simulation of color concept learning for the system of natural color concepts. Specifically, each part of our study consisted of the following steps:

1. Split up the materials into the natural color concepts and sample randomly from the chips in each concept, where the number sampled from each concept is greater than 0 but otherwise random (sampled uniformly from the number of chips in the given concept), thus obtaining a set of pairs

{⟨ ⟨L * c , u * c , v * c ⟩, C c ⟩} c ∈ s
representing the CIELUV coordinates of Munsell chip c in sample s as well as its label C c indicating the color concept under which it falls.7 2. Apply the learning method (APM or KNN) to the set of pairs and on that basis predict the labeling of the chips in the materials that were not sampled. 3. Measure the accuracy of the prediction-the best guess of what the system of concepts is-by comparing it with the actual system of natural color concepts. 4. Repeat the foregoing steps 50 times. 5. Run the same procedure for 10,000 systems of color concepts, each obtained by randomly picking 10 or 11 (see below) points in CIELUV space as prototypes and using these to generate a Voronoi tessellation of the space (so meaning that now the 320 chips in the materials are split up according to the random system). 6. Compare the accuracy scores obtained for the natural system with those obtained for the nonnatural systems.

For KNN, we followed the customary recommendation of setting k equal to the square root of the size of the sample, rounded to the nearest integer. Of course, given that the size of the sample was always random, k could have a different value in every simulation. Also, given a sample s of chips and a chip c not in the sample, the weight each of the k chips in s nearest to c had in voting on c's label was set equal to the inverse of their squared Euclidean distance (as measured in CIELUV space) to c. Both parts of the study were conducted twice over, once for 10 color concepts and once for 11. This was because some participants in color naming studies for English and French used all basic color terms in describing the colors of the 320 Munsell chips used for our study while other participants used only 10, leaving out "gray" (see [START_REF] Claidière | A colour sorting task reveals the limits of the universalist/relativist dichotomy: Colour categories can be both language specific and perceptual[END_REF]. Given that in the same color naming studies there was quite some interpersonal variability in how the chips were named, we used the kmeans clustering algorithm to obtain a kind of objective approximation of the natural color concepts (see [START_REF] Douven | Measuring graded membership: The case of color[END_REF]. The results are shown in the first two rows of Figure 7, while the bottom row of that figure shows an example of a randomly generated system of 10 color concepts. 8Accuracy was measured in terms of Normalized Mutual Information (NMI), which quantifies the mutual dependence of two variables. In our case, in which we are comparing clustering results, the two variables are two different clusterings, and the NMI quantifies how similar they are. That the measure is normalized means that its values are between 0 and 1, with 0 indicating no shared information (the clusterings are completely dissimilar) and 1 indicating perfect agreement (the clusterings are identical), making NMI values easy to interpret and compare. Another benefit of using the NMI measure for comparing clusterings is that it takes into account both the amount of shared information and the total information in each clustering, in contrast to most of the simpler measures (see, e.g., [START_REF] Pfitzer | Characterization and evaluation of similarity measures of pairs of clusterings[END_REF].

As mentioned above, we ran the study both for 10 and for 11 color concepts. As also explained, in each run, we sampled 50 times from each system of color concepts (the natural one plus the 10,000 randomly generated ones), the sample each time yielding a random number of examples from each concept in the system. And applying the learning method (APM in one part of the study, KNN in the other) gave us a clustering of the 320 chips from Figure 6, which could then be compared via the NMI measure with what the clustering was according to the system from which the sample was taken. Thus, for each system of concepts, we obtained 50 NMI scores per learning method, which we summarized via their means and standard deviations.

Learning via the APM method yielded an average NMI score of .92 (± 0.04; range = [.83, .97]) for the system of 10 natural color concepts (shown in the top row of Fig. 7) and also an average NMI score of .92 (± 0.03; range = [.84, .97]) for the system of 11 natural color concepts (middle row of Fig. 7). The same learning method achieved, on average (i.e., averaged over the 10,000 systems), an average (i.e., averaged over the 50 samplings per system) NMI score of .72 (± 0.04; range = [.53, .85]) for the nonnatural systems with 10 color concepts and an average NMI score of .73 (± 0.03; range = [.57, .84]) for the nonnatural systems with 11 color concepts. The left panel of Figure 8 shows a density plot of the average NMI scores for 10-concept nonnatural systems, with the average score for the natural system shown as a dashed line. The corresponding plot for 11-concept systems looks virtually identical and is not shown here.

For the KNN method, we registered an average NMI score of .84 (± 0.06; range = [.70, .93]) for the system of 10 natural color concepts and an average NMI score of .83 (± 0.06; range = [.72, .96]) for the system of 11 natural color concepts. The average score for the 10-concept nonnatural systems was .59 (± 0.03; range = [.47, .70]), on average, while for the 11-concept nonnatural systems it was .61 (± 0.03; range = [.48, .71]). The right panel of Figure 8 shows again the density plot of these scores for the 10concept systems, together with the average score for the system with 10 natural color concepts. Here, too, the corresponding 11-concept-systems plot is omitted, given that it looks almost the same.

For neither method need we run a statistical test to see that naturalness has a significant effect on learning: the accuracy with which the algorithms were able to predict under which concepts the chips fall that were not sampled was significantly better if the training sample came from the natural concepts than when it came from some system of nonnatural color concepts. While that is a positive for Learnability, it will be recalled from the previous section that [START_REF] Douven | What are natural concepts? A design perspective[END_REF] left open the possibility that their list of criteria contained redundancies. And in the same section, it was mentioned that [START_REF] Douven | Putting prototypes in place[END_REF] found support for the claim that the system of natural color concepts strikes a near to optimal balance between Representation and Contrast, which are meant to jointly guarantee that prototypes can be placed in a space such that they are (i) good representatives of the other items falling under the concept they are the prototype of and (ii) easily distinguishable from each other. It is thus legitimate to ask whether the fact that the algorithms learned color concepts most accurately when those concepts were the natural ones could be because those concepts satisfy criteria on Douven and Gärdenfors' (2020) list other than Learnability.

To address this question, specifically to examine whether Contrast and Representation might already predict learning accuracy, four linear regressions were conducted, one for each combination of number of concepts (10 or 11) and learning algorithm (APM or KNN). For this, we first calculated for each of the 10,000 nonnatural 10-concept systems as well as for each of the 10,000 nonnatural 11-concept systems their Contrast and Representation scores, following [START_REF] Douven | Putting prototypes in place[END_REF] proposal to operationalize Contrast as the sum of the Euclidean distances among all the generating points in CIELUV space and Representation as the sum of the distances of those from the center of gravity of the concept that they represent (where, in this case, the center of gravity is calculated by taking the average L * -, the average u * -, and the average v * -coordinate of those of the 1,625 Munsell chips that fall under the given concept). These Contrast and Representation scores then did duty as the predictors in the said regression analyses, all of which had average NMI score as dependent variable.

For the APM method, the β-coefficients for Contrast and Representation were, respectively, 0.13 (t = 14.80, p < .0001) and -0.50 (t = 58.57, p < .0001) for the 10-concept systems and 0.12 (t = 14.23, p < .0001) and -0.48 (t = 55.12, p < .0001) for the 11-concept systems. For the KNN method, the corresponding β-coefficients are -0.01 (t = 0.57, p = .5674) and -0.23 (t = 23.32, p < .0001) for the 10-concept systems and -0.02 (t = 1.61, p = .1066) and -0.21 (t = 21.14, p < .0001) for the 11-concept systems. Thus, for the APM method, Contrast and Representation were both highly significant predictors for accuracy and hence for the learnability of conceptual systems. Also note that they predicted accuracy in the way one would expect: an increase in Contrast, as measured by an increase of the sum of distances among the points generating the system, led, on average, to a significant increase of accuracy, while an increase in Representation, as measured by a decrease of the sum of distances of the chips falling under a concept to the center of gravity of that concept, led to a significant increase of accuracy as well. For the KNN method, we find that only Representation is a highly significant predictor of the accuracy achieved by the learning method; Contrast is not significant.

Our primary aim was to shed light on the relation between naturalness and learnability. However, the learning methods we implemented computationally are of independent interest, so it is worth briefly comparing their performance in the above study. In the toy example we used previously to illustrate the methods, one got the impression that the KNN method did somewhat better than the APM method. Comparing the average accuracy scores reported above, and also the two panels in Figure 8, gives a different impression. Indeed, in our study the APM method did significantly better than the KNN method, as was confirmed by running four t-tests on the results obtained by the two methods for each of the following: the 10-concept natural system, the 10-concept nonnatural systems, the 11concept natural system, and the 11-concept nonnatural systems. For the natural 10-concept system, we had t(98) = 7.67, p < .0001, with a value for Cohen's d of 1.51 (which counts as large); for the nonnatural 10-concept systems, the results were t(19998) = 266.84, p < .0001, Cohen's d = 3.77; for the natural 11-concept system, t(98) = 8.94, p < .0001, Cohen's d = 1.76; and finally, for the nonnatural 11-concept systems, we had t(19998) = 273.42, p < .0001, with Cohen's d = 3.87. [START_REF] Douven | What are natural concepts? A design perspective[END_REF] hypothesized that what distinguishes natural from nonnatural concepts is that the former, but not the latter, are represented by the cells of an optimally partitioned similarity space. They defined optimality in terms of a list of engineering constraints. In this paper, we focused on the constraint that optimally partitioned similarity spaces result in easily learnable concepts, a correlation for which empirical evidence was still missing, in contrast to most of the other constraints on the list. This motivated the study reported in this paper.

Conclusion

Our study implemented computationally two plausible models of concept learning, one proposed by Gärdenfors, the other closely connected to Nosofsky's Generalized Context Model. We applied these models to the learning of color concepts in order to examine whether natural color concepts are learned more readily than nonnatural ones. The outcomes of the study confirmed this hypothesis for both employed models, thus lending empirical support to the notion that learnability is a distinctive characteristic of natural concepts.

The results were still not entirely positive for Learnability. While learnability appears to be a feature of naturalness, the learnability of systems of natural concepts may, as far as the results from our study go, be accounted for already by the fact that these systems satisfy other criteria on Douven and Gärdenfors' (2020) list, in particular, Contrast and Representation. As mentioned, this would not necessarily be inconsistent with these authors' proposal, given that they explicitly left open the possibility that their list contains redundancies.

Our study had some clear limitations. First, it only involves computational models of learning, which will not fully reflect human learning. For reasons mentioned in Section 2.3, it is not straightforward to compare the learning of natural concepts with the learning of nonnatural concepts by human participants, given that the participants will already be familiar with the former but not with the latter. This is not to say that such experiments are impossible, and we hope that the present results, indicating a clear connection between learnability and naturalness, will inspire cognitive psychologists to come up with a paradigm that will allow the connection to be tested in humans in a way that is methodologically sound. A second limitation concerns the fact that our evidence is restricted to color concepts. In principle, it is easy to rerun the study using conceptual spaces other than color space. In practice, this is more difficult, because the number of conceptual spaces that can be downloaded or are otherwise available for running the kind of simulations we conducted is extremely limited, and we actually know of no conceptual space that is as well validated as CIELUV and CIELAB space. This situation may change, however, given that the conceptual spaces framework is an active area of research, and given also that researchers are now commonly making their data available for others to work with. Once more conceptual spaces are available for experimentation, it will be interesting to see not only whether the link between learnability and naturalness that we found to exist for color concepts generalizes, but also (if it does) whether that link can be more generally accounted for in terms of satisfaction of the Contrast and Representation criteria.9

Figure 1 :

 1 Figure 1: The 1,625 chips from the Munsell Book of Colors shown in CIELUV space (left panel); the 320 chromatic chips used for the World Color Survey highlighted (right panel).

Figure 2 :

 2 Figure2: Illustration of the Approximate Prototype Model of concept learning, using a two-dimensional "color" space, shown with prototypes (panel a); based on the sampling from each of the concepts (panel b), a best guess is made of the locations of the various prototypes in the space, to which corresponds a best guess of the conceptual structure, which is the Voronoi tessellation generated by the prototype estimates (panel c); this is compared with the actual space (panel d), the latter shown in dotted lines.

Figure 3 :

 3 Figure 3: Map created by means of the k-Nearest Neighbors algorithm on the basis of the same sample of exemplars used for the illustration in Figure 2.

Figure 4 :

 4 Figure 4: Best APM guess of conceptual structure and sample on which the guess is based: some sampled exemplars end up in a concept other than the one they exemplify.

Figure 5 :

 5 Figure 5: Markers sized according to the silhouette width of the exemplars.

Figure 6 :

 6 Figure 6: The 320 chromatic Munsell chips used for the World Color Survey.

Figure 7 :

 7 Figure 7: Clusterings of the 320 chromatic WCS chips into 10 and 11 natural color concepts (top and middle row) and clustering into 10 concepts according to randomly generated system of color concepts (bottom row; clusters have been colored by taking the center of mass of the CIELUV coordinates of the chips in the cluster).

Figure 8 :

 8 Figure8: Density plots of average NMI scores for 10-concept systems obtained for the APM method (left) and the KNN method (right), each panel also showing the average NMI score for the corresponding system of natural color concepts (dashed line).

Or by sets of convex regions, if we are dealing with a multi-domain concept. We take this qualification to be read from here on.

The RGB coordinates of the Munsell chips were downloaded from the website of the Munsell Color Science Laboratory of the Rochester Institute of Technology and converted to CIELUV coordinates using the Colors.jl package for the Julia language.

For instance, Gärdenfors' model has been used to derive prior probabilities for items falling under specific concepts[START_REF] Decock | A geometric principle of indifference[END_REF] and to formalize vagueness[START_REF] Douven | Vagueness: A conceptual spaces approach[END_REF][START_REF] Douven | Vagueness, graded membership, and conceptual spaces[END_REF][START_REF] Douven | Measuring graded membership: The case of color[END_REF] as well as analogical reasoning[START_REF] Douven | Conceptual spaces and the strength of similarity-based arguments[END_REF]. It is not clear to us how any of that could be accomplished using the GCM.

For similar results, see[START_REF] Kemp | Kinship categories across languages reflect general communicative principles[END_REF],[START_REF] Xu | Numeral systems across languages support efficient communication: From approximate numerosity to recursion[END_REF],[START_REF] Xu | Historical semantic chaining and efficient communication: The case of container names[END_REF],Zaslavsky et al. (2019), and[START_REF] Mollica | The forms and meanings of grammatical markers support efficient communication[END_REF].

In Nosofsky's GCM, there are parameters to model response bias, memory strength, and salience of dimensions, which help to model recency and context effects. For our computational implementation, these do not matter. For instance, the order in which data are stored in computer memory makes no difference to how easily they can be retrieved.

As an anonyous referee remarked, it would be interesting to conduct the simulations to be presented in the following also for the full set of 1,625 Munsell chips, especially given that the chips for the WCS were all selected precisely because they all show highly saturated colors, thus raising the question whether we would obtain the same results if also chips showing less saturated colors were included. A practical problem here is that we currently lack data on how people would carve up the full set of Munsell chips into the eleven basic color categories.[START_REF] Jraissati | Delving deeper into color space[END_REF] did use the full set in their study, but presented their participants with a free-naming task, meaning that they could use any name they liked for any of the chips they were shown.

Note that because a random number of chips was sampled randomly from each concept, there was no fixed sample size. It was empirically determined that the sample size was, on average, 165.02(± 31.98).

The nonnatural concepts have the color resulting from averaging the CIELUV coordinates of the chips that fall under the concept. Also, all color concepts that occurred in the study-both the natural ones and the nonnatural ones-were convex, by construction. That the charts shown in Figure6might suggest otherwise is simply because these charts are not meant to represent a perceptual color space.

I am grateful to Christopher von Bülow and to two anonymous referees for this journal for valuable comments on a previous version.

GitHub

repository which can be accessed via this link.


Igor Douven

Galina V Paramei

Optimality, Prototypes, and Bilingualism

There is an old debate about the status of basic color categories (such as blue, green, and red), the question being whether that status derives from the way the world is or whether it is culture-bound. In more scientific terminology, this amounts to the question of whether the categorical structure of color space (how it is carved up and where the color prototypes or foci are located) is fixed by the world or a matter of cultural conventions. Some recent work suggests that the categorical structure of color space is not a conventional matter without, however, being completely determined by the world; it is also subject to constraints deriving from the various ways in which our perceptual and cognitive capacities are limited. While there is recent evidence for this newer position, we report a study comparing Italian monolingual, English monolingual, and Italian-English bilingual speakers with regard to focal color choices in the blue region of color space suggesting that cultural and linguistic factors play a role in the categorical structuring of color space.

Introduction

Philosophers, psychologists, and linguists have long been debating the nature of natural kind terms. According to some theorists (called "realists"), these terms capture, or at least are meant to capture, the in-built structure of reality, while others ("nominalists") maintain that which terms we regard as picking out natural properties or concepts is essentially conventional, or at most motivated by pragmatic concerns. A less well-known and in a way middle position-sometimes called "conceptualism"-conceives of the world as imposing constraints on the terms we use to talk and think but maintains that, at the same time, there are constraints deriving from our particular cognitive makeup.

Recently, evidence for conceptualism has emerged, at least insofar as the position applies to the color domain, which has long been regarded as providing us with prime examples of natural kind terms ("blue," "green," "red," etc.). Most notably, [START_REF] Regier | Color naming reflects optimal partitions of color space[END_REF] offer evidence for [START_REF] Jameson | It's not really red, green, yellow, blue: An inquiry into perceptual color space[END_REF]also Jameson, 2005) so-called Interpoint-Distance Model, according to which color concepts (i.e., denotative meanings of color terms) stem from a combination of the irregularities to be found in perceptual color space (i.e., color similarity space, which is often taken to be CIELAB space; see Fairchild, 2013, and below) and a preference for informative naming systems, where this preference is perceptually and cognitively motivated, specifically, in terms of how our limited perceptual and cognitive capacities favor adding color categories, so that color differences between adjacent categories get maximized while color differences within the new contiguous categories are minimized. Regier and colleagues' (2007) work inspired [START_REF] Douven | What are natural concepts? A design perspective[END_REF] to propose that natural concepts (i.e., the meanings of natural kind terms) are given by the cells of optimally partitioned similarity spaces (such as color space when the kind terms concern color categories), where the notion of an optimal partition is defined by reference to principles of good engineering, that is, principles a good engineer would want to respect in designing a system of concepts for creatures with our perceptual and cognitive makeup. For instance, according to the principle Douven and Gärdenfors call "Representation," an optimal partition will allow prototypes [START_REF] Rosch | Natural categories[END_REF]-in the case of basic color categories referred to as "foci" or "focal colors"-to be placed such that each prototype is a good representative of all items falling under the concept. And according to the principle these authors call "Contrast," an optimal partition will allow prototypes of different concepts to be so chosen that they are easily distinguishable from each other. These and the other principles Douven and Gärdenfors propose are jointly meant to lead to conceptual systems that facilitate learning and memorization and help avoid classification errors. As [START_REF] Douven | What are natural concepts? A design perspective[END_REF] show, evidence that at least some of these principles have indeed been at work in shaping our concepts, in particular our color concepts, is already to be found in the literature (e.g., [START_REF] Jameson | Why GRUE? An Interpoint-Distance Model analysis of composite color categories[END_REF][START_REF] Kemp | Kinship categories across languages reflect general communicative principles[END_REF][START_REF] Xu | Numeral systems across languages support efficient communication: From approximate numerosity to recursion[END_REF][START_REF] Xu | Historical semantic chaining and efficient communication: The case of container names[END_REF]see Douven, 2019, specifically for evidence for Representation and Contrast in the context of color categorization).

But while there is evidence for a version of conceptualism, some of this same evidence also suggests that conceptualism may not be the whole story. For instance, Regier and co-authors (2007) show that their computational model of Jameson and D'Andrade's hypothesis does remarkably well in predicting color concepts in the languages with up to and including six basic color terms (BCTs; see below). As shown in [START_REF] Jraissati | Does optimal partitioning of color space account for universal color categorization?[END_REF], however, the same model does worse for languages with more than six BCTs, even much worse for languages with 11 BCTs, such as English (see also [START_REF] Douven | Clustering colors[END_REF]. Something similar holds for evidence documented in [START_REF] Douven | Putting prototypes in place[END_REF]. Douven looked at constellations of 11 possible color prototypes that do best, on balance, at satisfying Representation and Contrast. Since there is no unique best trade-off between these principles, the said constellations form what is known as a "Pareto front."1 And [START_REF] Douven | Putting prototypes in place[END_REF] found that, while the actual constellation of prototypes lies close to that front, it does not quite lie on the front.

To be sure, these discrepancies could just be a matter of not having discovered all principles of good engineering that define optimality in [START_REF] Douven | What are natural concepts? A design perspective[END_REF] proposal. For instance, the Pareto front in [START_REF] Douven | Putting prototypes in place[END_REF] corresponds to the best trade-off of two desiderata for optimality. And while the actual constellation of prototypes was not represented as a point on that front, it might well be on a front that results from taking into consideration further desiderata, perhaps ones yet to be discovered. Similarly, an extension of Regier and colleagues' (2007) computational model that also incorporates criteria beyond informativeness (where we imagine these further criteria to be equally motivated by reference to our perceptual and cognitive capacities) might be able to achieve the same good fit for languages with more than six BCTs that the extant model achieves for languages with up to and including six BCTs.

Another possibility, which is not necessarily inconsistent with the previous one, is the presence of more local or contingent effects on how we use color terms to carve up and furnish (by placing prototypes) color space, and more generally on how we use natural kind terms to carve up and furnish the relevant similarity spaces.2 Among such effects could be primacy effects (items encountered early on in the process of concept acquisition shape our concepts more than ones encountered later), recency effects (items encountered most recently have a relatively larger impact on the shape of our concepts), a combination of the two, and, crucially, cultural and linguistic effects, specifically, exposure to rival or partly rival conceptual systems as associated with other cultures or languages we happen to be acquainted with (see Ervin, 1961; Caskey-Sirmons & Hickerson, 1977). 3 In this paper, we look at the second possibility, in particular, at the possible effects of exposure to partly different conceptualizations of the color domain in bilingual Italian-English speakers. Specifically, we analyze a data set from a study with such speakers, as well as with monolingual English and monolingual Italian speakers to investigate the influence of immersion into a non-native language and cultural factors on the placement of prototypes for color terms as well as on concept extensions in color space. Still more specifically, our study focused on the blue region of color space, given that Italian speakers are known to require more than one BCT to name the blue colors. Thus, if cultural factors impact conceptualization, one might expect to find in Italian-English bilinguals some interaction between the different conceptualizations of the blue region associated with English and Italian, where this interaction might be revealed by comparing where in the blue region the bilingual speakers locate focal colors with where monolingual Italian and monolingual English speakers do. Before we present the study, we provide some background on the "Italian blues" in relation to the dominant view of BCTs.

2 Italian blues [START_REF] Berlin | Basic color terms[END_REF]Kay (1969/1991, p. 6) define a color term as basic if it is monolexemic, not included in any other BCTs, applied not only to a limited class of objects, and is psychologically salient for all informants. According to the Berlin and Kay model, languages with a developed color lexicon have maximally 11 BCTs. But the BCT upper limit tenet has been questioned in recent years (see [START_REF] Paramei | Language and psychology[END_REF]. In particular, the blue region of color space was demonstrated to require two BCTs in a number of languages, with the two terms differentiating light and dark(er) shades of blue. The classic examples are Russian sinij "dark blue" and goluboj "light blue," with the latter named by [START_REF] Berlin | Basic color terms[END_REF][START_REF] Berlin | Basic color terms[END_REF] as a potential 12 th BCT. Recent reviews of linguistic and psycholinguistic studies of the two "Russian blues" underscored Berlin and Kay's conjecture [START_REF] Paramei | Singing the Russian blues: An argument for culturally basic color terms[END_REF][START_REF] Paramei | Russian "blues": Controversies of basicness[END_REF].

The "blue challenge" encompasses also Italian. Earlier linguistic studies provided evidence of azzurro, blu, and celeste as salient terms in both spoken language [START_REF] Giacalone Ramat | Strutturazione della terminologia dei colori nei dialetti sardi. Italia linguistica nuova ed antica [A structure of color terminology in Sardinian dialects[END_REF] and written language [START_REF] Grossmann | Colori e lessico: Studi sulla struttura semantica degli aggettivi di colore in catalano, castigliano, italiano, romeno, latino ed ungherese [Colors and lexicon: Studies on semantic structure of color adjectives in Catalan, Castilian, Italian, Romanian, Latin and Hungarian[END_REF] across various Italian dialects and, thus, as BCT candidates.

Azzurro is deeply entrenched in Italian and has been attested already in the 9 th century, originally having denoted lapis lazuli [START_REF] Frison | Lapis lazuli, lazurite, ultramarine "blue," and the colour term "azure" up to the 13 th century[END_REF]. According to De Mauro (1983), initially azzurro belonged solely to the written language and was absent in Italian dialects; it entered the spoken language after the political unification of Italy during the Risorgimento (1815-1871), with school education having become affordable to the general population. Celeste originates from Latin caeruleus, derived from caelum "sky," and at the time did not have a color meaning. With a symbolic religious meaning, but also with a color sense, it is attested in the 13 th century, denoting light shades of blue and being common in Italian dialects [START_REF] Grossmann | Colori e lessico: Studi sulla struttura semantica degli aggettivi di colore in catalano, castigliano, italiano, romeno, latino ed ungherese [Colors and lexicon: Studies on semantic structure of color adjectives in Catalan, Castilian, Italian, Romanian, Latin and Hungarian[END_REF]. Blu was the last of the "blues" to enter Italian, at the end of the 17 th century and was deployed to lexicalize deep (dark/navy) blue, conceivably as a result of its use in the cloth trade (Pastoureau, 2001, p. 127). Nowadays it is considered to be the most widespread "blue" term in Italian [START_REF] Sandford | A cognitive linguistic usage perspective: What is Italian blu, azzurro, celeste? Do English speakers agree on blue semantics?[END_REF].

In addition to the linguistic studies addressed above, recent psycholinguistic studies carried out in different regions of Italy provide converging evidence that in Italian at least two BCTs are required for naming the blue region (e.g., [START_REF] Bimler | Clothed in triple blues": Sorting out the Italian blues[END_REF], 2018;[START_REF] Paramei | Italian blues": A challenge to the universal inventory of basic colour terms[END_REF], 2018;[START_REF] Uusküla | Linguistic categorization of blue in Standard Italian[END_REF]. Whether two suffices or whether three are needed is still a matter of controversy.

Across all linguistic and psycholinguistic studies, the authors are unanimous that blu is the counterpart of English "dark/navy blue." However, the second Italian BCT is argued to be either azzurro [START_REF] Paggetti | Color naming in Italian language[END_REF] or celeste [START_REF] Paramei | Diatopic variation in referential meaning of "Italian blues[END_REF]; in particular, in some of starting point for the optimization process that eventually led to the space as we know it, partitioned into eleven basic color concepts.

speakers' opinion azzurro denotes a shade in-between celeste "light-blue" and blu "dark blue" [START_REF] Grossmann | Colori e lessico: Studi sulla struttura semantica degli aggettivi di colore in catalano, castigliano, italiano, romeno, latino ed ungherese [Colors and lexicon: Studies on semantic structure of color adjectives in Catalan, Castilian, Italian, Romanian, Latin and Hungarian[END_REF], whereas for others its meaning is similar to celeste and both are in opposition to blu [START_REF] Albertazzi | Color names, stimulus color, and their subjective links[END_REF].

The degree of use of the three terms in the spoken language is known to be subject to diatopic, diastratic, and diaphasic variation (e.g., [START_REF] Paramei | Diatopic variation in referential meaning of "Italian blues[END_REF]. In particular, for Verona speakers (Veneto region) azzurro is the second BCT denoting the blue region and corresponds to "lightand-medium blue" [START_REF] Paramei | Italian blues": A challenge to the universal inventory of basic colour terms[END_REF], 2018). In comparison, for Alghero speakers of the Catalan-Algherese dialect (Sardinia) celeste appears to be the second BCT and denotes the light and medium shades of blue, while azzurro, with the meaning of "dark medium blue," is apparently not a basic term there [START_REF] Paramei | Diatopic variation in referential meaning of "Italian blues[END_REF]. Particularly relevantly for our study (given the provenance of the monolingual Italian participants we used for this paper; see below), for Florence speakers (Tuscany) the blue region is "clothed in triple blues": blu denotes "dark/navy blue," azzurro "medium blue," and celeste is reserved for "light blue" [START_REF] Bimler | Clothed in triple blues": Sorting out the Italian blues[END_REF][START_REF] Del Viva | Florence "blues" are clothed in triple basic terms[END_REF].

Study

We were interested in the effect of bilingualism on focal color judgments in the blue region of color space as this might provide evidence for cultural and/or linguistic effects on the placement of prototypes in color space. Studies on the effect of bilingualism on color conceptualization are sparse, but the few that exist [START_REF] Ervin | Semantic shift in bilingualism[END_REF][START_REF] Caskey-Sirmons | Semantic shift and bilingualism: Variation in the color terms of five languages[END_REF][START_REF] Athanasopoulos | Cognitive representation of colour in bilinguals: The case of the Greek blues[END_REF][START_REF] Paramei | Cross-linguistic similarity affects L2 cognate representation: blu vs. blue in Italian-English bilinguals[END_REF] suggest that, for Italian-English bilinguals, focal color choices in relation to color terms in one language could exert an effect on focal color choices in relation to color terms in the other language. The goal of the present study was to look for such an effect in comparing the responses from English monolingual speakers, Italian monolingual speakers, and Italian-English bilingual speakers to questions about focal colors in the blue region of color space.

Method

Participants

There were 92 participants in total, constituting three groups: British English monolingual speakers (EN), Italian monolingual speakers (IT), and Italian-English bilingual speakers (BI):

(i) EN speakers (N = 32; F = 20) were from West-North England, aged 30.1 (± 8.9) years. All had a college education, some at a graduate level, or obtained a doctoral degree. Their self-reported English-language proficiency scores were well above 9 for all categories (reading, writing, speaking, listening), on a scale from 1-10, with 10 being the highest.

(ii) IT speakers (N = 31; F = 17) were students or graduates of the University Florence, born in Tuscany, 22.9 (± 2.1) years old (for detailed demographic characteristics, see Del [START_REF] Del Viva | Florence "blues" are clothed in triple basic terms[END_REF]. Italianlanguage proficiency data were not available.

(iii) BI speakers (N = 30; F = 16) were aged 35 (± 9.8) years. All but one of them were born in Italy, and all but one resided in the United Kingdom at the time they participated in the study. Apart from one, all BIs had a university education, including 19 holding a doctoral degree. The majority of participants were originally from Central Italy, as well as from Lombardy, Tuscany, Puglia, and from Sardinia; one early bilingual was born in North-West England.

The mean age at which they had started to learn English was 9.8 (± 5.6) years, with the majority being late bilinguals (N lBI = 28), which are bilinguals who started learning English under the age of 6 years [START_REF] Wattendorf | Images of the multilingual brain: The effect of age of second language acquisition[END_REF]. The average age from which the BIs had lived in an Englishspeaking country was 23.5 (± 8.0); the duration of their residence in an English-speaking country was on average 7.6 (± 8.8) years. At the time they participated in the study, BIs used English on average 72.9 % (± 23.4 %) of the time. Their average self-reported English proficiency scores, on a scale from 1 to 10, were: 8.3 (± 1.0) for reading, 7.7 (± 1.3) for writing, 7.5 (± 1.2) for speaking, and 7.8 (± 1.3) for listening. Their average proficiency scores for Italian were, respectively, 9.7 (± 0.7), 9.2 (± 0.9), 9.5 (± 0.9), and 9.8 (± 0.6). All participants completed the Nation Vocabulary Test [START_REF] Nation | Teaching and learning vocabulary[END_REF], scoring an average of 80.2 (± 8.2) points out of 90.

All participants had normal color vision tested with the Ishihara Pseudoisochromatic Plates [START_REF] Ishihara | Test for colour-blindness (24 plates edition)[END_REF] and normal or corrected-to-normal visual acuity. None had reported any ocular disease, eye surgery, diabetes, or use of a medication that could have affected color vision.

Ethical approval was obtained from the Ethics Committees of the Departments of Psychology of Liverpool Hope University and the University of Florence. The study followed the ethical principles of the Declaration of Helsinki. All participants gave their written informed consent prior to participation in the study.

Materials

As stimuli served the 237 Munsell chips from eight glossy charts from The Munsell Book of Color [START_REF] Munsell | A color notation: An illustrated system defining all colors and their relations[END_REF], to wit, 7.5bg, 10bg, 2.5b, 5b, 7.5b, 10b, 2.5pb, and 5pb; together these charts encompass the blue region in Munsell color space. Value of the Munsell chips varied between 2 and 9, and Chroma varied (even number notation) between 2 and 10 or, for the charts 10b, 2.5pb, and 5pb, between 2 and 12.

For the visualizations and, also, for all of the statistics we carried out, we assumed the CIELAB coordinates of the Munsell chips, as online available at the website of the Munsell Color Science Laboratory from the Rochester Institute of Technology (RIT). While only a limited number of Munsell chips were used as stimuli (representing the blue region) in the study, Figure 1 gives readers unfamiliar with CIELAB space a sense of its shape by depicting the CIELAB locations of all Munsell chips (top row). The bottom row of that figure highlights the stimuli that were used in the study.

Procedure

At both testing locations, in Liverpool and Florence, we ensured identical procedure and illumination conditions. Participants were adapted to mesopic lighting in an otherwise dark room for at least 10 minutes, the temporal window ensuring dark adaption of cones [START_REF] Pirenne | Dark-adaptation and night vision[END_REF]. Following this, the charts were presented in a viewing booth under D65-metametric illumination (Just Normlicht Mini 5000; Fa. Colour Confidence) suspended 40 cm above the chart and delivering a 30 × 25 cm 2 light area. At the chart surface, luminance was 220 cd/m 2 (measured by the PR-650 SpectaScan Colorimeter; Photo Research, Inc.), corresponding to an illuminance of 1387 lux.

The response sheets consisted of a replica of the chip layout of the eight Munsell charts that constituted our materials. Participants were asked to write color names within white cells corresponding to the chart chips. The charts were presented one-by-one in a fixed order (as used in the description of the Materials). An unconstrained color-naming method was used for naming the Munsell chips, which, along with monolexemic hue terms, also allowed to use compound, modified or suffixed terms. Participants were further asked to indicate on the charts the most representative colors for azzurro, blu, and celeste in Italian or for blue and light blue in English.4 Bilinguals were tested on separate days in a counter-balanced order of Italian and English sessions. The experimenter provided instructions and communicated with the participant in the language corresponding to the session.

Results and discussion

The data were gathered in the context of a broader comparison of color naming by bilingual Italian-English speakers. For our present purposes, only focal color responses matter; an analysis of the full data from the free-naming task is relegated to future work. Figures 2 and3 give a visual summary of the focal color choices, the former exhibiting the blue and light blue foci, as designated both by the English monolinguals and the Italian-English bilinguals, and the latter doing the same for the Italian monolinguals' and Italian-English bilinguals' azzurro, blu, and celeste choices.

As a first step in our analysis, we used the QHull.jl library for the Julia language [START_REF] Bezanson | Julia: A fresh approach to numerical computing[END_REF]) to compute, per group of participants, convex hulls for their choices for each of the aforementioned color terms. Figure 4 shows the hulls for the bilinguals' blue and light blue focal color choices. Then, using the same library, we computed the volumes of the various hulls, which are given in Table 1. Here, one notices some quite remarkable differences already. In English, the convex hull of the bilinguals' choices of focal light blue covers more than three times the space covered by the convex hull of the EN monolinguals' choices for that focal color. In Italian, much the same is true for the convex hulls for the IT monolinguals' and bilinguals' choices of focal blu. And while for blue the convex hull of the bilinguals' choices is also larger than that of the EN monolinguals' choices, for azzurro the opposite is the case: the hull of the IT monolinguals' choices is about twice as large as that of the bilinguals' choices. We went on to calculate measures of central tendency for the various color-group combinations. Centroids (i.e., centers of mass) of the various convex hulls were calculated using the QHull.jl library once more. We also calculated centroids based on all focal color choices, both straight (i.e., unweighted)-by averaging the three coordinates (L * , a * , b * ) of all chips that had been designated by at least one participant as being focal for the given color-and weighted, by also taking into account how many participants had designated a chip as being focal for the given color. In the same way, we calculated straight and weighted medoids. (Medoids stand to centroids as medians stand to means.) The different calculations did not yield dramatically different results. In Figure 5, we are showing the weighted centroids in the full, threedimensional CIELAB space. Because three-dimensional graphics can be hard to interpret, Figure 6 also gives the three possible two-dimensional views of the space with centroids placed in it.

Just eye-balling the results, we see some clear patterns. The bilinguals' blue centroid appears closer to both the IT monolinguals' and the bilinguals' blu centroid than to the EN monolinguals' blue centroid. Similarly, the bilinguals' light blue centroid appears much closer to the IT monolinguals' celeste centroid than to the EN monolinguals' light blue centroid. These impressions are confirmed by looking at the distance matrix, given in Table 2. It is seen, for instance, that the bilinguals' focal blue centroid is about three times farther removed from the EN monolinguals' focal blue centroid than from the IT monolin- To find out whether the observable differences in the choices of the foci from the three groups have any statistical significance, we conducted four repeated-measures ANOVAs with color-group combination as a between-participants variable and the coordinate (L * , a * , b * ) as a within-participants variable. One ANOVA focused on the blue/blu comparison, comparing the combinations blue (EN), blue (BI), blu (IT), and blu (BI); the second ANOVA focused on a light blue/celeste comparison, comparing light blue (EN), light blue (BI), celeste (IT), and celeste (BI); the third ANOVA did the same for blue and azzurro; and the fourth did the same for light blue and azzurro.

The first ANOVA revealed a main effect of the color-group combination, F(3, 117) = 15.61, p < .0001, a main effect of the coordinate, F(2, 234) = 2615.84, p < .0001, as well as an interaction between the coordinate and the color-group combination, F(6, 234) = 14.88, p < .0001. As for the main effect of the color-group combination (the variable most directly of interest to our research), pairwise comparisons showed that the blue (EN) foci differed significantly from all other combinations, all ps < .0001, but that the other combinations did not differ significantly among each other, all ps > .46. See the left panel in the top row of Figure 7 for the estimated marginal mean coordinates for the various color-group combinations.

The second ANOVA also revealed a main effect of the color-group combination, F(3, 117) = 6.57, p < .0005, and a main effect of the coordinate, F(2, 234) = 7132.77, p < .0001, but the interaction between the coordinate and the color-group combination was only borderline significant, F(6, 234) = 2.12, p = .05. Pairwise comparisons showed that the light blue (EN) foci differed significantly from the light blue (BI) foci, p = .005, and from the celeste (IT) foci, p = .037; furthermore, the light blue (BI) foci differed significantly from the celeste (IT) foci, p = .004; finally, the celeste (IT) foci differed significantly from the celeste (BI) foci, p = .03. The right panel in the top row of Figure 7 shows the marginal mean coordinates for the various color-group combinations as estimated in this ANOVA.

The third ANOVA found, again, a main effect of both the color-group combination, F(3, 118) = 12.86, p < .0001, and the coordinate, F(2, 236) = 3443.40, p < .0001, as well as a significant interaction between these variables, F(6, 236) = 42.08, p < .0001. Pairwise comparisons revealed significant differences between blue (EN) and azzurro (IT), p = .003, between blue (BI) and azzurro (IT), p < .0001, and between blue (BI) and azzurro (BI), p = .0001. See the left bottom panel in Figure 7 for the corresponding estimated marginal mean coordinates.

The fourth ANOVA, finally, showed a very similar pattern, with a main effect of the color-group combination, F(3, 118) = 13.78, p < .0001, as well as of the coordinate, F(2, 236) = 5448.99, p < .0001; there was also a significant interaction between these variables, F(6, 236) = 6.97, p < .0001. Pairwise comparisons showed there to be significant differences between focal colors for light blue (EN) and each of the other color-group combinations, all ps < .005. The bottom panel in Figure 7 plots corresponding estimated marginal mean coordinates.

Discussion

We saw that bilinguals' choices of focal colors in English were more diffuse (i.e., spread out in color space) than the EN monolinguals' choices, in accord with instability and shifts of BCT prototypes in bilinguals reported previously by [START_REF] Ervin | Semantic shift in bilingualism[END_REF][START_REF] Caskey-Sirmons | Semantic shift and bilingualism: Variation in the color terms of five languages[END_REF][START_REF] Athanasopoulos | Cognitive representation of colour in bilinguals: The case of the Greek blues[END_REF]. By contrast, bilinguals' focal color choices in Italian were more concentrated than the IT monolinguals' choices for two out of the three Italian "blue" terms. That for blu the bilinguals' choices were actually much more diffuse than those of the IT monolinguals is plausibly due to the phonological and orthographic similarity between the English word blue and the Italian word blu (see [START_REF] Kroll | The Revised Hierarchical Model: A critical review and assessment[END_REF]. This may have led some bilinguals to identify colors as being typically blu because, speaking English, they would identify them as typically blue, and thereby to expand the convex hull for focal blue, which also encompassed colors more likely to be identified as being typically blu by IT monolingual speakers (see also [START_REF] Paramei | Cross-linguistic similarity affects L2 cognate representation: blu vs. blue in Italian-English bilinguals[END_REF]. The aforementioned similarity between homophone blue and blu may also explain the clear shift of the blue prototype as identified for the bilingual speakers, which was found to be significantly darker and closer to their and IT monolinguals' prototype for blu than to the blue prototype as determined for the EN monolingual speakers. We also saw evidence for a shift in the bilinguals' celeste prototype toward their light blue prototype and away from the IT monolingual speakers' celeste prototype. All in all, these findings point to cultural and linguistic effects in the mental representation of cognates and semantic equivalents of "blue" in Italian-English bilingual speakers.

General discussion

As mentioned in the Introduction, there is clear evidence in favor of the view that principles of rational design (e.g., principles having to do with the informativeness of naming systems) underlie the conceptual structure of color space, more specifically, how we carve up that space categorically and linguistically, and where, in it, we place the color category foci. But the same evidence also shows that the design principles that have been put forward as such in the literature may not tell the whole story about how perceptual color space gets its linguistically-defined structure. The whole story may, of course, encompass hitherto unidentified design principles. But the study presented in the foregoing gives reason to believe that, whatever the true collection of design principles may be, cultural and linguistic factors also play some part in the categorical structuring of color space. Some may want to see these further factors as noise, detracting from an ideal structure fully fixed by principles of rational design. Even then, it is important to be at least aware of them.

At a more methodological level, our study showed how work with bilingual speakers can help to identify cultural and linguistic influences on color conceptualization. In many ways, the present study just scratched the surface. One important limitation of the study was that our bilingual participants were all native speakers of Italian but not of English, even if they were highly to very highly proficient in English, and at the time this study was conducted tended to speak English most of the time. Ideally, the study would be complemented by a "symmetric" group of native English speakers highly proficient in Italian and residing in Italy for a number of years. Our results indicate that bilinguals' focal colors for English blue and light blue were closer to the Italian monolinguals' focal colors for blu and celeste, respectively, than to the English monolinguals' focal colors for blue and light blue. It appears reasonable to predict that the opposite would be the case for bilinguals of the kind just described, that is, native English speakers sufficiently immersed in Italian. We mention this here as an avenue for future research.

Figure 1 :

 1 Figure 1: Top row showing CIELAB space from different angles, with all Munsell chips from the RIT Munsell Color Science Laboratory placed into the space; bottom row highlighting the chips that were used as stimuli in the study.

Figure 2 :

 2 Figure 2: Focal light blue (top row) and blue (bottom row) choices of EN monolinguals (left column) and bilinguals (right column).

Figure 3 :

 3 Figure 3: Focal celeste (top row), azzurro (middle row), and blu (bottom row) choices of IT monolinguals (left column) and bilinguals (right column).

Figure 4 :

 4 Figure 4: Convex hulls of the bilinguals' focal blue and light blue choices, seen from different viewpoints.

Figure 5 :Figure 6 :

 56 Figure 5: Weighted centroids for all color-group combinations. (To facilitate orientation, we also show the locations of the blue region stimuli from the study, but not of all Munsell chips, as that would clutter the graphs too much in this case.)

Figure 7 :

 7 Figure 7: Marginal mean coordinates for the various color-group combinations as estimated in the repeated measures ANOVAs described in the text.

Table 1 :

 1 Volumes (in cubic units) of the convex hulls of focal color choices for the various color-group combinations.

		EN	BI	IT
	Blue Light blue Azzurro Blu Celeste	2877.9 3097.0	4006.2 11570.8 4089.6 6765.3 3326.4	8100.0 2270.7 4275.1

Note: EN = English monolinguals; BI = bilinguals; IT = Italian monolinguals. For reference, the volume of the convex hull encompassing all Munsell chips is 887469.5.

Table 2 :

 2 Distance matrix of weighted centroids of focal colors for the various color-group combinations.

			EN		IT		BI
			Light blue Blue Celeste Azzurro Blu Light blue Blue Celeste Azzurro Blu
	EN IT BI	Light blue Blue Celeste Azzurro Blu Light blue Blue Celeste Azzurro Blu	0.0 35.8 35.8 0.0 8.4 32.0 17.2 20.7 48.9 17.4 10.6 31.0 45.4 15.4 48.4 18.2 20.2 19.1 5.8 40.4	8.4 32.0 0.0 11.6 45.4 2.4 41.3 44.9 14.0 9.9	17.2 48.9 20.7 17.4 11.6 45.4 0.0 35.1 35.1 0.0 10.6 44.2 31.2 5.5 34.9 2.2 3.1 33.6 20.9 53.2	10.6 45.4 31.0 15.4 2.4 41.3 10.6 31.2 44.2 5.5 0.0 39.9 39.9 0.0 43.6 4.9 12.8 29.5 11.7 49.4	48.4 18.2 44.9 34.9 2.2 43.6 4.9 0.0 33.5 52.5	20.2 5.8 19.1 40.4 14.0 9.9 3.1 20.9 33.6 53.2 12.8 11.7 29.5 49.4 33.5 52.5 0.0 23.6 23.6 0.0

Note:

The numbers represent Euclidean distances (∆E) in CIELAB space.

Formally, the Pareto front is the curve or surface (or hyper-surface) in the space of possible solutions to a multi-objective optimization problem such that, for any solution represented on the curve/surface, one can only improve with regard to one of the given objectives by doing worse with regard to one or more of the others.

It is an open question whether all concepts can be represented in similarity spaces. We will leave this broader question aside here, our main interest being in the categorical structure of color space. For more on the broader program of using similarity spaces to represent concepts, see[START_REF] Gärdenfors | Conceptual spaces[END_REF].

An anonymous referee made the important observation that yet another possibility (compatible with the ones mentioned) is that, because color space will have evolved over time, and is thus not to be conceived as the product of an optimization process from scratch, there have been anchoring effects of older, less fine-grained partitions of the space, which have served as a kind

We are aware that light blue is not a BCT in English; however, we decided to assess focal color of this non-basic category as a proxy to Italian celeste. Note also, that light blue was found to be among the most frequently used English color terms in an online free-naming experiment[START_REF] Jraissati | Delving deeper into color space[END_REF].

Acknowledgments

The study was conducted in the framework of Erasmus Exchange Agreements between Liverpool Hope University with University of Sassari (2009Sassari ( -2011) ) and University of Florence (2019-2021). We thank Carmen De Caro, Maria Michela Del Viva, and Ilaria Mariani for providing data for Italian monolinguals (Florence), as well as students for assisting in data collection: Nadia Al-Mahrouky and Ricky Morton (English monolingual speakers); Cristina Stara and Ricky Morton (Italian-English bilinguals), both groups in North-West England (Liverpool and Manchester). We are greatly indebted to Panos Athanasopoulos for sharing the Nation Vocabulary Test; John Mollon for the advice on the standardized illumination; Deborah Roberson for guidance on the unconstrained color-naming method; Guido Frison for consultation on early use of Italian terms for "blue"; and Manila Soffici for consultation on the Florentine dialect and Standard Italian. We thank Robert Hewertson for technical assistance, Kaida Xiao for illuminance measurement, and all participants for their time and good will. We are also greatly indebted to two anonymous referees for valuable comments on a previous version of this paper.