A case against convexity in conceptual spaces

Abstract

The notion of conceptual space, proposed by Gärdenfors as a framework for the representation of concepts and knowledge, has been highly influential over the last decade or so. One of the main theses involved in this approach is that the conceptual regions associated with properties, concepts, verbs, etc. are convex. The aim of this paper is to show that such a constraint—that of the convexity of the geometry of conceptual regions—is problematic; both from a theoretical perspective and with regard to the inner workings of the theory itself. On the one hand, all the arguments provided in favor of the convexity of conceptual regions rest on controversial assumptions. Additionally, his argument in support of a Euclidean metric, based on the integral character of conceptual dimensions, is weak, and under non-Euclidean metrics the structure of regions may be non-convex. Furthermore, even if the metric were Euclidean, the convexity constraint could be not satisfied if concepts were differentially weighted. On the other hand, Gärdenfors’ convexity constraint is brought into question by the own inner workings of conceptual spaces because: (i) some of the allegedly convex properties of concepts are not convex; (ii) the conceptual regions resulting from the combination of convex properties can be non-convex; (iii) convex regions may co-vary in non-convex ways; and (iv) his definition of verbs is incompatible with a definition of properties in terms of convex regions. Therefore, the mandatory character of the convexity requirement for regions in a conceptual space theory should be reconsidered.

1 Introduction

The notion of conceptual space, proposed by Gärdenfors (2000) as a framework for the representation of concepts and knowledge, has been highly influential over the last fifteen years. Since his initial proposal, Gärdenfors (2014) has tried to extend the approach both to the modeling of actions and events, and to the semantics of verbs, prepositions and adverbs. One of the basic theses of his approach is that the conceptual regions associated with properties, concepts (or object categories), verb meanings, etc. are convex. The aim of this work is to show that such a constraint, that of the geometrical convexity of conceptual regions, is problematic; not only from a theoretical perspective, but also with regard to the inner workings of the theory itself.

In Sect. 2, after this brief introduction, I expound the main features of Gärdenfors’ theory of conceptual spaces, focusing on his definitions of property and concept. At the same time I aim to explain clearly the role played by the convexity constraint in the theory, as opposed to other possible criteria that could be imposed on the geometry of regions. In Sect. 3 I recap how the notion of similarity is characterized within a geometrical approach; and I introduce the distinction between standard and non-standard distances. There, and after introducing the notion of Voronoi partition, I hold that Gärdenfors is tacitly committed to the thesis that conceptual regions are the cells resulting from a Voronoi tessellation starting with a set of prototypes.

Nevertheless, the regions resulting from a Voronoi tessellation are only convex under very specific assumptions, namely, if the metric is Euclidean, and if distances are not differently weighted for each particular concept. Therefore, Gärdenfors may try to defend the convexity constraint through two different strategies: either arguing directly in favor of the convexity of the conceptual regions; and/or arguing for the assumptions which guarantee that the regions produced by a Voronoi tessellation are convex. I will try to show that: [I] Gärdenfors’ direct arguments in favor of convexity are not conclusive. [II] The assumption of a Euclidean metric is not guaranteed. [III] It is not implausible that the distances to the prototypes of distinct concepts are differently weighted.

Section 4 focuses on criticism of the major cognitive reasons for the convexity of conceptual regions: (a) co-implication with the prototype theory of concepts; (b) cognitive economy; (c) its perceptual foundations; and (d) effectiveness of communication. I aim to show that none of them compels us to accept the convexity constraint as compulsory.

Notwithstanding that, if the metric underlying conceptual spaces were the standard Euclidean metric (that is, if distances were Euclidean and non-weighted), then the convexity of regions would be guaranteed. With regard to this point, Gärdenfors argues that, for the case of integral dimensions, the Euclidean metric fits the empirical data better than the city-block metric. So, given that his definitions of property and concept are for domains constituted by sets of integral dimensions, it is possible to conclude that the metric of conceptual spaces is the Euclidean one. Section 5 brings into question the empirical evidence that supports such an argument; evidence supposedly in favor of the relation of mutual dependency between integral dimensions and the Euclidean metric. By questioning the evidence in this way, I intend to show that the metric underlying conceptual spaces can be non-Euclidean, and in that case the convexity constraint on regions does not hold, as I indicate in Sect. 6. After that, in Sect. 7, I allege that, even if conceptual spaces do function with a Euclidean framework, conceptual regions might be non-convex if distances of comparison in categorizations are weighted differently.

Finally, Sect. 8 is devoted to proving that the convexity requirement is brought into question by the very characterization of the inner workings of Gärdenfors’ conceptual space theory itself. From all these arguments, I conclude (Sect. 9) that the mandatory character of the convexity constraint should be rethought, perhaps in favor of a weaker (and non-mandatory) criterion for the geometry of regions.

2 Conceptual spaces and the convexity constraint

Gärdenfors proposal consists of a non-connectionist theory of conceptual spaces based on the notion of similarity. In general terms, a similarity space theory of concepts can be described by the following fundamental thesis (Gauker 2007): the mind is a representational hyperspace within which (a) dimensions represent ways in which objects can differ, (b) points represent objects, (c) regions represent concepts, and (d) distances are inversely proportional to similarities (between objects or concepts). Consequently, an object will belong to a concept if and only if its values in every dimension of that similarity space produce an n-tuple that lies inside the region associated with that concept.

2.1 Gärdenfors’ conceptual spaces

Nonetheless, there are important differences between the conceptual space theory proposed by Gärdenfors (2000) and this general framework. Firstly, (natural) properties are convex regions of a given domain (Criterion P), and are typically associated with the meaning of adjectives. Secondly, (natural¹) concepts are bundles of properties (or, alternatively, sets of convex regions) in a number of domains, together with the salience weights of those domains and information concerning how their regions are correlated (Criterion C); and they typically represent the meaning of nouns. Finally, in this framework, the notion of domain is critical: it is defined as a set of integral² dimensions that are separable from all other dimensions.

In accordance with this general scheme, concepts are a result of the division of the similarity space into convex regions (constituted by the sets of points representing those objects that exhibit the sensory properties characteristic of such regions). The convex regions are identified by Gärdenfors precisely with concepts.

Lastly, in his most recent work, Gärdenfors and his collaborators have tried to extend this basic framework from the cases of properties and concepts (or, alternatively, from adjectives and nouns) to the representation of states, changes, actions and events; through these, to the semantics of verbs, adverbs and prepositions; and ultimately to apply it to the case of human communication (Gärdenfors and Warglien 2012, 2013; Warglien et al. 2012; Gärdenfors 2014). Very briefly sketched, Gärdenfors takes as a starting point the thesis that verbs typically represent dynamic properties of objects; are parts of events; and involve actions which are constituted by forces (commonly exerted by agents). In virtue of this, his proposal for verbs consists, in effect, of a holistic model of actions, forces, events and verbs, characterized by means of conceptual spaces. Within such a framework, verbs denote changes in properties (that is, movements in the representation of objects or concepts within the conceptual space), and they both refer to and are represented by convex regions of vectors.

2.2 The convexity constraint

As is evident from the previous subsection, the requirement for the convexity of regions runs through all the conceptual space theory defended by Gärdenfors. Not only properties and concepts (or object categories), but also the semantics of verbs, adverbs and prepositions (Gärdenfors 2014) are conceived and represented within his theory by convex regions.

The convexity requirement can be thought of as a generalized definition of the conception of a natural kind as a qualitatively spherical region, expounded by Quine (1969) in his discussion of the definition of natural kinds in terms of (comparative) similarity. Gärdenfors’ aim is to characterize the geometrical form of natural properties and concepts, when acting as (optimal) evolutionary tools in tasks such as problem-solving, memorizing, planning, communicating, etc. To that end, he distinguishes three possible criteria which could constrain the geometry of a region (see Fig. 1):

Fig. 1

Representation of the three different criteria for the geometry of conceptual regions: connectedness, star-shapedness and convexity. Paths containing exclusively points belonging to the considered regions are represented by solid lines. Paths also containing points outside the regions are represented by dashed lines. The problematic points not belonging to the regions are represented by crosses. a Representation of a disconnected region $R_{1}$ where the point A is not reachable from the point B following a continuous path of points belonging to $R_{1}$. b Representation of a connected region $R_{2}$ where every point x in $R_{2}$ is reachable from every other point y in $R_{2}$, following a continuous path of points belonging to $R_{2}$. c Representation of a connected but non-star-shaped region $R_{2}$ (the same as in graph b), where there is no point with respect to which $R_{2}$ satisfies the star-shapedness constraint (for instance, $P_{1}$ cannot be that point because between $P_{1}$ and C there are points not belonging to $R_{2})$. d Representation of a connected and star-shaped region $R_{3}$, where there is a point $P_{2}$ in $R_{3}$ such that, for every point x in $R_{3}$, all the points between $P_{2}$ and x also belong to $R_{3}$. e Representation of a star-shaped but non-convex region $R_{3}$ (the same as in graph d), where there are points D and E in $R_{3}$ such that not all the points between them also belong to $R_{3}$. f Representation of a star-shaped and convex region $R_{4}$, where for every two points x and y in $R_{4}$, all the points between x and y also belong to $R_{4}$

• connectedness constraint: it must be possible to reach every point in the region from every other point by following a continuous path consisting only of points belonging to the region;

• star-shapedness constraint (with respect to a point P): for every point x in the region, all the points between³ x and P must belong to the same region;

• convexity constraint: the region must satisfy the star-shapedness constraint with respect to all the points in the region, that is, for every two points in the region, all the points between them must also belong to the same region.

The strength of these three criteria increases in order: every star-shaped region is a connected region and (trivially) every convex region is a star-shaped region.

3 Similarity measures and Voronoi diagrams

3.1 Geometric similarity measures

There are four main approaches to characterizing the notion of similarity (Goldstone and Son 2005): geometrical, feature based, alignment based and transformational. Here, in the spirit of Gärdenfors’ conceptual spaces, I focus on geometric characterizations of similarity. Models based on this approach define similarity as a measure that is inversely proportional to distance, which is usually determined according to a Minkowski metric. Let us remember the expression for the distance (in a generic Minkowski metric) between two objects (and/or prototypes of concepts) A and B located within an n-dimensional space, where $X_i^{[Y]}$ represents the value of the i-th dimension associated with the concept Y:

$$\begin{aligned} d(A,B)=\left( {\sum _{i=1}^n {\left| {X_i^{[A]} -X_i^{[B]} } \right| ^{p}} } \right) ^{1/p} \end{aligned}$$

The value of the parameter p determines the type of metric and distance: if $p=1$, they are called Manhattan (or city-block); when $p=2$, they are called Euclidean.

This expression corresponds to ordinary Minkowski distances. Nevertheless, these distances can be weighted differently according to various criteria. For example, in the case of conceptual spaces, the weights could be a function of the number of particular cases (instances or examples) on which a given concept is based. In such a case, the distance-of-comparison in categorizations of a certain object, O, with respect to a particular concept, $C_{i}$ (represented within the conceptual space by the prototype $P_{{C}_{i}} )$, referred to as $d_{Ci} (O,P_{{C}_{i}} )$, could be expressed, under a multiplicatively weighted scheme,⁴ as follows⁵:

$$\begin{aligned} d_{Ci} (O,P_{{C}_{i}} )=w_i d(O,P_{{C}_{i}} ) \end{aligned}$$

where $w_{i}$ represents the weight assigned to that concept. In Sect. 7 below, I will show the implications of non-standard weighting with regard to the convexity requirement.

3.2 Voronoi tessellations of a conceptual space

Once assumed a geometric similarity measure, Gärdenfors’ proposal can be characterized by means of Voronoi diagrams, inasmuch as concepts and properties can be conceived as the cells resulting from a Voronoi tessellation of the conceptual space.

A Voronoi diagram is a partition of an n-dimensional space into regions, based on the distances between each point and the points belonging to a particular subset G of that n-dimensional space. The points belonging to G are commonly called seeds or generators, and in Gärdenfors’ theory those points are the prototypes of concepts. The general idea is that for each generator $g_{i}$ there exists a region constituted by those points nearest to $g_{i}$ than to any other seed belonging to G. The points equidistant from their two closest generators will constitute the boundaries of regions. Thus, for example, in the case of a standard Euclidean metric where both concepts and dimensions were equally weighted, the boundaries of regions would be determined by the bisectors of the segments connecting each pair of generators.

On my view, Gärdenfors is strongly committed to the thesis that the shapes and boundaries of conceptual regions are produced by a Voronoi tessellation of the conceptual hyperspace, whose input are the prototypes of the relevant concepts. Regarding this, it could be objected that Gärdenfors never expresses openly his commitment to the thesis that conceptual regions are the cells resulting from a Voronoi tessellation starting with a set of prototypes (Thesis V). Nevertheless, although that is true, it is also clear that he accepts Thesis V tacitly when he explains how significant elements of his theory work. For instance, in The geometry of meaning (2014), his explanations with regard to conceptual learning (Gärdenfors 2014, p. 42), conceptual change (ib., p. 43), categorization (ib., pp. 27–28), communication and language (ib., pp. 274–275), and vagueness (ib., pp. 45–46) are all of them based on the assumption of Thesis V.

In fact, if Gärdenfors considered that conceptual regions might be something different to cells resulting from a Voronoi tessellation, then he should have provided a distinct kind of explanation. More specifically, if he considered that Voronoi partitions are only one possible way by which his conceptual space theory could be articulated, he ought to have explained those phenomena for a characterization K of his theory such that Voronoi tessellations were a particular case of K. The fact that he provides no explanations other than the ones based on Thesis V is indicative of his strong (although tacit) commitment to that thesis. In consequence, throughout all my paper I will assume that Gärdenfors accepts Thesis V.

4 Gärdenfors’ arguments for the convexity constraint

In his work, Gärdenfors does not provide any ‘definitive’ argument in favor of the convexity constraint, but he offers a series of reasons that suggests a high degree of plausibility for it. My aim in this section is to show that none of those arguments is compelling: we do not have to accept convexity as a mandatory requirement for the geometry of regions.

4.1 Mutual dependence with the prototype theory

One of the six basic principles that Gärdenfors considers to be embodied by the cognitive approach to semantics is that concepts show prototypical effects which cannot be explained from the standpoint of classical theories of concepts. In fact, one of the main advantages of Gärdenfors’ approach is that his conceptual spaces provide a natural explanation of prototypical effects for many concepts⁶ (Rosch 1978; Lakoff 1987). Let us see why.

According to the prototype theory, concepts are prototypes, that is, representations whose structure encodes information about the properties that their members tend to have. However, there are different ways in which the prototype theory can be articulated (Smith and Medin 1981): (a) featural models: an object is classified under a given category if it possesses a sufficient number of the properties associated to that concept; (b) dimensional models: an object is classified under a given category if it possesses to some degree a sufficient number of those properties. (Gärdenfors’ conceptual spaces are a particular type of dimensional models, where distances in a certain dimension are indicative of the degree-of-possession of the properties associated to that dimension.) In both cases an object will be categorized or not under a particular concept in function of the similarity between the object and the prototype of that concept. Their similarity will be determined by virtue of their shared properties (either possessed, or possessed to some degree, depending on whether the model is featural or dimensional). In consequence, the more prototypical a member of a category is: (i) the more attributes it shares with the other members of the category; and (ii) the fewer features it shares with the members of other categories (Rosch and Mervis 1975).

And, with regard to the question of how the prototypes of concepts are acquired, it is reasonable to think that those prototypes are the result of a process of maximization of similarities (or, alternatively, minimization of distances) between the evaluated objects, and the tentative prototype of a particular category. The final set of prototypes will be the one which maximizes intra-group similarity and minimizes inter-group similarity. Therefore, the prototype of a concept arises from the generalization of the properties of the objects chosen as tentative members of a given category. Hence, prototypes are those members (whether real, or not⁷) of a category that best reflect the similarity structure of the category as a whole. Next, a Voronoi partition of the conceptual hyperspace can be produced, whose input would be the prototypes of the relevant concepts.

In addition, in order to categorize a new object under a particular concept, that object will only need to be evaluated with respect to the prototypes of the relevant categories, which makes the categorization process very efficient (with no need to resort to the shapes or boundaries of regions).

Based on this, it is possible to say that Gärdenfors’ conceptual space theory provides a natural explanation of the typicality effects we expect a prototype theory of concepts to exhibit, given that: (i) in a conceptual space the centers of gravity (or mass centroids) of the regions associated with each concept can be identified with the most representative members endorsed by the prototype theory of concepts; and (ii) prototypicality can be characterized as a measure that is inversely proportional to the distance (of an object) from those centers of gravity.⁸

With regard to this first claim, Gärdenfors defends the notion that those who adopt the prototype theory of concepts should expect a representation of concepts and properties as convex regions; and contrariwise, that if concepts are characterized as convex regions, then prototypical effects should be expected (Gärdenfors 2000, pp. 86–87; 2014, pp. 26–27).

It is my view that this is the main argument offered by Gärdenfors in support of the convex geometry of regions. However, neither of the two assertions constitutes a reason in favor of the convexity requirement, given that both of them could also be applied to a star-shaped region resulting from a Voronoi tessellation, as I now explain.

• [A] If properties and concepts were defined as star-shaped regions (produced by a Voronoi tessellation) then prototypical effects would also be expected: in this case the typicality of an object with regard to a given category is also a function of the distance between the point representing that object and the prototype of its category.

• [B] The only thing that should be expected by a consistent prototype theorist is the star-shapedness of conceptual regions: a prototype theorist should expect that if an object belongs to a certain category, then all the objects with the same proportional distances from the prototype but more similar to it (that is, all the objects between the object under consideration and the prototype), should also belong to that category. This is exactly what happens under the star-shapedness constraint.⁹

Thus, Gärdenfors’ alleged mutual dependence between the prototype theory and the convexity of regions (within a conceptual space approach articulated by means of Voronoi tessellations), also takes place between the theory of prototypes and the star-shapedness of regions. Consequently, such a relationship cannot be a crucial reason in favor of the convexity constraint.¹⁰

4.2 Cognitive economy

When Gärdenfors originally defined properties in terms of convex regions, he mainly based his decision on the argument provided by Shepard (1987, p. 1319). Shepard argued that evolution would have led to consequential regions (in our psychological space) in a way such that the boundaries of those regions were not oddly shaped. Next Gärdenfors (2000, p. 70) maintained that such an evolutionary preference could be supported by a principle of cognitive economy in terms of memory, learning and processing.

However, the cognitive economy argument depends on the assumption that the handling of convex sets of points requires less memory, learning and processing resources than the handling of regions with capricious forms. In this case, Gärdenfors’ argument can be structured as follows.

• (i) Properties and concepts are determined by convex regions within a conceptual hyperspace. (Criteria P and C)

• (ii) Those convex regions can be the result of a Voronoi tessellation starting with a set of prototypes.¹¹ (Thesis V)

• (iii) A Voronoi tessellation could support, in a cognitively efficient way, psychological processes such as concept learning, categorization, communication and language.¹² Additionally, Voronoi tessellations can also explain other phenomena, such as conceptual change and vagueness.

• (iv) Therefore, the handling of convex regions could explain cognitive efficiency in all those tasks and processes.

The problem is that, as shown in Sects. 6 and 7 below, the conceptual regions can be non-convex and yet compatible with a Voronoi tessellation. Moreover, given that the reasons offered by Gärdenfors are not exclusive to convex regions, an independent argument for the greater cognitive efficiency of handling convex regions (over non-convex ones) is still required.

I entirely support Gärdenfors’ aim of basing the conceptual space theory on reasons related to cognitive economy; in fact, I think that most of his arguments are thoroughly valid with regard to such a general point. That is, a conceptual space theory is highly efficient from a cognitive point of view, given that:

• only prototype locations have to be memorized;

• in categorization tasks the only distances evaluated would be those between the evaluated objects and the prototypes associated with the categories considered; and

• concepts could be learned from a very small number of particular examples.

Nonetheless, all these facts are common to every conceptual space theory which assumes that concepts are represented by regions resulting from a Voronoi tessellation, independently of the geometrical structure (convex or non-convex) of those conceptual regions. Consequently, cognitive efficiency cannot be a crucial reason to support the convexity criterion.

4.3 Perceptual foundation

Gärdenfors usually contends that many perceptually grounded domains, such as color, taste, vowels, etc., are convex, based on evidence in favor of the convexity of the regions associated with numerous typical properties of all those domains (Fairbanks and Grubb 1961; Sivik and Taft 1994). The color domain, however, seems to be his preferred example of integral dimensions. In the case of color, the problem is that the work that Gärdenfors refers to as evidence is entirely associated with sensory dimensions; there is no guarantee that things work in the very same way in non-perceptual domains. This last point is explicitly recognized by Gärdenfors (2014, p. 137) when he acknowledges that the evidence (mainly associated with the color domain) does not provide automatic support for the convexity constraint in other domains.

4.4 Effectiveness of communication

One of the most recent arguments offered by Gärdenfors (2014, p. 26) for the convexity requirement is that the convexity of conceptual regions is decisive in effective communication. In this case Gärdenfors argues that Jäger’s research has shown that in language, convex regions are a result of cultural evolution (Jäger 2007). However, Jäger’s work does not constitute an argument either in favor of or against the geometrical structure of conceptual spaces, given that he assumes the standard¹³ Euclidean metric (Jäger 2007, p. 554); so semantic categories have to be convex (see below for discussion of this point). Due to this, when Gärdenfors cites Jäger’s research in support of his thesis that conceptual regions are convex, he falls into petition principii: since Jäger starts from the standard Euclidean metric, his results simply do not disprove the convexity constraint thesis; but they cannot confirm it.

To sum up, on the one hand, under the assumption of Thesis V neither the mutual dependence with the prototype theory of concepts, nor the economy cognitive argument are definitive reasons in favor of convexity. That was so because those arguments are equally valid for all the conceptual regions resulting from a Voronoi tessellation, independently that their shapes are convex or not. On the other hand, none of the two other arguments for convexity (perceptual foundation and effectiveness of communication) are convincing enough to accept that concepts have to be convex.

5 Integral dimensions, Euclidean metric, and convexity

As stated above, Gärdenfors’ conceptual spaces rest on the assumption that regions are convex and, if concepts are not differently weighted, this convexity is guaranteed under a Euclidean metric (Gärdenfors 2000, p. 88; Okabe et al. 1992, p. 57). By virtue of this, the theory requires that the metric underlying our psychological space is Euclidean. On this occasion, the main argument in favor of a Euclidean metric is that in the case of integral dimensions, a Euclidean metric fits the empirical data better than a city-block metric (the latter would be more appropriate in the case of separable dimensions). Furthermore, given that Gärdenfors’ definitions of property and concept are for domains constituted by sets of integral dimensions, it is possible to conclude that the conceptual spaces underlying them function with a Euclidean metric and, consequently, that their associated regions are convex.

However, this argument presents several problems, mainly due to the alleged mutual dependency relationship between integral dimensions and the (standard) Euclidean metric.

• First, Gärdenfors adduces a sort of co-implication between integral domains and the Euclidean metric: “If the Euclidean metric fits the data best, the dimensions are classified as integral; $(\ldots )$ when the dimensions are integral, the dissimilarity is determined by both dimensions taken together, which motivates a Euclidean metric” (Gärdenfors 2000, p. 25). The first implication is true, because if the metric is not city-block, the dimensions are non-separable (that is, they are integral). Nonetheless, the second conditional is false, inasmuch as the non-separability of dimensions does not necessarily imply that the metric is Euclidean.¹⁴ Thus, the Euclidean character of the metric structure cannot be based on the integral character of domains. Gärdenfors assumes that dimensions are integral, but that is not sufficient to guarantee that the metric is Euclidean. In consequence, the Euclidean structure of a metric space needs empirical evidence independent from the one associated to the non-separability of its constitutive dimensions.¹⁵

• Second, and even assuming (as Gärdenfors does) that the co-implication between integral dimensions and the Euclidean metric were the case, the empirical evidence referred to in favor of the integral or separable character of a particular set of dimensions is tied to perceptual domains,¹⁶ such as color, sound, size, shape, etc. (Garner 1974; Maddox 1992; Melara 1992). All that work faces a threefold difficulty, when taken as evidence in favor of Gärdenfors’ theses, as I now explain.

  • [A] All the work is based on classification experiments and judgments of similarity at a conscious level, in which the integral character of dimensions (and, in consequence, the Euclidean character of the metric) could depend on how those classifications and judgments are consciously carried out, and not on the geometrical structure of the perceptual space.

  • [B] The experiments were developed over a small number of perceptual domains, so accepting them as evidence of the geometry of conceptual spaces requires the assumption that the behavior of the metric structure is the same across all perceptual and conceptual domains.¹⁷ That is, it is necessary to assume that such behavior extends not only from the perceptual domains studied to all other perceptual domains, but also to all conceptual domains (in general not related to any of the perceptual domains studied), which might not be the case.

  • [C] This kind of work is used to contrast Euclidean and city-block metrics, and shows that the former fits integral sets of dimensions better, while the latter provides a better fit when the dimensions are separable. In the case in hand, however, the problem is that both metrics provide good fits, but not perfect fits. This ultimately means that the best metric is neither the Euclidean nor the city-block one; but is something between the two. For example, in Handel and Imai (1972, p. 110) the optimal parameter p for integral dimensions in a general Minkowski metric is 1.7, which may be acceptable as reasonably close to a Euclidean space, but with non-convex regions¹⁸ (given that their convexity would require a value of p equal or much closer to 2). Therefore, what can be derived from this work is not that the (standard) Euclidean metric is warranted for integral dimensions, but only that the expected metric for integral domains will be closer to the (standard) Euclidean metric than to the (standard) city-block metric.

To sum up, all this evidence appears to be controversial; both that supporting the integral character of conceptual dimensions, and that which allegedly backs up the relationship between the integral character of dimensions and the Euclidean metric. The consequence is that, in both cases, the underlying metric could be non-Euclidean and, hence, conceptual regions could be non-convex.

6 Conceptual spaces under a non-Euclidean metric

Merely from attending to the basic requirements of a similarity space theory of concepts, it can be seen that the convexity constraint is unnecessary: nothing in the general conception of this kind of theories demands a Euclidean metric, and under a non-Euclidean metric the conceptual regions resulting from a Voronoi tessellation can be non-convex. Nonetheless, a constant throughout all of Gärdenfors’ work is that he explicitly adopts a Euclidean metric which apparently guarantees the convexity of the conceptual regions. The problem is that if the conceptual space metric is non-Euclidean, then regions may be non-convex. The goal of this section is to describe what the consequences would be if the assumption was of a non-Euclidean metric.

As introduced in Sect. 3.1 above, the formula for the (standard) distance, given a generic Minkowski metric, between two objects (and/or prototypes of concepts) A and B located within an n-dimensional space, is given by the expression:

$$\begin{aligned} d(A,B)=\left( {\sum _{i=1}^n {\left| {X_i^{[A]} -X_i^{[B]} } \right| ^{p}} } \right) ^{1/p} \end{aligned}$$

where the value of p determines the specific type of distance ($p=1$, Manhattan; $p=2$, Euclidean), and it could take any positive real value (not only integer). The boundaries of conceptual regions will then depend on the specific metric chosen, and so will the convex or non-convex character of those regions (as illustrated by the graphs in Fig. 2).

Fig. 2

Boundaries of the conceptual regions resulting from a maximization process implementing the prototype theory of concepts, for four distinct possible metrics. The final prototypes are represented by the four black dots, whose coordinates are (1.5, 1), (1.8, 2.7), (2, 1.5) and (3, 1). The boundaries of the conceptual regions are drawn as dotted dark-grey lines. a Boundaries for the city-block metric (parameter $p =$ 1). b Boundaries for the Euclidean metric (parameter $p =$ 2). c Boundaries for a conceptual space that fits the Euclidean metric better than the city-block one (with parameter $p =$ 1.7), as happened in Handel and Imai’s (1972, p. 110) experiments. d Boundaries for a higher-order Minkowski metric (parameter $p =$ 3)

As is evident from Fig. 2, only the (standard) Euclidean metric satisfies the convexity requirement,¹⁹ while the other metrics generate regions that are more or less non-convex.

Consequently, if the metric of conceptual spaces is not Euclidean in a strong sense, then the convexity constraint on regions cannot be mandatory in that very same strong sense; contradicting what Gärdenfors’ theory requires of them.

7 Conceptual spaces under a weighted Euclidean metric

Nonetheless, even if the metric of conceptual spaces was Euclidean, it is possible that conceptual regions would not be convex. Obviously, this would not be case, as just explained in the section above, under the standard Minkowski distance which, for the Euclidean case $(p=2)$, is defined as:

$$\begin{aligned} d(A,B)=\sqrt{\sum _{i=1}^n {\left( {X_i^{[A]} -X_i^{[B]} } \right) ^{2}} } \end{aligned}$$

But let us think for a moment about how, in a theory such as Gärdenfors’, concepts are produced. In a first instance, if a particular concept is not innate, then it should have been learnt sometime in the past from a set of particular examples. Additionally, it could be argued that the size of the sample of examples has an effect on how objects are categorized under a particular concept.

For example, let us imagine a subject who had been exposed to hundreds of instances of the concept dog, but only a few cases of the concept fox. Then it could be thought that if that same subject were exposed to one new instance of fox, different from all the foxes already encountered and with a certain resemblance to the concept dog already acquired, a judgment of the new instance as fitting the concept dog could be more confidently reached than one of it fitting the concept fox. That would mean (within a conceptual space theory of the mind), that the subject could ascribe a greater weight to the concept dog than to the concept fox (see Fig. 3).

A phenomenon such as this could occur even under a Euclidean metric (that is, even if the underlying conceptual space were Euclidean), where base distances were calculated using the formula above for d(A, B). If objects were categorized as just been described, the distances associated with each concept would be differently weighted depending on the number of examples on which that concept were based. These differently weighted distances would correspond with the non-standard multiplicatively-weighted distances introduced in Sect. 3.1 above. Consequently, the formula for the distance-of-comparison, $d_{Ci} (O,P_{C_{i}} )$, in categorizations of a particular object O with regard to a given concept $C_{i}$ (represented by a prototype $P_{C_{i}} )$ would be:

$$\begin{aligned} d_{Ci} (O,P_{C_{i}} )=w_i d(O,P_{C_{i}} ) \end{aligned}$$

Here, the value of $w_{i}$ represents the weight associated with each concept, which would be a function of the number of examples, $n_{i}$, on which such a concept is based. Indeed, the greater the number of examples, $n_{i}$, the greater the relative similarity claimed between O and $P_{C_{i}}$ (or, equivalently, the lower the distance-of-comparison $d_{Ci} (O,P_{C_{i}} ))$, and hence, the lower the weight of the distances $w_{i}$. The weight $w_{i}$ could be, for example, a function ranging from two (if the number of examples is very small) to one (when that number is large enough), as given by $w_i =1+1/n_i$ (see Fig. 4).

Fig. 3

Boundaries of the conceptual regions resulting from a maximization process implementing the prototype theory, for different weightings of concepts. The three considered concepts are dog, cat and fox, whose prototypes are represented by the black dots, with coordinates (3.5, 2), (0.5, 0.5) and (3, 1). The boundaries of the conceptual regions are drawn as dotted dark-grey lines. a Boundaries for the standard Euclidean space, where the weights of all the prototypes are equal to 1. b Boundaries for a non-standard (prototype-weighted) Euclidean space, where the distances-of-comparison for the concepts dog, cat and fox are multiplicatively-weighted by 0.25, 0.4 and 1 respectively (what could happen if the subject had been exposed [i] to a great number of examples of dogs, [ii] to a smaller number of cats, and [iii] only to a very few number of foxes)

The point is that a conceptual space which functioned in this way would produce conceptual regions whose shapes are different from the ones produced by the standard Euclidean metric. Those shapes will be commonly non-convex, what contradicts the assumption with regard to the convexity constraint. The graphs in Fig. 3 contrast the boundaries of convex regions in the standard Euclidean space, with the boundaries of non-convex regions in a prototype-weighted Euclidean space.

Consequently, if concepts were weighted differently (depending on the sizes of their sets of examples) then, even within a Euclidean space, conceptual regions could be non-convex.²⁰ Of course, the foregoing requires empirical contrast via psychological research, which will have to decide whether concepts are different weighted or not. Nonetheless, at least from a theoretical perspective, the size of the set of examples from which a certain concept is learnt could influence the reliability of such a concept. On my view, this possibility is significant by itself, beyond the fact that at present there exists or not empirical evidence about it

Therefore, there are important reasons to think that not every concept has the same weight in the conceptual space structure. If this were indeed the case, then those distinct weights would lead to a non-standard Euclidean space, which would result in non-convex conceptual regions.

Fig. 4

Representation of the weight function $w_i =1+1/n_i$, that could underlie a non-standard multiplicatively-weighted Euclidean space

8 Inner problems of convexity in Gärdenfors’ theory

So far I have shown the following. [1] None of the arguments provided by Gärdenfors for the convexity constraint constitutes a compelling reason in favor of that requirement, given that all of them rest on controversial assumptions. [2] His argument for the integral character of conceptual dimensions (in support of a Euclidean metric) is weak; while under a non-Euclidean metric, the structure of regions can be non-convex. [3] Even if the metric were Euclidean the convexity constraint might be not satisfied; if, for example, distinct concepts were differently weighted in terms of the number of examples on which each of them is based.

However it could be the case that, despite all of this, conceptual regions are in fact convex (as assumed by Gärdenfors). In this section, I show that Gärdenfors’ convexity constraint is brought into question by his own characterization of conceptual spaces. On the one hand, I will prove that in some cases the regions associated with the properties of a concept are not convex (either taken individually, or as the result of their combination in that concept); while in other cases, the composition of convex regions associated with properties can lead to non-convex concepts (depending on how the properties co-vary over those concepts). On the other hand, I will show that Gärdenfors’ definition of properties in terms of convex regions is not compatible with his characterization of verbs as convex regions of vectors from one point to another.

Fig. 5

Inner form of the apple conceptual space, as a product space of different quality properties. The apple space is represented by the bigger rounded rectangle. Properties (such as red, green, epicycloid, etc.) are convex regions represented by the ellipses; for example, the property green corresponds to a convex region of the color space, or color domain. Quality domains (such as color, shape, taste, etc.) are represented by the smaller rounded rectangles (adapted from Fiorini et al. 2014, p. 132)

8.1 On the convexity of properties and concepts

One of the most recent papers co-authored by Gärdenfors (Fiorini et al. 2014) provides a detailed description, absent from previous work, of the inner workings of conceptual spaces. In that paper, Gärdenfors and collaborators represent the inner structure of the apple concept by the product space resulting from the properties in those quality domains that form such a conceptual space (as shown in Fig. 5).

Difficulty 1 Some of the properties are not convex.

This difficulty could be summed up as follows. There are non-convex physical properties, and it is not easy to conceive a convex approach for the representation of some of those non-convex properties. The first point is largely uncontroversial, given that the physical shape of many objects is not convex, as happens with the shape of an apple.

With regard to the shape properties, Gärdenfors proposes different models for representing them, suitable for different kinds of shapes. Nevertheless, none of them is proper for the characterization of general shapes and, in particular, for a convex representation of the shape of an apple, as it will be shown in the following points:

• (A) The approach followed to represent rectangles (Gärdenfors 2000, pp. 93–94; 2014, pp. 35–36) by the conditions satisfied by their quadruples of points in $R^{2}$ can only be applied to very basic geometrical shapes (not including the epicycloid).

• (B) The model proposed for the analysis of general shapes (Gärdenfors 2000, pp. 95–96; 2014, pp. 121–122), based on the work of Marr and Nishihara (1978), could be more or less applicable to the case of the shapes of animals, as a combination of cylinders (associated with their different parts) together with information about how those cylinders are joined, but not to the shapes of arbitrary objects.

• Therefore, neither of these two models allows us to represent the shape of an apple, distinguishing it from the shape of a lemon, pear or melon. And, although the second is useful for characterizing movements and actions, neither of them is compelling as a model for general shapes.

Fig. 6

Epicycloid curves representing the ideal two-dimensional (2D) contour of an apple. The apple’s ideal shape will be the three-dimensional (3D) surface resulting from the rotation of any of these curves around the horizontal axis. a Epicycloid curve with ratio $n =$ 1. b Epicycloid curve with ratio $n =$ 2

• (C) Thirdly, his approach to locative prepositions (Gärdenfors 2014, pp. 205–214) leads to an accurate formalization of the meaning of near, far, inside, outside, beside, etc. in terms of a polar coordinate system. In light of this, it seems that Gärdenfors’ aim is to transfer how these prepositions are applied to shapes in the physical world, to the shapes of their associated conceptual spaces.²¹ And the same can be said regarding to his description of the meaning of bumpy, as a structure in physical space constituted by “an even (but continuous) distribution of values on the vertical dimension of a horizontally extended object” (Gärdenfors 2014, p. 246). However, a direct translation of shapes from the physical space to a convex representation within a conceptual hyperspace is only possible if the shape of the considered object is convex. The problem is that the shape of many objects is not convex, as happens with the epicycloid ²² for the case of apples. An epicycloid is a plane curve generated by the path of a point on a smaller circle (with radius r) as that circle rolls around a larger fixed circle (with radius nr, where n is an integer). The epicycloid is given by the following parametric equations:

  $$\begin{aligned} x(\theta )=r(n+1)\cos \theta -r\cos [(n+1)\theta ] \\ y(\theta )=r(n+1)\sin \theta -r\sin [(n+1)\theta ] \end{aligned}$$

  An apple shape could be associated with an epicycloid with a value of n equal to 1 or 2 (see Fig. 6). Nonetheless, neither of these 2D curves (and consequently, none of their associated 3D surfaces) is convex, because in both cases it is possible to find pairs of points within the regions they bound such that some points between them do not belong to the same region.

Finally, this problem extends from the apple example to many other object categories whose shapes are not convex. And, although the fact that Gärdenfors is not able to provide a method for the convex representation of non-convex shapes (such as the shape of an apple) does not constitute a proof that no method exists, it is an evidence for the difficulty of conceiving a natural way of representing a non-convex shape by means of a convex conceptual region.

Difficulty 2 The conceptual region resulting from the combination of convex properties (belonging to the same domain) can be non-convex.

This characterization of the apple space sheds a great deal of light on how conceptual spaces are supposed to work internally, especially regarding the following point: different properties in the same domain can be associated with the same concept (for example, the properties red and green in the color domain, or sweet and sour in the taste domain, for the case of the apple concept).²³

Here the problem is that two properties from the same domain (red and green, for instance) cannot be composed into a product space. Let us recall here that the product space, R, of a set of constitutive properties $Q_{1}$, $Q_{2}$, ..., $Q_{n}$, is equal to the set of objects²⁴ belonging simultaneously to $Q_{1}$, $Q_{2}$, ..., and $Q_{n}$. For instance, if the apple space were constituted only by the shape and texture spaces previously shown, then a particular object would be an apple if it were epicycloid and smooth. Or, from a logical point of view, for an object to be categorized as an apple, it is necessary (but not sufficient) that the following conjunction of properties is satisfied over those quality domains:

$$\begin{aligned} (shape=\hbox {EPICYCLOID})\wedge (texture=\hbox {SMOOTH}) \end{aligned}$$

All this works if the properties considered belong to different domains. The problem is that when two (or more) properties belong to the very same domain, they cannot be composed into a product space, because in this case the product space would not include the desired set of objects. For instance, if we now included the color domain for the case of the apple concept, the resulting product space would have to satisfy the condition:

$$\begin{aligned} (color_1 =\hbox {RED})\wedge (color_2 =\hbox {GREEN})\wedge (shape=\hbox {EPICYCLOID})\wedge (texture=\hbox {SMOOTH}) \end{aligned}$$

This condition would certainly include all the red-and-green apples,²⁵ but not the green (but not red) apples, nor the red (but not green) apples. This is so because the kind of composition required when two or more properties belong to the very same domain is not their product space, but their addition space, that is, the region resulting from the union of the regions associated with those properties. This kind of composition could be identified with a logical disjunction, so the condition associated with the apple space could be better expressed (with a unique color dimension) as:

$$\begin{aligned}{}[(color=\hbox {RED})\vee (color=\hbox {GREEN})]\wedge (shape=\hbox {EPICYCLOID})\wedge (texture=\hbox {SMOOTH}) \end{aligned}$$

The problem is that the conceptual space resulting from the addition of the red and green properties is not convex, given that the orange color establishes a discontinuity between them; as is obvious from their representation in the color spindle (Fig. 7). Consequently, the resulting color space (associated with the apple concept) is not convex.

Fig. 7

Representation of the color spindle and its constitutive dimensions: hue, intensity and brightness. The relevant colors for the examples provided are denoted by their initials: R $=$ red, G/G$'$ $=$ green, O $=$ orange, Y $=$ yellow, W $=$ white, B $=$ black (adapted from Churchland 2005, p. 536). (Color figure online)

An even clearer case would be that associated with the swan conceptual space, which would be constituted (following Gärdenfors’ approach) by the product space resulting from a set of properties in the quality domains color, shape, etc. In this case, two different properties (black and white) are represented in the color domain. Those two properties should be represented by convex regions (in fact, points) within the color domain, but there is no path within the color spindle between them that only passes through points representing the colors of a swan. In this case, the combination of the black and white properties determines a disconnected region, so it cannot be convex (or even star-shaped).²⁶

Difficulty 3 The space resulting from the covariation of convex regions can be non-convex.

Even if a concept C is constituted by properties represented by convex regions (as Gärdenfors assumes), depending on how the properties of the instances (or particular cases) of C co-vary, the conceptual region R associated to C might be convex or not.

Let us now consider a concept C composed by two properties $F_{1}$ and $F_{2}$. If $F_{1}$ and $F_{2}$ were properties located in domains constituted by only one dimension ($d_{1}$ and $d_{2}$, respectively), then the region R representing C will be situated in a conceptual space composed by those two dimensions $d_{1}$ and $d_{2}$. For instance, this is the case of the mountain concept, whose properties (width and height) are represented by one-dimensional domains (Adams and Raubal 2009, p. 258; quoted in Gärdenfors 2014, p. 29).

Let us assume further that those properties $F_{1}$ and $F_{2}$ are represented by the regions (more specifically, intervals) $Q_{1}$ and $Q_{2}$ within the dimensions $d_{1}$ and $d_{2}$, respectively. For the case of the mountain concept, if $d_{1}$ and $d_{2}$ represent the width and height of the mountain, respectively, those regions could be given by the intervals $Q_{1}$ = (1500, 13000) and $Q_{2}$ = (1200, 8000) (all of them expressed in meters).

However, it could happen that C were represented by a region R which did not cover completely the Cartesian product $Q_{1}\times Q_{2}$ of its constituent properties. In that case, there will exist pairs of points $(q_1 ,q_2 )$ which belong to $Q_{1}\times Q_{2}$, but do not belong to R, that is, $\exists (q_1 ,q_2 )((q_1 \in Q_1 )\wedge (q_2 \in Q_2 )\wedge ((q_1 ,q_2 )\notin R))$. For instance, the mountain’s conceptual region might not be rectangular, but triangular (see Fig. 8a):

Fig. 8

Two different ways in which the covariation between the width and height dimensions (constitutive of the mountain concept) can happen. a Covariation resulting in a triangular conceptual region. b Covariation resulting in a triangle with a concave curve as the hypotenuse

If a formation is very high, its width will not matter much; it will still be a mountain. However, a lower and very wide formation might not be called a mountain. Thus, the region in the product space that represents mountain has more or less a triangular shape. (Gärdenfors 2014, p. 29)

In such a case, an upward projection of the earth’s surface whose width and height were given by the pair (9000, 4000) (represented as K in Fig. 8a), would belong to $Q_{1}\times Q_{2}$, but would not be called a mountain.

Withal, if the conceptual space associated with mountain is triangular, the convexity of the mountain’s conceptual region is guaranteed.

Nonetheless, if the region R (representing C) did not cover completely the Cartesian product $Q_{1}\times Q_{2}$ of the constituent properties of C, it could happen that R is not even convex. For instance, in Adams and Raubal’s example the hypotenuse of the triangle which delimits the conceptual region of mountain may not be a straight line, but a concave curve (as shown in Fig. 8b). Therefore, depending on how the properties of a concept C co-vary, its conceptual region will be convex or not.

8.2 On the compatibility of the convexity of properties and verbs

Lastly, when Gärdenfors extends his conceptual space theory to the semantics of verbs, such an extended framework introduces a general problem (which could be described as structural). It is associated with his definition of verbs, and closely related to his basic conception of property. In this case, the problem is that his characterization of verb meanings, as vectors from one point to another, is not compatible with a definition of properties and concepts in terms of convex regions.

In his extended theory, Gärdenfors identifies states and changes with zero and non-zero vectors; and based on them he defines events as changes in the state of a patient (usually due to the action of an agent). The problem is that, strictly speaking, states and changes cannot be identified with points and single vectors, respectively, if properties are not represented by points, but by (allegedly) convex regions. In virtue of this, states should be represented by regions; and changes therefore ought to be represented by sets of vectors from every point in the region associated with the initial property, to every point in the region associated with the final property.²⁷

The same can be said with regard to the result vectors associated with a given verb. Gärdenfors defines a verb as a change in the properties of an object; that is, as a movement in its representation within the conceptual space. Based on this, such a change is represented by means of a vector from the position of the initial object to that of the final object. However, and given that a state is, in fact, not represented by a point (but by the region associated with the property described by such a state), a verb cannot be represented merely as a vector (or a mapping from one point to another), but must be represented as a mapping from one region to another.

Thus, a verb should be represented not by a vector (or convex set of paths, with only one origin and one endpoint), but by a set whose elements are convex sets of paths (each with a different origin and/or end).

Obviously, here I have not proved that those sets of vectors which should represent verbs cannot be convex. To my knowledge, it is hard to see in which sense it may be said that a set C constituted by all the pairs of points (x, y) such that, $x\in X$ and $y\in Y$, where X and Y are convex sets, is convex. Notwithstanding, the burden of proof lies on the side of Gärdenfors. If he wants (i) to provide a unifying framework for the semantics of verbs, nouns and adjectives (Warglien et al. 2012), and (ii) to explain the semantics of verbs by changes of states (ib.), he has to prove that a set C as the one just described is convex.

9 Conclusions

One of the main theses of Gärdenfors’ (2000, 2014) conceptual space theory is the convexity constraint on the geometry of the conceptual regions associated with properties, concepts (or object categories), actions, verbs, prepositions and adverbs. Nonetheless, in this paper I have shown that such a convexity constraint is problematic; both from a theoretical perspective, and with regard to the inner workings of the theory itself.

On the one hand, I have shown that none of Gärdenfors’ arguments in favor of the convexity requirement compels us to accept it as a mandatory criterion for the geometry of regions. [1] With regard to his first argument, everything that can be said concerning the co-implication of the prototype theory and the convexity of regions, could also be said regarding the star-shapedness constraint on regions. [2] In relation to the cognitive economy argument, it depends on the controversial assumption that handling convex regions requires fewer computational resources than handling regions with capricious forms. [3] Regarding the perceptual foundation argument, it relied on the hypothesis that perceptual and conceptual domains share the same geometrical structure, which might not be the case. [4] Finally, his argument concerning a more effective communication is based on studies that assume the standard Euclidean metric, but such an assumption is far from trivial.

On the other hand, and with regard to the kind of metric underlying conceptual spaces, under the standard Euclidean metric assumed by Gärdenfors (that is, Euclidean and non-weighted distances), the convexity of regions would indeed be guaranteed. However, the question regarding the type of metric that can underlie conceptual spaces is an empirical one; and all of the evidence provided by Gärdenfors in support of the standard Euclidean metric is controversial. Firstly, the Euclidean metric cannot be based on the integral character of domains, requiring empirical evidence independent from the one associated to the non-separability of dimensions. Secondly, the empirical evidence referred to in favor of the integral character of domains (and, in consequence, in favor of the Euclidean metric and the convexity of regions) comes from a very small number of perceptual domains; things might not work in exactly the same way in other perceptual and in non-perceptual domains. Thirdly, none of the work cited identifies integral domains with the Euclidean metric perfectly, but rather with a metric that is more similar to the Euclidean than to the city-block; and such a kind of metric does not lead to convex conceptual regions. Due to all of this, if the metric underlying conceptual spaces were standard, it may be that it would not be Euclidean in a strong sense; and in that case, it has been shown that the convexity constraint on regions is not valid.

Additionally, it has been proved that, even if the metric underlying conceptual spaces were Euclidean, regions could be non-convex if the distances-of-comparison in categorizations were differently weighted; depending, for example, on the number of examples on which each concept is based. That is, convexity is guaranteed only under the standard Euclidean metric: not under a weighted Euclidean metric. The problem is that, even if the psychological space is Euclidean, there are good reasons in favor of a non-standard multiplicatively-weighted determination of distances; under which, conceptual regions could be non-convex.

Finally, even if none of the above problematic possibilities were the case, Gärdenfors’ convexity constraint is brought into question by his own characterization of the inner workings of conceptual spaces. The problems could be summed up as follows. [I] Some of the allegedly convex properties of concepts are not convex, as happens with those associated with the shape domain, and it is not clear how they could be represented in a convex way. [II] The conceptual region resulting from the combination of two (or more) convex properties belonging to the same domain can be non-convex, and the same happens for its associated concept. [III] The space resulting from the covariation of different convex regions could be non-convex; as a theoretical possibility that needs for more empirical research. [IV] Gärdenfors definition of verbs, as vectors from one point to another, is not compatible with a definition of properties and concepts in terms of convex regions.

Based on all this, I conclude that the mandatory character of the convexity requirement for regions in any similarity space theory of concepts (and so in Gärdenfors’ conceptual spaces) should be reconsidered, in favor of a weaker constraint, such as a non-obligatory version of the star-shapedness constraint. Notwithstanding, this paper should not be seen as a defense of a mandatory star-shapedness requirement, given that such a constraint has its own problems, most of which are not mentioned here.²⁸