Modelling Comparative Concepts in Conceptual Spaces

Transcription

1 Modelling Comparative Concepts in Conceptual Spaces Lieven Decock 1, Richard Dietz 2, and Igor Douven 3 1 Faculty of Philosophy, VU University Amsterdam 2 Department of Philosophy, University of Tokyo 3 Faculty of Philosophy, University of Groningen l.b.decock@vu.nl, rdietz@l.u-tokyo.ac.jp, i.e.j.douven@rug.nl Abstract. The conceptual spaces approach has emerged as a new and powerful way of thinking about concepts. In earlier work, the present authors have addressed the question of how to model vague concepts in the conceptual spaces framework. That in turn was instrumental in Decock s and Douven s account of a graded membership relation in further work. In this paper, we make use of the account of graded membership to present a way of modelling comparative concepts in conceptual spaces. Finally, Dietz alternative account of comparative concepts is contrasted with the presented Decock Douven type account. Keywords: comparative concepts, conceptual spaces, graded membership, Voronoi diagrams. Over the past fifteen years or so, the conceptual spaces approach has emerged as a new and powerful way of thinking about concepts. The single most characteristic feature of this approach is that it represents concepts geometrically, as regions of metrical spaces. This has brought unprecedented precision to the study of concepts. The approach also enjoys considerable empirical support. Given that the conceptual spaces approach is a relative newcomer, it will not be surprising to learn that the approach is still very much in a state of flux. While its basic machinery is more or less in place, much work is still expended on refining and enriching that machinery, mostly with an eye toward widening the scope of the approach. For it is generally recognized that, at least in its basic form, the conceptual spaces approach can only deal with a very limited range of concepts. Indeed, in that form the approach may well be restricted to concepts closely tied to sensory perception, like color concepts and auditory concepts. Moreover, in its basic form it may not even be able to represent perceptual concepts in a completely adequate manner, for in that form, concepts must be represented as having sharp boundaries, which is clearly false for many, possibly even most, concepts. However, Gärdenfors and various co-authors have recently made progress in extending the framework to other than perceptual concepts; see Gärdenfors [2007], Gärdenfors and Warglien [2012], and Gärdenfors and Zenker [2011], [2012]. And Douven et al. [2013] propose certain emendations to the conceptual spaces framework to accommodate the fact that concept boundaries can be vague. Y. Motomura, Y. Butler, and D. Bekki (Eds.): JSAI-isAI 2012, LNAI 7856, pp , c Springer-Verlag Berlin Heidelberg 2013

2 70 L. Decock, R. Dietz and I. Douven But there is still more work to be done if the conceptual spaces approach is to offer a general account of concepts. Among other things, it is not obvious how to model comparative concepts with the help of conceptual spaces. Our aim in the present paper is to address this issue. We present a way of modelling the main type of comparative concepts in terms of a recent theory of graded membership proposed by Decock and Douven [2013]. For another, we discuss an alternative conceptual spaces account of comparative concepts that is developed in Dietz [2013]. To begin, we briefly describe the basic features of the conceptual spaces framework as well as some recent add-ons of the framework. 1 The Conceptual Spaces Framework The most basic idea underlying the conceptual spaces approach is that concepts can be modelled by means of metrical spaces. These spaces are one-dimensional or multi-dimensional structures whose dimensions correspond to fundamental qualities in terms of which objects may be compared with each other. Objects are mapped onto points in these spaces, and the distance between the representations of two objects in a given space is supposed to be inversely proportional to the similarity between the objects in the respect modelled by the space. This may be made clearer by considering the example of color space. This space is generally taken to be three-dimensional, with one dimension representing hue think of a color circle with yellow, green, blue, violet, red, and orange (neighboring yellow again) lying in that order on the circle one dimension representing saturation the intensity of the color and one representing brightness, which ranges from white to black, through all shades of gray. To say that objects a and b are more similar to each other as far as their colors are concerned than objects c and d is to say, in terms of the conceptual spaces approach, that the representations of a and b in color space are closer to each other than are the representations of c and d in the same space, where closeness is measured by means of the metric defined on color space. Like most of the other better-known conceptual spaces, color space is equipped with a Euclidean metric. Gärdenfors, who has contributed to the development of the conceptual spaces approach as much as anyone, has shown that one obtains a particularly elegant and economical account of categorization by combining conceptual spaces with prototype theory and the mathematical technique of Voronoi diagrams. According to prototype theory, among the members of a category, some are more representative of the category than others. The most representative members are called the prototypes of the category. 1 Prototypes tend to play various special roles in our cognitive lives, the most important one probably being in learning the use of category nouns: in teaching a child the use of such a word, we mostly point it at prototypical instances of the category designated by the word. Voronoi diagrams offer a way of carving up metrical spaces. More precisely, a Voronoi diagram divides a metrical space into cells such that each cell has a center and further contains all and only those points that lie no closer to the center of any 1 For more on prototype theory, see Murphy [2002].

3 Modelling Comparative Concepts in Conceptual Spaces 71 other cell than to its own center. Given an m-dimensional space S with associated metric δ S and a sequence p 1,...,p n of pairwise distinct points in S, wesaythat the region v(p i ) := { p δ S (p, p i ) δ S (p, p j ), for all j {1,...,n} with j i } is the Voronoi polygon/polyhedron associated with p i. Together the elements of {v(p i )} 1 i n constitute the Voronoi diagram generated by p 1,...,p n. 2 To see how this combines with prototype theory to yield an account of categorization, let the points representing prototypes in a space serve as the centers of a Voronoi diagram. This divides the space into separate regions, each of which represents the concept whose prototype is the center of the corresponding Voronoi polygon/polyhedron. Given that all Voronoi polygons/polyhedrons of Voronoi diagrams based on a Euclidean metric are convex (Okabe et al. [2000:58]), concepts representable in Euclidean metrical spaces the vast majority of concepts that have so far been studied in the conceptual spaces approach automatically come out as convex regions. This is important, given that, for those concepts, convexity is supported by both empirical and theoretical considerations. All this pertains to categorical concepts, such as green and salty, which divide a class of individuals on the basis of the presence or absence in those individuals of a given property. However, these concepts do not exhaust the class of concepts. Next to categorical concepts, there are comparative concepts, such as greener than and saltier than, which order a class of individuals with respect to a given property. How are we to model these in the framework of the conceptual spaces approach? Fig. 1. Point a is closer to prototypical point p than point b is, yet b does, but a does not, fall under the concept of which p represents the prototype At first, the answer might seem easy, namely: by explicating i is more C than i is as i is closer to the C prototype than i is. Unfortunately, this cannot hold generally for comparative concepts. For consider Figure 1, which gives an example of a Voronoi diagram of a bounded two-dimensional Euclidean space. In this figure, point a is clearly closer than point b to prototypical point p. Yet while b falls 2 See Okabe et al. [2000, Ch. 2] for a detailed presentation of the technique of Voronoi diagrams.

4 72 L. Decock, R. Dietz and I. Douven under the concept of which p is the prototype, a does not. And, whichever other constraints one may wish to place on an account of comparative concepts, one will certainly want such an account to validate the following monotonicity principle for comparative concepts C-er and corresponding categorical concepts C: (MP) If x is C-er than y, thenify is C, thenx is C as well. Clearly, the current proposal does not validate (MP): in Figure 1, let C designate the concept that has p as its prototype; then a is more C than b is, yet b is C while a is not, contradicting (MP). In the following, we argue that an account of graded membership proposed in Decock and Douven [2013] offers a way of modelling comparative concepts that does validate (MP). 2 Graded Membership The account of graded membership to be used capitalizes on Douven et al. s [2013] proposal of how to model vague concepts in conceptual spaces. We start by summarizing this proposal. It has frequently been remarked that most natural language predicates are vague. Vagueness of predicates is standardly taken to consist in the existence of borderline cases, cases that, colloquially put, neither seem to belong clearly to a given category nor seem to belong clearly not to that category. For example, a given color shade may strike us as being not quite green nor quite blue but as something in between green and blue, or as being both green to some extent and blue, to some extent. To state in general and precise terms what a borderline case is, is regarded as a central question in the debate about vagueness. The version of the conceptual spaces approach we are considering the version with prototypes and Voronoi diagrams suggests a rather straightforward answer to this question: borderline cases of a concept C are cases that are represented by a point that lies as far from the C prototype as it lies from at least one other prototypical point in the relevant space. But this answer cannot be quite right. For surely there can be, say, red/orange borderline cases such that any very small change in their color would again result in a red/orange borderline case. However, this could not happen if borderlines were really just one point thick, as they are in the picture at issue. Almost all small changes as small as one likes will result either in a clear case of red or in a clear case of orange. In this picture, after all, all borderline cases are immediately adjacent to non-borderline cases (although they are also adjacent to some borderline cases). To overcome this problem, Douven et al. [2013] propose an extension of the conceptual spaces approach that yields conceptual spaces carved up by diagrams with thick borderlines. In a first step, they observe that concepts need not have unique prototypes. For instance, there is not just one shade of red that will strike us as being typically red (Berlin and Kay [1969/1999] provide empirical support for this claim). Thus, Douven et al. assume that conceptual spaces may have prototypical regions rather than prototypical points. Then, in the second

5 Modelling Comparative Concepts in Conceptual Spaces 73 step, they propose an extension of the technique of Voronoi diagrams to go with the idea of prototypical regions. For details, we refer the reader to Douven et al. s paper. Here, we confine ourselves to giving an informal characterization of this technique. The basic idea is the following. Given a conceptual space S with prototypical regions r 1,...,r n,consideralln-tuples of points that pick precisely one point from each r i.eachsuchn-tuple defines a Voronoi diagram on S. Now take the set of all those diagrams and project them onto each other. That gives what Douven et al. call the collated Voronoi diagram on S. It can be proved that if each r i is connected (in the topological sense), then the boundary region of the collated Voronoi diagram is full, that is, it does not contain holes (this is made precise in Douven et al. [2013], but see Figure 2 for a suggestive illustration). This means that, given fairly weak conditions, each concept that is naturally thought of as permitting of some vagueness will have borderline cases which are fully surrounded by other borderline cases. This is enough to solve the problem described in the previous paragraph. As an aside, we mention that these amendments to the standard conceptual spaces framework do not by themselves make it any easier to define comparative concepts simply in terms of distances. The standard way to measure the distance of a point from an area makes use of the so-called Hausdorff metric, which in its most general form measures distances between two sets of points. For present concerns, we only need to consider this metric insofar as it measures the distance of a point p from a set T of points in a space S with associated metric δ S.For these purposes, the metric can be defined as follows: h S (p, T ):=inf { δ S (p, x) x T } ; informally put, in a Euclidean space, the Hausdorff distance between a point and a region is given by the length of the shortest line connecting the point with the region. Letting δ be a Euclidean metric defined on the space represented in Figure 2, it is evident just by looking at that figure that h(a, r) < h(b, r). Yet, here too, b falls under the concept associated with r while a does not. Again, we have a violation of (MP). To arrive at a more adequate model of comparative concepts, we rely on a further extension of the conceptual spaces framework, to wit, on the account of graded membership offered in Decock and Douven [2013]. This account embeds a proposal by Kamp and Partee [1995] into the version of the conceptual spaces approach with prototypical regions and collated Voronoi diagrams. Kamp and Partee s proposal is an attempt to formulate a semantics for vague terms. They start by considering a language with simple predicates (roughly, predicates that have monolexemic expressions in English) which may be vague. For this language, they define a partial model which consists of a universe of discourse (a set of individuals) and an interpretation function which, for each predicate in the language, divides the universe of discourse into three parts: one part that contains the clear instances of the predicate (the positive extension), one that contains the clear non-instances of the predicate (the negative extension), and one that contains the remaining objects (if this part is empty, the predicate is crisp). Correspondingly, and in the obvious way, they define a partial truth predicate for this language.

6 74 L. Decock, R. Dietz and I. Douven Fig. 2. A two-dimensional collated Voronoi diagram with point a being closer to r than point b, while b, but not a, falls under the concept associated with r The partial model is supplemented with a class of completions, which are ways to eliminate truth value gaps in the partial model by splitting up, for each predicate, the set of individuals that belong neither to the predicate s clear instances nor to its clear non-instances. Importantly, they do not consider all possible ways of splitting up these non-clear instances, but only the ones which respect typicality rankings, which means informally put that if a completion groups a non-clear instance of C with the clear C instances, then it should not group anything that is at least as similar to the prototype of C with the clear non-instances of C. Their idea then is to let the degree to which an individual i falls under a given concept be given by the proportion of completions that group i with the clear instances of the concept. As Kamp and Partee admit, however, the constraint to respect typicality rankings is, even in combination with plausible formal constraints, not strong enough to ensure uniqueness of the membership function. They do not see how to arrive at a unique membership function without making arbitrary decisions. Decock and Douven show that, by embedding Kamp and Partee s proposal in a conceptual spaces framework, the geometry of conceptual spaces will suffice to complete the proposal in a principled manner. The upshot is a unique membership function (that is, a unique membership function for each concept). The crucial step forward that the conceptual spaces framework allows one to take is that the key ingredients of Kamp and Partee s proposal partial models and completions can both be construed as geometrical objects which can themselves be represented in a space of the right dimensionality. A measure on that space then yields the unique membership function. The following explains this in a bit more detail. Given a conceptual space S with set {r 1,...,r n } of prototypical regions, Douven et al. [2013] distinguish between the restricted and the expanded collated Voronoi polygon/polyhedron associated with a given r i. These notions receive formal definitions in Douven et al. [2013], but informally put, the former consists of the points that are closer to all points in r i than they are to any point in r j, for all j i, while the latter consists of the restricted collated Voronoi polygon/polyhedron together with the points that are as close to some point

7 Modelling Comparative Concepts in Conceptual Spaces 75 in at least one of the r j (j i) as they are to some point in r i.thesettheoretic difference between the restricted and the expanded collated Voronoi polygon/polyhedron associated with r i is the boundary region of r i. In these terms, we can let a conceptual space with corresponding collated Voronoi diagram define a partial model for the relevant part of the language (e.g., color words if the space is color space). Specifically, we can let restricted collated Voronoi polygons/polyhedrons play the part of Kamp and Partee s positive extensions, the complements of expanded collated Voronoi polygons/polyhedrons play the part of their negative extensions, and the boundary regions of the prototypical regions play the part containing the indeterminate instances. The role of completions is played by the n-tuples that generate the Voronoi diagrams which together make up the collated Voronoi diagram. A moment s reflection suffices to see that a (simple) Voronoi diagram splits the boundary region of any concept represented in the relevant space into two parts: one part containing the borderline cases that are grouped with the clear cases and the other part containing the remaining borderline cases. Given that Voronoi diagrams are functions of similarity rankings, the splitting-up of the boundary region is clearly in the spirit of Kamp and Partee s proposal. It is now rather straightforward to implement in the current setting Kamp and Partee s idea of determining graded membership in terms of proportions of completions. The measure is most easily introduced by first considering a space whose prototypical regions consist of only finitely many points. For in this case the degree to which an individual falls under a concept represented in the space simply amounts to the ratio between the n-tuples that generate Voronoi diagrams in which the individual is grouped with the clear cases and the total number of n-tuples. However, we are not aware of any concepts that could be realistically modelled by means of this kind of space. So, we will have to consider spaces whose prototypical regions consist of infinitely many points. Decock and Douven generalize the measure for Euclidean spaces with prototypical regions with finitely many points to a measure that pertains to all Euclidean conceptual spaces by exploiting the fact that completions thought of now as n-tuples generating Voronoi diagrams can themselves be represented as points in a space. Specifically, given an m-dimensional conceptual space with n prototypical regions, each completion can be represented as a point in an m n-dimensional space. Then the degree to which an individual falls under a concept is defined to equal the volume of the set of points representing completions that group the individual with the clear instances of the concept relative to the volume of the total m n-dimensional space. As Decock and Douven show, the resulting membership functions of the n concepts represented in the m-dimensional space have a number of attractive features. For instance, they are in a clear sense S-shaped, which is in accordance with experimental data on membership functions (see Hampton [2007]). Also because of their shape, these functions account nicely for the phenomenon of higher-order vagueness, that is, the fact that there seem to be no sharp transitions from clear cases to borderline cases and from borderline cases to clear

8 76 L. Decock, R. Dietz and I. Douven non-cases. A virtue not mentioned in Decock and Douven [2013] is that these membership functions can also serve to give a semantics for comparative concepts, as we now want to show. 3 Modelling Comparative Concepts The basic clause of our proposal is entirely straightforward. Let M C designate the membership function for concept C. Then (CC) For all individuals i and i and all comparative concepts C-er than and corresponding categorical concepts C, i is C-er than i iff M C (i) >M C (i ). Supposing that i may be said to fall under the concept C iff M C (i) > 0, (CC) validates (MP): because the range of any membership function is the [0, 1] interval, it follows from (CC) that M C (i) > 0ifM C (i) > M C (i ). Also note that this approach to the semantics of comparatives swiftly extends to kindred concepts, like (what are sometimes called) equatives and weak comparative concepts, to wit, by defining that i and i are equally C (or that i is as C as i ) iff M C (i) =M C (i ), and also that i is at least as C as i iff M C (i) M C (i ). Although not directly related to the issue of comparative concepts, it is further worth noting that graded membership can also be used to define semantics for expressions stating or denying determinateness: i is determinately (definitely, fully) C iff M C (i) =1;i is determinately not C iff M C (i) = 0; and i is indeterminately C iff 0 <M C (i) < 1. Indeed, we can even define such locutions as i is twice as C as i which is true iff M C (i) =2M C (i ) but, obviously, not all features of our model need have psychological reality. 3 Although straightforward, this proposal may be expected to raise some concerns. For one, it has been argued, plausibly we think, that not all concepts have prototypes. For example, it is unclear what the prototype of old, orof tall,orofcheap could be, even if we take these concepts to be relativized to a certain class of individuals (e.g., it is not even clear what the prototype of tall is when this concept is relativized to the class of basket ball players). Because our account of comparative concepts relies on prototype theory, it is limited to comparative concepts whose associated categorical concept has one or more prototypes. As far as we can see, however, this is hardly a drawback of our proposal. At least, we are unable to think of any concept lacking a prototype that is not representable in a one-dimensional space. And it seems that such concepts give rise to comparative concepts whose semantics can be modelled purely in terms of locations in the relevant space. For instance, age, height, price, all being measured on an interval scale, the semantics of older than, taller than, and cheaper than can simply be stated in terms of the < relation. 3 Though it does seem psychologically realistic to model locutions like i is much C-er than i asm C(i) θm C(i ), for some threshold value θ (which may have to be different for different concepts, and possibly also for different contexts).

9 Modelling Comparative Concepts in Conceptual Spaces 77 For another concern, it is an immediate consequence of our proposal that if individuals i and i are both fully C, then neither of them can be C-er than the other; the same holds if they are fully not C. But might one not compare, say, crimson and vermillion with respect to their redness? Is vermillion not redder than crimson, even if both are determinately red? Equally, is an orange cup not redder than a purple cup, even if both are determinately not red? In response to this, we start by noting that something may be 100 % C, or fully C, without being a prototype of C. For instance, an albino tiger is 100 % a tiger, but it is not a typical tiger. Similarly, a flamingo is 100 % a bird, but it is not a typical bird. Further note that to claim that of two things which are both 100 % C, one is more C than the other, sounds incomprehensible, and would invite the question of whether the thing that is more C than the other is (say) 110 % C. So,ifi is C-er than i,thenatmosti can be 100 % C. 4 Now, while neither crimson nor vermillion is prototypically red, both would seem to be 100 % red. Thus, it makes little sense to claim that one of these colors is redder than the other. This is not to say that we cannot compare them with respect to redness. It would make perfect sense to assert that vermillion is more typically red than crimson. More generally, to compare two things both of which are fully C with regard to C, the comparison must be in terms of how typically C they are. Something very similar holds for comparisons of things that are determinately not C. Surely, to say of such things that one is C-er than the other is to say that the second is less C than the first, which seems to warrant the question whether the first is perhaps 10 % C. Insofar as it still makes sense to compare such things with regard to C, we believe the proper expression must be of the form a is more (or less) C-ish than b. For instance, it might be said that an orange cup is more reddish than a purple cup, while the latter is more bluish than the former. Naturally, instances of the schematic forms a is more typically C than b and a is more C-ish than b refer to comparative concepts as much as those of a is C-er than b do. If our aim is to model comparative concepts generally, we must also account for comparative concepts of the former type. This task seems relatively easy, however, for here the Hausdorff metric seems to be all we need. That is to say, we can define both a is more typically C than b and a is more C-ish than b to hold iff the Hausdorff distance from a to the prototypical C 4 This is true for the kind of concepts that our proposal is meant to pertain to, namely, concepts that have prototypes. At least, we cannot think of any counterexamples to the claim that comparing two things with respect to C implies that they are to some extent C if these counterexamples are to involve concepts that have prototypes. We suggested earlier that concepts that lack prototypes are ones that can be represented in a one-dimensional space. For such concepts, or at least for some of them, the said claim is false. For instance, Mary may be younger than Kate without either Mary or Kate being young. Similarly, Kevin may be shorter than Paul while both are tall. As mentioned, however, the comparative concepts that go with concepts such as young and short may have an entirely straightforward semantics in terms of locations in the appropriate conceptual space.

10 78 L. Decock, R. Dietz and I. Douven region is smaller than the Hausdorff distance from b to that region. Obviously, there can be no problem here that one individual may be closer to a prototypical region than another individual, yet the latter belongs to the category associated with the region while the former does not. In the present proposal, the Hausdorff metric is applied to individuals that either both belong to the clear instances of the category to the prototypical region of which their distance is measured, or to the clear non-instances of that category. Also note that, in this proposal, all C prototypes are equally typically C. After all, the Hausdorff distance of any point in a given prototypical region to that region itself is 0. Surely that is in accordance with pre-theoretical intuition. Could one not use the more typical and more -ish types of construction also to compare borderline cases of the concept at issue with one another, or with a clear instance or non-instance of the concept? Perhaps one could. But we venture that a speaker that did use them for any such purpose would be unable to tell any difference in meaning between either of the said constructions and the -er than construction. Hence, when used in any of the said ways, the more typical and -ish types of construction may not call for a semantics different from the semantics for the -er than type of construction. Summing up thus far, our semantics of comparative concepts distinguishes between three types of comparative concepts: those expressed by locutions of the form a is C-er than b ; those expressed by locutions of the form a is more typically C than b ; and those expressed by locutions of the form a is more C- ish than b. The second and third type make sense for comparing either clear C instances or clear C non-instances, and probably make sense only for these cases unless they are understood as having the same meaning as the corresponding comparative concept of the first type. The semantics of the second and third type of comparative concepts is given in terms of differences in the Hausdorff distance to the prototypical C region. The first type of comparative concept makes sense only for comparing individuals at least one of which is a borderline C case. If there are prototypical C cases, then the semantics of this type of concepts is given in terms of graded membership, understood in the manner of Decock and Douven [2013]. If C has no prototypes, the semantics is given by the ordering relation induced by the one-dimensional space in which the associated categorical concept is represented. 5 A worry one may have about this semantics that unlike the worries considered above does not concern its scope is that the semantics leaves no room for verdicts to the effect that i is C-er than i to be vague, as it seems they can 5 Note that it would seem to make no sense to assert that a is more typically C than b if there are no typical C cases to begin with. It would also strike us as positively odd if someone were to assert that Mary is more oldish than Kate, or that John is more tallish than Harry. And if someone were to assert that, we suppose that he or she would be unable to explain in which way what he or she asserted differed in meaning from Mary is older than Kate or John is taller than Harry. So, if one does want to use -ish comparative constructions involving categorical concepts lacking prototypes, we propose that their semantics does not differ from the semantics for the more standard comparative constructions involving these concepts.

11 Modelling Comparative Concepts in Conceptual Spaces 79 be. After all, either M C (i) >M C (i ), or M C (i) <M C (i ), or M C (i) =M C (i ). As explained in Gärdenfors [2000], however, there attaches some inherent uncertainty to psychological metrics, in particular to the similarity measures that are at the root of the current proposal. It is not as though we had perfect access to such measures. Accordingly, there attaches some inherent uncertainty to the membership functions of the concepts we use. To sense that it is vague whether i is C-er than i may just manifest that the M C values of i and i are so close that we are unable to discern which is greater than the other, or whether they are perhaps equal. Even if there is some uncertainty attaching to psychological metrics, this uncertainty is small enough for our proposal to have clear empirical content. Of course, especially insofar as tokens of the more -ish than type of construction are concerned, we may be dealing with a part of discourse that is not very well regimented. Usage of such tokens is very rare; we ourselves are unsure whether we ever used them in daily life (it is hard to think of something that might give one a reason to compare an orange and a purple cup with respect to their redness). Moreover, the present authors are presumably no exceptions in (as far as they recall) not having received any tutoring from their educators regarding when to use this type of construction and when not to do so. That being said, however, it should not be so hard to put the semantics to the test at least insofar as it pertains to the more standard, -er than type of construction. For instance, from the locations of various pairs of borderline color shades in color space it should be possible to derive whether or not someone would agree that one member of the pair is (say) redder than the other one. However, checking this and other empirical consequences of our account must be left to another occasion. Finally, the outlined account of comparatives concepts deriving from Decock and Douven [2013] is compared with Dietz [2013], which is, to our knowledge, the only alternative conceptual spaces account of comparative concepts that has been proposed so far. 4 An Alternative Approach Dietz approach may be best understood as a way of generalizing the conceptual spaces account of Douven et al. [2013] one step further. In Douven et al. [2013], Gärdenfors idea of modelling categorical concepts in terms of Voronoi diagrams is generalized for the case that prototypes form extended areas in a space. On the more refined categorization rule of collated Voronoi diagram (suggested by Douven et al.), distances to the prototype of a relevant concept and distances to prototypes of competing concepts still factor equally into categorization. In Dietz [2013], the idea of collated Voronoi diagrams is furthermore generalized in a way that allows distances to prototypes to receive different weights. To be more specific, consider a conceptual space S with a set R = {r 1,...,r n } of prototypical regions. Then for any distribution of points P = {p 1,...,p n } with p i r i for any 0 i n, the Voronoi region associated with p i is the

12 80 L. Decock, R. Dietz and I. Douven set of all points whose distance to p i is not larger than their distance to any point from P that is distinct from p i. On Dietz account, the basic notion of a Voronoi region associated with a prototypical point p is relativized to weights (or scaling factors) taken from the unit interval [0, 1] where the weight for the distance to the distinguished point p and the weight for the distance to alternative prototypical points add up to one. Specifically, for any distribution of points P = {p 1,...,p n } with p i r i for any 0 i n, the graded Voronoi region associated with p i and a weight λ is the set of all points whose distance to p i scaled by the factor λ is not larger than their distance scaled by the factor 1 λ to any point from P that is distinct from p i. By collating graded Voronoi regions associated points of a particular region and with a particular weight (analogously to collations of ungraded Voronoi regions as proposed in Douven et al. [2013]), one receives accordingly a graded collated notion of a Voronoi region associated with a prototype region and a weight. 6 The graded collated diagram generated by R relative to a weight λ is accordingly the set of disjoint collated Voronoi regions associated with r i and λ, for0 i n. AsDietz observes, we can interpret collated Voronoi diagrams as encoding the following categorization rule: a point falls into the category of the prototype area the supremum distance to which is still not larger than the infimum distance to the union of other prototype areas. Graded collated Voronoi diagrams accordingly encode a categorization rule of the following form: a point falls into the category of the prototype area associated with a weight λ the supremum distance scaled by λ to which is not larger than its infimum distance scaled by 1 λ to the union of prototypical areas distinct from r. 7 It can be shown that graded collated Voronoi regions are in general nested, in the following sense: Let M,d be any metric space with a distribution of prototypical areas R = {r 1,...,r n }. For any r R, and any weights λ and λ [0, 1], let u(r, R, λ) andu(r, R, λ ) be the graded collated Voronoi regions associated with r and λ and with r and λ respectively. Then if λ>λ,then u(r, R, λ) u(r, R, λ ). The nestedness property of graded collated Voronoi regions suggests modelling comparative concepts in terms of the ordering relation of set-theoretic inclusion between graded collated Voronoi regions. For instance, suppose C is the concept associated with a prototype area r from a set of prototype areas R. Then the suggestion is to say that a point x is C-er than y just in case for some weight λ, x u(r, R, λ), but y u(r, R, λ). 8 6 One may wonder whether there is any difference between collating Voronoi regions that are relativized to a weight or, conversely, relativizing a collated Voronoi region to a weight. In fact, these two methods come to the same. With view to this, it is not misleading to speak of graded collated Voronoi regions here. 7 For the equivalence results in support of these reinterpretations, see Dietz [2013, Sect. 3]. 8 In words, for some weight λ, x s supremum distance to r scaled by λ is still not larger than the corresponding infimum distance scaled by 1 λ to the union of other prototype areas, while for y, the corresponding supremum distance is larger than the corresponding infimum distance.

13 Modelling Comparative Concepts in Conceptual Spaces 81 Let us have a closer look at the properties of comparative concepts in Dietz s sense and check them against the properties of comparative concepts in the above suggested sense, in the line of Decock and Douven [2013]. 9 It was noted that on the account that derives from Decock and Douven [2013], a comparative concept C-er can only apply to a pair of objects x and y if at least one object falls into the borderline area for C-ness; for, otherwise, x agrees with y in degree of membership, which implies, according to the Decock Douven account, that it cannot be C-er than y. Speaking more generally, on this account, clearly true (false) instances of a concept C are maximal (minimal) elements of the domain that is partially ordered by C-er (that is, they are not smaller (larger) than any other element in C-ness). On Dietz account, by contrast, this constraint is not valid. For example, take a distribution of prototypical regions R = {r 1,...,r 3 } in the Euclidean space of reals R, wherer 1 =[0, 1], r 2 =[3, 4], and r 3 =[7, 8]. Let B, C, andd be the collated Voronoi regions associated with r 1, r 2,andr 3 respectively that is, the areas which (according to Douven et al. [2013]) represent the clear instances of the concepts associated with these prototypical areas. Then if x =3andy =4,x and y are both elements of the prototype area r 2 and also clearly true instances of the concept C. However, on Dietz account, it follows that y is C-er than x. 10 On the face of it, it may be regarded as an advantage of Dietz account that it allows for a unified account of comparisons for cases where either at least one object is a borderline case and other cases where no object is a borderline case. For whereas on the account along the lines of Decock and Douven [2013] we need to introduce other ordering relations ( being more typically C, and being more C-ish ) in order to accommodate comparisons outside the borderline area, Dietz account supplies sufficient means for accommodating such comparison with one single ordering relation. However, this potential advantage comes with some other features, which may be found objectionable. For one, as the above toy example brings out, Dietz account makes even room for comparisons within a prototype area. Whether this result makes for an account of orderings that is too permissive,isintheendanempiricalquestion. At least the findings discussed in Hampton [2007] seem to indicate that there is no evidence for prototypes of a concept C being different with respect to C- ness. In view of these considerations, one may object that some features of Dietz account are rather artifacts, which do not reflect any real features of the data to be modelled. On a related potential problem, Dietz account is modelling facts of the form x is C-er than y. It does not have any implications on comparative facts of 9 In what follows, we expound the discussion in Dietz [2013] in a more informal way. For the underlying technical details, the reader is referred to the said paper. 10 Whereas the supremum distances of x and y to r 2 are the same (namely one), their infimum distances to the union of other prototype areas are different: for x it is two, whereas for y it is three. Hence for some weight λ, the supremum distance of x to other prototypes scaled by λ is still smaller than the infimum distance to r 2 scaled by λ, whereasforx the same does not hold.

14 82 L. Decock, R. Dietz and I. Douven the form x is C-er than y, to a greater extent than v is C-er than w. This lack of specificity may be only desirable insofar as there are cases where a concept C allows us to order objects meaningfully with respect to C-ness, without allowing us order also pairs of objects with respect to distance in C-ness. 11 On the other hand, there are clearly cases of concepts which make room also for comparisons of the latter type (e.g., consider the domain of color concepts). It is this very kind of concepts that is of central interest in Decock and Douven [2013], which discusses cases where graded membership maps on similarity in an S-shaped way: for, according to this, we have to admit of cases where some patch x is redder than a patch y to an extent that is larger than the extent to which some other patch v is redder than a patch w, even though x and y are as similar in color as v and w. That is, insofar as the account of more comparisons in distance is a desideratum, Dietz s account as it stands is of no avail. This leaves it open yet whether the account may be somehow carried over to comparisons of distances. We have to leave this question open here. Having highlighted some potential objections against Dietz account of comparative concepts, lastly, on a positive point, we turn to some properties of this account which are of special interest. Dietz develops his account against the background of Gärdenfors theory on what makes categorical concepts natural (or not gerry-mandered ). Specifically, Gärdenfors submits a constraint for the special case of properties, that is, primitive concepts, such as color concepts, with only one domain in other words, the relevant dimensions are not perceivable (cognizable) separately. The restriction to properties is essential insofar as it relates to the choice of the metric. As Gärdenfors [2000:25] points out, dimensions are classified as non-separable in case a Euclidean metric fits the data about subjects judgments of similarity (otherwise, the relevant dimensions are treated as separable). The said constraint suggested by Gärdenfors is the convexity criterion: (CP) For any x and y, ifx and y both are F, then for any z that is between x and y, z is F as well. ( Betweenness is to be understood here in the standard metric sense: z is between x and y iff the distances between x and z and between z and y add up to the distance between x and y.) Dietz [2013, Sect. 2] submits two convexity criteria which carry over Gärdenfors criterion from categorical to comparative concepts: 11 Likewise, one may add that in view of such cases, there is a problem with the Decock Douven type account. To wit, given a function of graded membership for a concept, if we can compare objects with respect to their membership values, we can equally compare pairs of objects in distance in value. However, insofar as there is no evidence for meaningful comparisons of pairs with respect to distance in C-ness, the Decock Douven type approach fails to distinguish between cases where we can only meaningfully order objects, and cases where we can also meaningfully order pairs of objects in distance.

15 Modelling Comparative Concepts in Conceptual Spaces 83 (C 1 ) A strict partial ordering C-er referring to one domain in a conceptual space is a natural comparative concept only if for all points x in the space, the corresponding set {y y is C-er than x} is a convex region. 12 (C 2 ) A strict partial ordering C-er referring to one domain in a conceptual space is a natural comparative concept only if for all points x in the space, the corresponding set {y y is C-er than x (x is not C-er than y y is not C-er than x)} is a convex region. Informally put, C 1 says that for any individual, the set of individuals that are C-er than x form a convex region. C 2 is the analogue to C 1 for the class of individuals that are no less C than x. Importantly, either criterion has intuitive force independently of the criterion CP (note that C 1 and C 2 pertain to comparative concepts, whereas CP pertains to categorical concepts). As importantly, as Dietz [2000, Sect. 2] argues, some empirical findings on color categorization may be interpreted as evidence in favor of these convexity criteria. 13 That is, insofar as a theory of comparative concepts can describe conditions on which these convexity criteria are met and insofar as CP is derivable from these criteria, a theory of comparative concepts may provide some independent motivation for Gärdenfors convexity criterion. On Dietz account, comparative concepts that refer to one domain indeed satisfy conditions, on which CP follows from C 1 and C 2. More specifically, in order to establish a connection between instances of CP and instances of C 1 /C 2, Dietz focusses on comparative concepts C that are associated with a strict partial ordering C-er, in the sense that C satisfies both the monotonicity constraint MP (see Section 1) and the following difference constraint, saying that if C distinguishes between two objects, then the object which is C is C-er than the other one: (DF) If x is C and y is not C, thenx is C-er than y. With these preliminaries in place, Dietz argument for CP from C 1 to C 2 may be reconstructed as follows. (1) On Dietz account, it follows for any metric space, for any categorical concept C that is associated with a given comparative concept C-er, that for 12 Strict partial orderings are relations that are asymmetric and transitive. 13 Dietz refers to Sivik and Taft [1994]. In Sivik and Taft s experiments, test persons were asked to assess color samples as to how well they corresponded to a particular Swedish color term, on a seven-graded scale. The areas that received the same means in the subjects assessments formed connected areas, so-called isosemantic lines, which demarcated areas that approximated convex areas. The union of isosemantic lines and the areas they demarcated were nested, in accordance with the ordering of the scale. As Dietz argues, it seems natural to interpret the respectively demarcated areas as areas of colors that were assessed as fitting better (or at least, as fitting no less good) a particular color term which accords with what should be expected according to the two convexity criteria for comparative concepts suggested by Dietz.

16 84 L. Decock, R. Dietz and I. Douven some point p in the relevant space, C is representable either as the class of points that are C-er than p, or as the class of points that are no less C than p. 14 (2) For any Euclidean space, comparative concepts do validate the convexity criteria C 1 and C 2, yet only restrictedly. 15 More precisely, suppose R = {r 1,...,r n } is the relevant distribution of prototypical areas and r R is the prototype area for a concept C. Then, on Dietz account, the comparative concept C-er validates C 1 and C 2 for any point p in the space, if for some factor λ 5, p s supremum distance to r scaled by λ is still not larger than the infimum distance to prototypical areas distinct from r scaled by 1 λ. Informally put, a comparative concept being C-er than behaves normally (i.e., satisfies C 1 and C 2 ) with respect to points that are closer to the furthest prototype area for C-ness than to the nearest point in any other prototype area. 16 With respect to other points, however, the same comparative concept may behave abnormally : the class of individuals that is C-er than a particular individual from points of the said latter type may fail to be convex. This general result can be strengthened for the special case where prototypes are representable by single points in a space. In this case, comparative concepts in general behave abnormally with respect to points that are closer to the nearest member of some alternative prototype area than to the furthest member of the distinguished prototype area. (3) If r is the prototypical area for a concept C, then for any point p in the space, there is a weight λ such that the class of points that is C-er than p is representable by the graded collated Voronoi region associated with r and λ. 17 The results (1) (3) directly bear on categorization rules. To wit, by (1) and (2), if a categorical concept C that is associated with a comparative concept is representable by a particular point that is at least as close to the furthest prototype point for C as to the nearest point in an alternative prototype area, it follows that C must be convex. That is, we have some independent, conditional motivation for CP. By (3) then, it follows furthermore that C is also representable as a graded collated Voronoi region. That is, we have some independent motivation for categorization rules that are representable as an instance of graded collated Voronoi diagram. A further result follows for the simple case where prototypes are representable by single points in a space. In this case, graded collated Voronoi regions are 14 The result holds independently of the metric we choose. It only hinges on some proviso on the structure of the domain which is standardly satisfied in conceptual spaces. 15 Recall that the choice of a Euclidean metric is well-motivated, assuming that we deal with concepts that refer to only one domain. 16 This informal way of putting things is of course a bit rough, since infima or suprema do not need to be maximal or minimal elements. Our informal characterizations are to be taken with this caveat. 17 The set of points that are no less C than p is the set of points p such that for any λ [0, 1], if p u(r, R, λ) thenalsop (u, R, λ). Suppose the supremum distance of a point p to r is the real x and that its infimum distance to the union of other prototype areas is y. Thenforλ = y/(x + y), the set of points that are no less C than p is the graded collated Voronoi region associated with r and λ.