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A case against convexity in conceptual spaces

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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.

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Notes

  1. Although these two definitions are for natural properties and concepts, as a matter of fact Gärdenfors applies them almost universally: he does not distinguish between natural and non-natural properties or concepts (except in order to discriminate artificial non-convex properties or concepts, such as those associated with Goodman’s term grue: green before a given date and blue after that date).

  2. Integral dimensions are those processed in a holistic and unanalyzable way, where the assignation of a value to a particular dimension requires a value to be given to the others. If dimensions are not integral, then they are separable. Gärdenfors’ main domain example is the case of color, which would be constituted of three dimensions: hue, intensity and brightness.

  3. Although the second and third constraints require the definition of betweenness [an axiomatic definition of which can be found in Gärdenfors (2000, p. 15)], I will not discuss that topic in this paper.

  4. For a detailed review of approaches to weighting that are distinct from the multiplicative one, see Okabe et al. (1992, pp. 119–134).

  5. Ordinary distances used simply to be called distances (or standard distances), and that is what I will do in this work; conversely, I will use the terms weighted distance and non-standard distance indistinctly.

  6. Prototypical effects are associated with the fact that some members of a category are considered more representative of it than others. For instance, robins are considered more representative of the bird category than eagles, and eagles more than chickens.

  7. That is, with or without real instances of them.

  8. Here it must be said that, within the conceptual space framework, prototypicality effects in a region can happen regarding two main different referents:

    1. [I]

      With regard to the centers of gravity, or mass centroids, of the regions: this option is quite straightforward (and the one adopted by Gärdenfors), because the center of gravity of a convex region is the most typical member of that region. This alternative also implies a strong commitment to the literal reading that properties and concepts are represented by convex regions (and not merely by their prototypes). This is so because, in order to determine the center of gravity it is necessary to evaluate the whole region associated to the concept; and the region must be convex so that its center of gravity belongs to the region.

    2. [II]

      With regard to the prototypes, or Voronoi generators, of the regions: this alternative has a crucial difference with respect to option [I]: the prototype is the most typical member, not of the region resulting from a Voronoi tessellation based on that prototype, but of the instances (particular cases or examples) from which the prototype of that concept was determined (for example, by means of a cluster analysis). This approach is more compatible with the principle of cognitive economy than the first, because in alterative [I] the evaluation of typicality requires (i) calculating or storing the whole conceptual region, and (ii) determining therefrom the gravity center of that region. In contrast, in option [II] typicality evaluation only needs the determination of the distance to the prototype.

    And, even though Gärdenfors chooses option [I], both approaches are equally acceptable in order to explain typicality effects in conceptual regions. In fact, option [II] is the one chosen by me when I argue below that the star-shaped regions which result from a Voronoi tessellation can also explain prototypicality.

  9. The argument which connects the prototype theory (articulated by means of Voronoi tessellations) with the star-shapedness of regions can be summed up as follows:

    • Premise 1: If an object, O, belongs to a concept, C (characterized by a prototype P), this entails that the ball B(O, OP), centered at O and with radius OP, does not contain any other prototype distinct from P. [Premise 1 is equivalent to the thesis that concepts are the result of a Voronoi tessellation (Thesis V)].

    • Premise 2: Minkowski metric with \(p\ge 1\) and non-weighted prototypes.

    • Conclusion: Conceptual regions are star-shaped.

    The general idea is that for every object A, between O and P, it is possible to prove that the ball B(A, AP) is included within B(O, OP). Therefore, P is the nearest prototype to A; that is, the object A also belongs to C and, in consequence, conceptual regions are star-shaped. [For the specific details of this proof see Lemma 5 in Lee (1980, p. 608)].

  10. Withal, this should not be seen as a defense of a mandatory star-shapedness constraint, because it could be accepted the different weighting of prototypes, and in that case the conceptual regions might not be star-shaped (as shown in Fig. 3b).

  11. In order that the resulting regions may be convex, the Voronoi tessellation will have to meet some conditions (Euclidean metric, and non-weighted prototypes).

  12. In Gärdenfors’ words: “The [Voronoi] tessellation mechanism provides important clues to the cognitive economy of concept learning. If the categorization of each point in a space had to be memorized, this would put absurd demands on human memory. However, if the partitioning of a space into categories is based on a Voronoi tessellation, only the relative positions of the prototypes need to be remembered” (Gärdenfors 2014, pp. 27–28). Gärdenfors makes use of similar arguments for the cognitive efficiency of applying Voronoi tessellations in his explanation of how human communication works (ib., pp. 274–275).

  13. According to the notions of standard and non-standard distances given in footnote 5 above.

  14. On the one hand, the city-block metric is accepted when dimensions are separable because, in that case, the dimensions are the most meaningful element: they contribute independently to the total distance, and must remain invariant (unrotated) to keep the same conceptual space structure (Garner 1974, p. 119). On the other hand, the Euclidean metric is proposed when dimensions are non-analyzable because, in this case, distance (which remains the same for all rotations of axes) is the most relevant element.

    However, it is possible that [i] although the dimensions did not contribute independently to the value of distances (and, therefore, distance were the most meaningful element); [ii] it happened that, despite of [i], the conceptual space structure were not irrelevant, and distances were not invariant under rotations of dimensions. In that case, dimensions would be non-separable (due to [i]) but, at the same time, they would not be secondary (by virtue of [ii]). Therefore, the non-separability of dimensions does not imply a Euclidean structure of the underlying conceptual space, whose metric could be non-Euclidean (for instance, with parameter p equal to 1.7 or 3) and still able to explain the non-separability of domains.

  15. Thanks to an anonymous reviewer whose comments led me to rethink this point.

  16. As happened with the perceptual foundation argument in favor of the convexity constraint (see Sect. 4).

  17. This behavior of the metric structure can be summed up as follows: separable dimensions are better characterized by a city-block metric, while the Euclidean metric is the best for integral dimensions.

  18. See Sect. 6 below for the meaning of the p parameter within the standard Minkowski metric. That section also contains a chart (Fig. 2) which shows that for a parameter p equal to 1.7 the conceptual regions are not convex.

  19. For a formal demonstration of this, see Okabe et al. (1992, p. 57).

  20. For a summary of the properties of a weighted conceptual space, see Okabe et al. (1992, pp. 120–123). One of those properties is that the regions resulting from a multiplicatively weighted Voronoi tessellation do not need to be convex (as shown in Fig. 3), or even connected; and that they can also contain holes. Additionally, according to this kind of approach, the region associated with a particular concept \(C_{i}\) will be convex if and only if the weights of all its adjacent regions are smaller than \(w_{i}\) (in Fig. 3 that is the case of the region associated with the prototype P).

  21. The problem is that, for the convexity constraint to be met by the regions characterizing these prepositions, the convexity of the objects to which they apply is necessary.

  22. Although Fiorini, Gärdenfors and Abel describe the apple’s shape as a cycloid, in fact the shape corresponds to an epicycloid.

  23. In this case it could be argued that, because it is evident that the disjunctive combination of convex properties in the same domain can fail to be convex, Gärdenfors could not have been unaware of it. This leads to the question of to what extent Gärdenfors is committed to convexity. Here I will try to show that Gärdenfors is strongly committed to the convexity of concepts, but before a terminological clarification is needed.

    In Gärdenfors (2000) he distinguished between properties (defined as convex regions) and concepts (defined as sets of convex regions), but it is not possible to find there an explicit assertion about the convexity or non-convexity of concepts. However, things change in his latter works, where we find statements like “as proposed in Gärdenfors (2000), concepts can be modeled as convex regions of a conceptual space” (Warglien and Gärdenfors 2013, p. 2171), or “the convexity of concepts is also crucial for ensuring the effectiveness of communication” (Gärdenfors 2014, p. 26). Nonetheless, it is not clear that in these last quotes Gärdenfors is referring with the term “concept” the same as in Gärdenfors (2000), because in Gärdenfors (2014) the notion “object category” began to play the role played by the notion “concept” in Gärdenfors (2000). Withal, in Gärdenfors (2014), he does not explicitly assert that object categories are represented by convex regions.

    Notwithstanding this, in these recent works (Warglien and Gärdenfors 2013; Gärdenfors 2014) he tries to explain human communication via a “meeting of minds”, using a fix-point argument. Such an argument requires the convexity of concepts: “the concepts in the minds of communicating individuals are modeled as convex regions in conceptual spaces (...) If concepts are convex, it will in general be possible for interactors to agree on joint meaning even if they start out from different representational spaces.” (Warglien and Gärdenfors 2013, p. 2165). But, this argument is expected to apply at least to noun phrases (ib., p. 2170) and, thus, to object categories (as the apple concept), which should also be represented by convex regions. On my view, this proves that Gärdenfors is firmly committed to the convexity of both properties and concepts.

  24. It may be thought that the conceptual region R was equal to the (not a) set of objects belonging simultaneously to \(Q_{1}\), \(Q_{2}\), ..., and \(Q_{n}\). However, that is not the approach followed by Gärdenfors (2014, p. 29), who accepts the possibility of a non-rectangular conceptual space, which does not contain the whole set of points belonging simultaneously to all its constitutive properties (as is the case of the graphs shown in Fig. 8). That is the reason why the following logical conditions are necessary, but not sufficient.

  25. At the expense of considering two different color dimensions (color \(_{1}\) and color \(_{2})\) as constitutive of the apple conceptual space, given that if both color dimensions were the same, the set of objects satisfying this condition would be void. Here I will not enter into the discussion of the problems associated with such implications.

  26. Therefore, the swan conceptual space is a problem for any theory which attributes a mandatory character to the connectedness requirement for the geometry of conceptual regions. Obviously, this applies to any criterion stronger than the connectedness one (as is the case with the star-shapedness and convexity requirements).

  27. This is the same notion of change that Gärdenfors has in mind when he says the following:

    In general, a change of state is not represented by a specific vector. Instead, it can be represented by a category of changes of state. (...) If the start point is set as the origin, one can represent a category of change events as a region of end points. (...) For example, going “upwards” in a two-dimensional space will correspond to a convex region of points located in a cone to the “north” of the origin. (Warglien et al. 2012, p. 162)

    However, this is not the only possible way to generalize the notion of change. Another possibility would be that the knowing subject S knew neither the initial point nor the final point of the change, and that the initial point could not be set as the origin. This would happen if John said to S, “the leaves were yellowed by disease”, but S does not know the tree whose leaves were referred to. In this case S knows neither the exact start point (a kind of green) nor the exact end point (a kind of yellow) of the change expressed by “yellowed”, so he will have to represent the properties associated to the initial and final states by regions (the ones associated to the green and yellow colors). Consequently, the change expressed by “yellowed” will be represented as the set of vectors going from every point in the region representing green to every point in the region representing yellow.

  28. Nonetheless, these problems would also be associated with the convexity constraint (given that every convex region is also a star-shaped region).

References

  • Adams, B., & Raubal, M. (2009). Conceptual space markup language (CSML): Towards the cognitive semantic web. In Proceedings of the third IEEE international conference on semantic computing (ICSC 2009) (pp. 253–260). Berkeley, CA: IEEE Computer Society.

  • Churchland, P. (2005). Chimerical colors: Some phenomenological predictions from cognitive neuroscience. Philosophical Psychology, 18(5), 527–560.

    Article  Google Scholar 

  • Fairbanks, G., & Grubb, P. (1961). A psychophysical investigation of vowel formants. Journal of Speech and Hearing Research, 4(3), 203–219.

    Article  Google Scholar 

  • Fiorini, S. R., Gärdenfors, P., & Abel, M. (2014). Representing part–whole relations in conceptual spaces. Cognitive Processing, 15(2), 127–142.

    Article  Google Scholar 

  • Gärdenfors, P. (2000). Conceptual spaces: The geometry of thought. Cambridge, MA: MIT Press.

    Google Scholar 

  • Gärdenfors, P. (2014). The geometry of meaning: Semantics based on conceptual spaces. Cambridge, MA: MIT Press.

    Google Scholar 

  • Gärdenfors, P., & Warglien, M. (2012). Using conceptual spaces to model actions and events. Journal of Semantics, 29(4), 487–519.

    Article  Google Scholar 

  • Gärdenfors, P., & Warglien, M. (2013). Semantics, conceptual spaces, and the meeting of minds. Synthese, 190(12), 2165–2193.

    Article  Google Scholar 

  • Garner, W. R. (1974). The processing of information and structure. Potomac, MD: Erlbaum.

    Google Scholar 

  • Gauker, C. (2007). A critique of the similarity space theory of concepts. Mind & Language, 22(4), 317–345.

    Article  Google Scholar 

  • Goldstone, R. L., & Son, J. Y. (2005). Similarity. In K. J. Holyoak & R. G. Morrison (Eds.), The Cambridge handbook of thinking and reasoning (pp. 13–36). Cambridge: Cambridge University Press.

    Google Scholar 

  • Handel, S., & Imai, S. (1972). The free classification of analyzable and unanalyzable stimuli. Perception and Psychophysics, 12(1), 108–112.

    Article  Google Scholar 

  • Jäger, G. (2007). The evolution of convex categories. Linguistic and Philosophy, 30(5), 551–564.

    Article  Google Scholar 

  • Lakoff, G. (1987). Women, fire, and dangerous things: What categories reveal about the mind. Chicago: University of Chicago Press.

    Book  Google Scholar 

  • Lee, D. T. (1980). Two-dimensional Voronoi diagrams in the L\(_{\rm{p}}\)-metric. Journal of the Association for Computing Machinery, 27(4), 604–618.

    Article  Google Scholar 

  • Maddox, W. T. (1992). Perceptual and decisional separability. In G. F. Ashby (Ed.), Multidimensional models of perception and cognition (pp. 147–180). Hillsdale, NJ: Lawrence Erlbaum.

    Google Scholar 

  • Marr, D., & Nishihara, H. K. (1978). Representation and recognition of the spatial organization of three dimensional shapes. Proceedings of the Royal Society of London, Series B, Biological Sciences, 200(1140), 269–294.

    Article  Google Scholar 

  • Melara, R. D. (1992). The concept of perceptual similarity: From psychophysics to cognitive psychology. In D. Algom (Ed.), Psychophysical approaches to cognition (pp. 303–388). Amsterdam: Elsevier.

    Chapter  Google Scholar 

  • Okabe, A., Boots, B., Sugihara, K., & Chiu, S. N. (1992). Spatial tessellations: Concepts and applications of Voronoi diagrams. New York: Wiley.

    Google Scholar 

  • Quine, W. V. (1969). Natural kinds. Ontological relativity and other essays (pp. 114–138). New York: Columbia University Press.

  • Rosch, E. (1978). Principles of categorization. In E. Rosch & B. Lloyd (Eds.), Cognition and categorization (pp. 27–48). Hillsdale, NJ: Erlbaum Associates.

    Google Scholar 

  • Rosch, E., & Mervis, C. B. (1975). Family resemblances: Studies in the internal structure of categories. Cognitive Psychology, 7(4), 573–605.

    Article  Google Scholar 

  • Shepard, R. N. (1987). Toward a universal law of generalization for psychological science. Science, 237(4820), 1317–1323.

    Article  Google Scholar 

  • Sivik, L., & Taft, C. (1994). Color naming: A mapping in the NCS of common color terms. Scandinavian Journal of Psychology, 35(2), 144–164.

    Article  Google Scholar 

  • Smith, E. E., & Medin, D. L. (1981). Categories and concepts. Cambridge, MA: Harvard University Press.

    Book  Google Scholar 

  • Warglien, M., & Gärdenfors, P. (2013). Semantics, conceptual spaces, and the meeting of minds. Synthese, 190(12), 2165–2193.

    Article  Google Scholar 

  • Warglien, M., Gärdenfors, P., & Westera, M. (2012). Event structure, conceptual spaces and the semantics of verbs. Theoretical Linguistics, 38(3–4), 159–193.

    Google Scholar 

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Acknowledgments

This research was carried out under a PhD scholarship from the University of the Basque Country, and financially supported by the Spanish Ministry of Economy and Competitiveness research projects FFI2011-30074-C02-02 and FFI2014-52196-P. I wish to give special thanks to Agustin Vicente for his continued advice and encouragement. A partial version of this paper was presented at the SPE8 Colloquium (2015). I am grateful to the audience there for valuable comments. Finally, I am also thankful to Toffa Evans for his assistance with the manuscript.

Funding

This study was financially supported by the Spanish Ministry of Economy and Competitiveness (research projects FFI2011-30074-C02-02 and FFI2014-52196-P), and was carried out under a PhD scholarship from the University of the Basque Country.

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  1. Department of Linguistics and Basque Studies, Faculty of Arts, University of the Basque Country, Paseo de la Universidad 5, 01006, Vitoria-Gasteiz, Álava, Spain

    José V. Hernández-Conde

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Hernández-Conde, J.V. A case against convexity in conceptual spaces. Synthese 194, 4011–4037 (2017). https://doi.org/10.1007/s11229-016-1123-z

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