I’ve already introduced the notion of a concept in two older blog posts (see here and here) to set the stage for the conceptual spaces framework and to motivate why concepts are useful. In short, a concept is a mental representation of a category of things in the world. For example, the concept apple ties together all knowledge we have about apples in general, such as their typical shapes and sizes, as well as what they can be used for (e.g., eating or throwing). In order to embed the conceptual spaces framework a bit more in the overall area of concept research, I will today sketch four psychological theories about concepts (based on the great overview by Murphy ) and show how they can be related to the conceptual spaces framework.
The Classical View
The classical view of concepts dates back to Aristotle and has been the predominant approach held implicitly by many psychologists in the beginning of concept research. It can be summarized as follows: Concepts are mentally represented as definitions, which provide a list of necessary and jointly sufficient conditions for membership in the category: If one of the conditions does not apply to an object, it cannot be a member of the category (necessity) and if an object fulfills all of the listed conditions, then it must be a member of the category (joint sufficiency). In other words, the definition includes everything that belongs to the category and excludes everything that does not. The classical view of concepts is tightly connected to formal logics, where concepts can be interpreted as well-defined sets and logical connectives can be used to combine them.
The classical view has been challenged on theoretical grounds, for instance by Wittgenstein who noted that it is very difficult to provide a definition of many everyday concepts (such as sports) based on a set of necessary and jointly sufficient conditions. If concepts were represented mentally as such definitions, it should however be relatively straightforward to verbalize these definitions. The classical view also fails to account for many empirical observations related to concept learning and concept use. It has therefore been essentially abandoned in the field of psychology.
The Prototype View
The prototype theory of concepts dates back to the pioneering work of Rosch. It assumes that each concept can be described by prototypical member. Category membership is then not based on the fulfillment of a list of conditions, but on the similarity of the object to the category prototype. The prototype view is able to explain why some examples are judged to be more typical representatives of a given category than others (e.g., a robin is a very typical bird, while a penguin is rather atypical), one of the strongest and most consistent effects observed with respect to concepts.
The most straightforward implementation of the prototype theory consists in representing each concept by a list of weighted features with the weights indicating the relative importance of the respective features. Classification can then be conducted by comparing the features of the observed object and the feature list used to define the concepts.
A more advanced model is based on schemata: A schema is a set of slots with possible fillers. Each slot can also have restrictions on the possible fillers, both with respect to their general type and with respect to their actual values. As each slot can only be filled with a single filler, the different fillers applicable to one slot compete with each other. This prevents concrete examples from containing conflicting information such as having both properties “flies” and “does not fly”. The different slots can furthermore be connected to each other by constraining each others values, thus encoding correlations.
The Exemplar View
In contrast to both the classical and the prototype view, the exemplar theory of concepts rejects the idea that concepts are represented in the form of some summary representation of the whole category. Instead, the exemplar theory argues that each concept is represented as the set of all observations of concept members that have been made so far. For instance, the dog concept is represented by the set of all encounters with actual dogs. A clear argument for this exemplar view is that when one starts to learn a concept, i.e., when the first example is observed, one basically has to memorize this example itself since there is not enough information for forming an abstraction. The exemplar theory however goes further by postulating that forming an abstraction is not necessary at all.
Concept membership in the exemplar view is based on the similarity of the object to the stored exemplars of the concept. Generally, it is better to have a high overlap with few exemplars than moderate overlap with many exemplars. The overall similarity can then be compared to a threshold to define concept membership. It is interesting to note that this exemplar-based definition of conceptual similarity is also capable of reproducing typicality effects since typical category members tend to be similar to many exemplars while atypical category members are only similar to very few exemplars.
The Knowledge View
The fourth approach to representing concepts we will consider here can be called the knowledge view. It emphasizes that concepts do not occur in isolation, but always stand in relations to other concepts and to our general knowledge of the world. Individual concepts are often interpreted as mental “micro-theories” about specific aspects of the world. These micro-theories are often incomplete and only partially integrated, but they provide not only a definition of category membership but also explanations and relations to other micro-theories.
Since the knowledge view does not propose concrete mechanisms for judging similarity like the prototype and the exemplar view, it can be seen as a complement to these approaches. It focuses on the influence of background knowledge on learning and reasoning which is usually ignored by exemplar and prototype approaches, but which is required in many contexts.
How do Conceptual Spaces fit in?
Gärdenfors  argues that the convexity requirement for conceptual regions allows us to relate the conceptual spaces framework to the prototype theory of concepts. If concepts are represented by convex regions, one can assign a degree of centrality to each of the points in this region by measuring its distance from the center of the region. Thus, a prototype (in the “best example” sense) can be obtained by computing the center of gravity for the conceptual region. Conversely, Gärdenfors shows that by assuming a prototype-based representation, one can easily generate convex region. For instance, if color properties such as red and orange are represented by their prototypical points in the color space (e.g., their corresponding focal colors), one can assign each point in the space to its closest prototype and thus obtain a partitioning of the overall space into convex regions. Using this prototype-based interpretation of conceptual spaces, one can easily model concept learning by defining the prototype as an average across all examples seen for the corresponding concept.
More recently, Lieto et al.  have used the conceptual spaces framework in order to build a computational model of conceptual categorization which unifies prototype theory, exemplar approaches, and the classical view of concepts. In their dual-PECCS system, they use a hybrid knowledge base which employs conceptual spaces for representing information about prototypes and exemplars (both of which are stored as points in the conceptual space) and which uses the OpenCyc ontology to represent classical definitional information about concepts in description logics.
In order to classify a new data point, Lieto et al. first make use of the conceptual spaces representation: If an exemplar is close enough to the given query point, the category associated to this exemplar is used as candidate response. If no matching exemplar is found, the system also takes into account all prototypes. It then picks the closest match among all prototypes and exemplars as candidate response. This candidate response (which is based on conceptual similarity) is then validated by checking whether the query and the proposed category match according to the definition from the ontology. One can thus say that prototypes and exemplars are used to generate candidate responses while the ontology is used to filter them.
Moreover, there was a recent proposal for replacing the ontology used in dual-PECCS with a theory-based approach . This essentially corresponds to replacing the classical view with the knowledge-based view. The resulting updated system then uses its background knowledge in the form of theories to determine a degree of consistency of similarity-based categorization with this theory. This replaces the binary decision based on necessary and sufficient criteria from the ontology.
Overall, you can hopefully see that the conceptual spaces framework can be seen as a geometric implementation of the prototype and exemplar approaches, which can also be combined with the classical and the knowledge-based view. This gives further support to conceptual spaces as a model of human conceptual representation.
 Murphy, G. The Big Book of Concepts. MIT Press, 2002
 Gärdenfors, P. Conceptual Spaces: The Geometry of Thought. MIT Press, 2000
 Lieto, A.; Radicioni, D. P. & Rho, V. Dual PECCS: a cognitive system for conceptual representation and categorization. Journal of Experimental & Theoretical Artificial Intelligence, Taylor & Francis, 2017, 29, 433-452
 Lieto, A. Heterogeneous Proxytypes Extended: Integrating Theory-Like Representations and Mechanisms with Prototypes and Exemplars. Biologically Inspired Cognitive Architectures 2018, Springer International Publishing, 2019, 217-227