A Geometric Model of Concept Formation

The classical account of concepts within philosophy is Aristotle’s theory of necessary and sufficient conditions (see [19] for a presentation of this and other theories of concept formation). His view on how concepts are determined has had an enormous influence throughout the history of philosophy. During this century, the Aristotelian notions became part of the program of the logical positivists who demanded that all scientific concepts should ideally be defined in terms of a limited number of observational terms. If a concept can’t be defined by necessary and sufficient conditions, it is not a proper scientific concept, at least according to the early positivist program. And, to go to a recent phenomenon, within expert systems in AI, most methods presume that the concepts of the experts can be delimited by necessary and/or sufficient conditions. However, one encounters several problems when trying to apply the Aristotelian theory. One drawback is that concepts have no sharp boundaries, and their domain of application is more or less vague.1 Furthermore, as Wittgenstein and others have noted, some concepts like ‘game’ don’t seem to have a common core which could be circumscribed by a set of necessary conditions. As a result of a growing dissatisfaction with the classical theory of concepts, an alternative theory was developed within cognitive psychology. This is the called prototype theory where Eleanor Rosch is one of the main proponents.2 The main idea of prototype theory is that within a category of objects, like those instantiating a property, certain members are judged to be more representative of the category than others. For example robins are judged to be more representative of the category ‘bird’ than are ravens, penguins and emus; and desk chairs are more typical instances of the category ‘chair’ than rocking chairs, deck-chairs, and beanbag chairs. The most representative members of a category are called prototypical members. It is well-known that many properties, like ‘red’ and ‘bald’ have no sharp boundaries, and for these it is perhaps not surprising that one finds prototypical effects. However, such