Shepard’s (1987) universal law of generalisation (ULG) illustrates that an invariant gradient of generalisation across species and across stimuli conditions can be obtained by mapping the probability of a generalisation response onto the representations of similarity between individual stimuli. Tenenbaum and Griffiths (2001) Bayesian account of generalisation expands ULG towards generalisation from multiple examples. Though the Bayesian model starts from Shepard’s account it refrains from any commitment to the notion of psychological similarity to explain categorisation. This chapter presents the (...) class='Hi'>conceptualspaces theory (Gärdenfors 2000, 2014) as a mediator between Shepard’s and Tenenbaum & Griffiths’ conflicting views on the role of psychological similarity for a successful model of categorisation. It suggests that the conceptualspaces theory can help to improve the Bayesian model while finding an explanatory role for psychological similarity. (shrink)
The conceptualspaces approach has recently emerged as a novel account of concepts. Its guiding idea is that concepts can be represented geometrically, by means of metrical spaces. While it is generally recognized that many of our concepts are vague, the question of how to model vagueness in the conceptualspaces approach has not been addressed so far, even though the answer is far from straightforward. The present paper aims to fill this lacuna.
This paper offers a novel way of reconstructing conceptual change in empirical theories. Changes occur in terms of the structure of the dimensions—that is to say, the conceptualspaces—underlying the conceptual framework within which a given theory is formulated. Five types of changes are identified: (1) addition or deletion of special laws, (2) change in scale or metric, (3) change in the importance of dimensions, (4) change in the separability of dimensions, and (5) addition or deletion (...) of dimensions. Given this classification, the conceptual development of empirical theories becomes more gradual and rationalizable. Only the most extreme type—replacement of dimensions—comes close to a revolution. The five types are exemplified and applied in a case study on the development within physics from the original Newtonian mechanics to special relativity theory. (shrink)
During the last decades, many cognitive architectures (CAs) have been realized adopting different assumptions about the organization and the representation of their knowledge level. Some of them (e.g. SOAR ) adopt a classical symbolic approach, some (e.g. LEABRA[ 48]) are based on a purely connectionist model, while others (e.g. CLARION ) adopt a hybrid approach combining connectionist and symbolic representational levels. Additionally, some attempts (e.g. biSOAR) trying to extend the representational capacities of CAs by integrating diagrammatical representations and reasoning are (...) also available . In this paper we propose a reflection on the role that ConceptualSpaces, a framework developed by Peter G¨ardenfors  more than fifteen years ago, can play in the current development of the Knowledge Level in Cognitive Systems and Architectures. In particular, we claim that ConceptualSpaces offer a lingua franca that allows to unify and generalize many aspects of the symbolic, sub-symbolic and diagrammatic approaches (by overcoming some of their typical problems) and to integrate them on a common ground. In doing so we extend and detail some of the arguments explored by G¨ardenfors  for defending the need of a conceptual, intermediate, representation level between the symbolic and the sub-symbolic one. In particular we focus on the advantages offered by ConceptualSpaces (w.r.t. symbolic and sub-symbolic approaches) in dealing with the problem of compositionality of representations based on typicality traits. Additionally, we argue that ConceptualSpaces could offer a unifying framework for interpreting many kinds of diagrammatic and analogical representations. As a consequence, their adoption could also favor the integration of diagrammatical representation and reasoning in CAs. (shrink)
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 conceptualspaces because: some of the allegedly convex properties of concepts are not convex; the conceptual regions resulting from the combination of convex properties can be non-convex; convex regions may co-vary in non-convex ways; and 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. (shrink)
We present an account of semantics that is not construed as a mapping of language to the world but rather as a mapping between individual meaning spaces. The meanings of linguistic entities are established via a “meeting of minds.” The concepts in the minds of communicating individuals are modeled as convex regions in conceptualspaces. We outline a mathematical framework, based on fixpoints in continuous mappings between conceptualspaces, that can be used to model such (...) a semantics. 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. Language is discrete, while mental representations tend to be continuous—posing a seeming paradox. We show that the convexity assumption allows us to address this problem. Using examples, we further show that our approach helps explain the semantic processes involved in the composition of expressions. (shrink)
The aim of this paper is to present a general method for constructing natural tessellations of conceptualspaces that is based on their topological structure. This method works for a class of spaces that was defined some 80 years ago by the Russian mathematician Pavel Alexandroff. Alexandroff spaces, as they are called today, are distinguished from other topological spaces by the fact that they exhibit a 1-1 correspondence between their specialization orders and their topological structures. (...) Recently, Ian Rumfitt (apparently not being aware of Alexandroff’s work) used a very special case of Alexandroff’s method to elucidate the logic of vague concepts in a new way. Elaborating his approach, the color circle’s conceptual space can be shown to define an atomistic Boolean algebra of regular open concepts. In a similar way Gärdenfors’ geometrical discretization of conceptualspaces by Voronoi tessellations also can be shown to be a kind of geometrical version of Alexandroff’s topological construction. More precisely, a discretization à la Gärdenfors is extensionally equivalent to a topological discretization constructed by Alexandroff’s method. Rumfitt’s and Gärdenfors’s constructions turn out to be special cases of an approach that works much more generally, namely, for Alexandroff spaces. For these spaces (X, OX) the Boolean algebras O*X of regular open sets are still atomistic and yield natural tessellations of X. (shrink)
We approach the semantics of prepositions from the perspective of conceptualspaces. Focusing on purely spatial locative and directional prepositions, we analyze both types of prepositions in terms of polar coordinates instead of Cartesian coordinates. This makes it possible to demonstrate that the property of convexity holds quite generally in the domain of prepositions of location and direction, supporting the important role that this property plays in conceptualspaces.
We argue that a cognitive semantics has to take into account the possibly partial information that a cognitive agent has of the world. After discussing Gärdenfors's view of objects in conceptualspaces, we offer a number of viable treatments of partiality of information and we formalize them by means of alternative predicative logics. Our analysis shows that understanding the nature of simple predicative sentences is crucial for a cognitive semantics.
We examine Gärdenfors’ theory of conceptualspaces, a geometrical form of knowledge representation (Conceptualspaces: The geometry of thought, MIT Press, Cambridge, 2000 ), in the context of the general Creative Systems Framework introduced by Wiggins (J Knowl Based Syst 19(7):449–458, 2006a ; New Generation Comput 24(3):209–222, 2006 b ). Gärdenfors’ theory offers a way of bridging the traditional divide between symbolic and sub-symbolic representations, as well as the gap between representational formalism and meaning as perceived (...) by human minds. We discuss how both these qualities may be advantageous from the point of view of artificial creative systems. We take music as our example domain, and discuss how a range of musical qualities may be instantiated as conceptualspaces, and present a detailed conceptual space formalisation of musical metre. (shrink)
This introductory chapter provides a non-technical presentation of conceptualspaces as a representational framework for modeling different kinds of similarity relations in various cognitive domains. Moreover, we briefly summarize each chapter in this volume.
There is a great deal of justified concern about continuity through scientific theory change. Our thesis is that, particularly in physics, such continuity can be appropriately captured at the level of conceptual frameworks using conceptual space models. Indeed, we contend that the conceptualspaces of three of our most important physical theories—Classical Mechanics, Special Relativity Theory, and Quantum Mechanics —have already been so modelled as phase-spaces. Working with their phase-space formulations, one can trace the (...) class='Hi'>conceptual changes and continuities in transitioning from CM to QM, and from CM to SRT. By offering a revised severity-ordering of changes that conceptual frameworks can undergo, we provide reasons to doubt the commonly held view that CM is conceptually closer to SRT than QM. (shrink)
A computational theory of induction must be able to identify the projectible predicates, that is to distinguish between which predicates can be used in inductive inferences and which cannot. The problems of projectibility are introduced by reviewing some of the stumbling blocks for the theory of induction that was developed by the logical empiricists. My diagnosis of these problems is that the traditional theory of induction, which started from a given (observational) language in relation to which all inductive rules are (...) formulated, does not go deep enough in representing the kind of information used in inductive inferences. As an interlude, I argue that the problem of induction, like so many other problems within AI, is a problem of knowledge representation. To the extent that AI-systems are based on linguistic representations of knowledge, these systems will face basically the same problems as did the logical empiricists over induction. In a more constructive mode, I then outline a non-linguistic knowledge representation based on conceptualspaces. The fundamental units of these spaces are "quality dimensions". In relation to such a representation it is possible to define "natural" properties which can be used for inductive projections. I argue that this approach evades most of the traditional problems. (shrink)
This paper is concerned with a version of Kamp and Partee's account of graded membership that relies on the conceptualspaces framework. Three studies are reported, one to construct a particular shape space, one to detect which shapes representable in that space are typical for certain sorts of objects, and one to elicit degrees of category membership for the various shapes from which the shape space was constructed. Taking Kamp and Partee's proposal as given, the first two studies (...) allowed us to predict the degrees to which people would judge shapes representable in the space to be members of certain categories. These predictions were compared with the degrees that were measured in the third study. The comparison yielded a test of the account of graded membership at issue. The outcome of this test was found to support the conceptualspaces version of Kamp and Partee's account of graded membership. (shrink)
Why is a red face not really red? How do we decide that this book is a textbook or not? Conceptualspaces provide the medium on which these computations are performed, but an additional operation is needed: Contrast. By contrasting a reddish face with a prototypical face, one gets a prototypical ‘red’. By contrasting this book with a prototypical textbook, the lack of exercises may pop out. Dynamic contrasting is an essential operation for converting perceptions into predicates. The (...) existence of dynamic contrasting may contribute to explaining why lexical meanings correspond to convex regions of conceptualspaces. But it also explains why predication is most of the time opportunistic, depending on context. While off-line conceptual similarity is a holistic operation, the contrast operation provides a context-dependent distance that creates ephemeral predicative judgments that are essential for interfacing conceptualspaces with natural language and with reasoning. (shrink)
Recent years have witnessed a revival of interest in the method of explication as a procedure for conceptual engineering in philosophy and in science. In the philosophical literature, there has been a lively debate about the different desiderata that a good explicatum has to satisfy. In comparison, the goal of explicating the concept of explication itself has not been central to the philosophical debate. The main aim of this work is to suggest a way of filling this gap by (...) explicating ‘explication’ by means of conceptualspaces theory. Specifically, I show how different, strictly-conceptual readings of explication desiderata can be made precise as geometrical or topological constraints over the conceptualspaces related to the explicandum and the explicatum. Moreover, I show also how the richness of the geometrical representation of concepts in conceptualspaces theory allows us to achieve more fine-grained readings of explication desiderata, thereby overcoming some alleged limitations of explication as a procedure of conceptual engineering. (shrink)
Actions and events are central to a semantics of natural language. In this article, we present a cognitively based model of these notions. After giving a general presentation of the theory of conceptualspaces, we explain how the analysis of perceptual concepts can be extended to actions and events. First, we argue that action space can be analyzed in the same way as, for example, colour space or shape space. Our hypothesis is that the categorization of actions depends, (...) to a large extent, on the perception of forces. In line with this, we describe an action as a pattern of forces. We identify an action category as a convex region of action space. We review some indirect evidence for this representation. Second, we represent an event as an interaction between a force vector and a result vector. Typically an agent performs an action—that is, exerts a force—that changes the properties of the patient. Such a model of events is suitable for an analysis of the semantics of verbs. We compare the model to other related attempts from cognitive semantics. (shrink)
Our aim in this article is to show how the theory of conceptualspaces can be useful in describing diachronic changes to conceptual frameworks, and thus useful in understanding conceptual change in the empirical sciences. We also compare the conceptual space approach to Moulines’s typology of intertheoretical relations in the structuralist tradition. Unlike structuralist reconstructions, those based on conceptualspaces yield a natural way of modeling the changes of a conceptual framework, including (...) noncumulative changes, by tracing the changes to the dimensions that reconstitute a conceptual framework. As a consequence, the incommensurability of empirical theories need not be viewed as a matter of conceptual representation. (shrink)
The dominating models of information processes have been based on symbolic representations of information and knowledge. During the last decades, a variety of non-symbolic models have been proposed as superior. The prime examples of models within the non-symbolic approach are neural networks. However, to a large extent they lack a higher-level theory of representation. In this paper, conceptualspaces are suggested as an appropriate framework for non- symbolic models. Conceptualspaces consist of a number of 'quality (...) dimensions' that often are derived from perceptual mechanisms. It will be outlined how conceptualspaces can represent various kind of information and how they can be used to describe concept learning. The connections to prototype theory will also be presented. (shrink)
In the framework of set theory we cannot distinguish between natural and non-natural predicates. To avoid this shortcoming one can use mathematical structures as conceptualspaces such that natural predicates are characterized as structurally nice subsets. In this paper topological and related structures are used for this purpose. We shall discuss several examples taken from conceptualspaces of quantum mechanics (orthoframes), and the geometric logic of refutative and affirmable assertions. In particular we deal with the problem (...) of structurally distinguishing between natural colour predicates and Goodmanian predicates like grue and bleen. Moreover the problem of characterizing natural predicates is reformulated in such a way that its connection with the classical problem of geometric conventionalism becomes manifest. This can be used to shed some new light on Goodman's remarks on the relative entrenchment of predicates as a criterion of projectibility. (shrink)
Knowledge representation is a central issue in a number of areas, but few attempts are usually made to bridge different approaches accross different fields. As a contribution in this direction, in this paper I focus on one such approach, the theory of conceptualspaces developed within cognitive science, and explore its potential applications in the fields of philosophy of science and formal epistemology. My case-study is provided by the theory of truthlikeness, construed as closeness to “the whole truth” (...) about a given domain, as described in the underlying language. I show how modeling propositions and their relations within a conceptual space has interesting implications for two issues in truthlikeness theory: the so called problem of language dependence, and that of measure sensitivity. I conclude by pointing at some open issues arising from the application of conceptualspaces to the analysis of philosophical problems. (shrink)
By understanding laws of nature as geometrical rather than linguistic entities, this paper addresses how to describe theory structures and how to evaluate their continuity. Relying on conceptualspaces as a modelling tool, we focus on the conceptual framework an empirical theory presupposes, thus obtain a geometrical representation of a theory’s structure. We stress the relevance of measurement procedures in separating conceptual from empirical structures. This lets our understanding of scientific laws come closer to scientific practice, (...) and avoids a widely recognised deficit in current philosophy of science accounts, namely to risk a collapse of the physical into the mathematical. (shrink)
A new construction of a certain conceptual space is presented. Elements of this conceptual space correspond to concept elements of reality, which potentially comprise an infinite number of qualities. This construction of a conceptual space solves a problem stated by Dietz and his co-authors in 2013 in the context of Voronoi diagrams. The fractal construction of the conceptual space is that this problem simply does not pose itself. The concept of convexity is discussed in this new (...)conceptual space. Moreover, the meaning of convexity is discussed in full generality, for example when space is deprived of it, its substitutes for concept domains are considered. (shrink)