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A theory of concepts and their combinations II: A Hilbert space representation

Diederik Aerts (Center Leo Apostel for Interdisciplinary Studies, Department of Mathematics and Department of Psychology, Vrije Universiteit Brussel, Brussels, Belgium)
Liane Gabora (Center Leo Apostel for Interdisciplinary Studies, Vrije Universiteit Brussel and Department of Psychology, University of California at Berkeley, Berkeley, California, USA)


ISSN: 0368-492X

Article publication date: 1 January 2005




To develop a theory of concepts that solves the combination problem, i.e. to deliver a description of the combination of concepts. We also investigate the so‐called “pet fish problem” in concept research.


The set of contexts and properties of a concept are embedded in the complex Hilbert space of quantum mechanics. States are unit vectors or density operators and context and properties are orthogonal projections.


The way calculations are done in Hilbert space makes it possible to model how context influences the state of a concept. Moreover, a solution to the combination problem is proposed. Using the tensor product, a natural product in Hilbert space mathematics, a procedure for describing combined concepts is elaborated. This procedure also provides a solution to the pet‐fish problem, and it allows the modeling of an arbitrary number of combined concepts. By way of example, a model for a simple sentence containing a subject, a predicate and an object, is presented.


The combination problem is considered to be one of the crucial unsolved problems in concept research. Also the pet‐fish problem has not been solved by earlier attempts of modeling.



Aerts, D. and Gabora, L. (2005), "A theory of concepts and their combinations II: A Hilbert space representation", Kybernetes, Vol. 34 No. 1/2, pp. 192-221.



Emerald Group Publishing Limited

Copyright © 2005, Emerald Group Publishing Limited

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