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


Purpose - 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.

Design/methodology/approach - 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.

Findings - 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.

Originality/value - 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.

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