Skip to main content
Open Access Publications from the University of California

UC Berkeley

UC Berkeley Previously Published Works bannerUC Berkeley

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

  • Author(s): Aerts, Diederik
  • Gabora, Liane M
  • et al.

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.

Many UC-authored scholarly publications are freely available on this site because of the UC's open access policies. Let us know how this access is important for you.

Main Content
Current View