Abstract
Quantum models of concept combinations have been successful in representing various experimental situations that cannot be accommodated by traditional models based on classical probability or fuzzy set theory. In many cases, the focus has been on producing a representation that fits experimental results to validate quantum models. However, these representations are not always consistent with the cognitive modeling principles. Moreover, some important issues related to the representation of concepts such as the dimensionality of the realization space, the uniqueness of solutions, and the compatibility of measurements, have been overlooked. In this paper, we provide a dimensional analysis of the realization space for the two-sector Fock space model for conjunction of concepts focusing on the first and second sectors separately. We then introduce various representation of concepts that arise from the use of unitary operators in the realization space. In these concrete representations, a pair of concepts and their combination are modeled by a single conceptual state, and by a collection of exemplar-dependent operators. Therefore, they are consistent with cognitive modeling principles. This framework not only provides a uniform approach to model an entire data set, but, because all measurement operators are expressed in the same basis, allows us to address the question of compatibility of measurements. In particular, we present evidence that it may be possible to predict non-commutative effects from partial measurements of conceptual combinations.
Highlights
In addition to providing a suitable mathematical framework for cognitive models, quantum cognition offers a different perspective on cognitive phenomena: uncertainty is described by means of superposed states (Aerts et al, 2011b), non-logical coherence involves interference (Aerts, 2009), order effects are revealed by incompatible measurements (Wang and Busemeyer, 2013), and certain “verb-noun” conceptual combinations mimic the structure of physically entangled particles (Aerts and Sozzo, 2014)
We have made some advances on the representational aspects of the quantum model for concept combinations
We proved that the first and second sectors of the two-sector Fock space model of concept conjunctions can be concretely represented in C3 and C3 ⊗ C3, respectively
Summary
The application of quantum models to cognitive phenomena is an emergent field known as quantum cognition (Aerts, 2009; Pothos and Busemeyer, 2013). Let μ(A), μ(B), and μ(AB) be the membership weights of an exemplar p with respect to a pair of concepts A and B and their conjunction AB We say that these membership weights are classical conjunction data if there exists a Kolmogorovian probability space ( , σ ( ), P), and events EA, EB ∈ σ ( ) such that. A large body of experimental evidence and a considerable amount of data analysis indicate that the membership of exemplars with respect to concept combinations does not form classical conjunction data (Fodor and Lepore, 1996; Hampton, 1997a,b; Aerts and Gabora, 2005a,b). The membership with respect to the conjunction of concepts is generally larger than the membership of one of the former concepts, and violates either conditions (2) or (3) The phenomenon of overextension has been demonstrated for membership estimations, and in typicality (Smith and Osherson, 1981; Hampton, 1996; Storms et al, 1998), property relevance (Fodor and Lepore, 1996; Hampton, 1997a,b; Aerts and Gabora, 2005a,b), and probability estimations (Tversky and Kahneman, 1983; Moro, 2009)
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