The authors convincingly demonstrate the greater potential of quantum probability as compared with classical probability in modeling situations of human cognition, giving various examples to illustrate their analysis. In our commentary, we provide additional arguments to support their claim and approach. We want to point out, however, that it is not just quantum probability, but also much more specific quantum structures, quantum states, observables, complex numbers, and typical quantum spaces – for example, Fock space – that on a deep level provide a modeling of the structure of human thought itself.

A first insight about quantum structure in human cognition came to us with the characterizations of classical and quantum probability following from the hidden-variable investigation in quantum theory – that is, the question of whether classical probability can model the experimental data of quantum theory (Bell Reference Bell1964; Einstein et al. Reference Einstein, Podolsky and Rosen1935). From these investigations, it follows that when probability is applied generally to a physical system, classical probability models the lack of knowledge of an underlying deterministic reality, whereas non-classical probability, and possibly quantum probability, results when indeterminism arises from the interaction between (measurement) context and system, introducing genuine potentiality for the system states (Aerts Reference Aerts1986). This allowed the identification of situations in macroscopic reality entailing such non-classical indeterminism and therefore being unable to be modeled by classical probability (Aerts Reference Aerts1986; Aerts et al. Reference Aerts, Durt, Grib, Van Bogaert and Zapatrin1993). It shows that opinion polls, where human decisions are intrinsically influenced by the context, constitute such situations, and therefore entail non-classical probability (Aerts & Aerts Reference Aerts and Aerts1995). Our first argument to support and strengthen the authors' claim is that a generalization of classical probability is necessary whenever intrinsically contextual situations evoking indeterminism and potentiality are present (Aerts Reference Aerts1986). We believe this to be commonly the case in situations of human cognition, and believe quantum probability to be a plausible description for this indeterminism and potentiality.

Another result followed from studying the structure and dynamics of human concepts themselves: how concepts combine to form sentences and carry meaning in human thought. An investigation into the relation of concepts to their exemplars allowed for the devising of a Gedankenexperiment violating Bell's inequalities, identifying the presence of quantum entanglement (Aerts et al. Reference Aerts, Aerts, Broekaert and Gabora2000). Considering a combination of concepts and its relation to exemplars led to an experimental violation of Bell's inequalities, proving that concepts entangle when they combine (Aerts & Sozzo Reference Aerts and Sozzo2011a). We studied the guppy effect: Whereas a guppy is considered a very typical “pet-fish,” it is regarded as neither a typical “pet” nor a typical “fish.” The study of this effect, proved to be impossible to model with classical fuzzy sets by Osherson and Smith (Reference Osherson and Smith1981), led us to develop a quantum-based concept theory presenting the guppy effect as a non-classical context influence. Concepts are modeled as entities in states in a complex Hilbert space, and quantities such as typicality are described by observables in the quantum formalism (Aerts & Gabora Reference Aerts and Gabora2005a; Reference Aerts and Gabora2005b; Gabora & Aerts Reference Gabora and Aerts2002). Our second argument to support and strengthen the authors' approach is that next to quantum effects such as entanglement and contextuality, typical quantum representations of states and observables appear in the combination dynamics of human concepts.

An abundance of experimental data violating set theoretical and classical logic relations in the study of the conjunctions and disjunctions of concepts (Hampton Reference Hampton1988b; 1998b) led to the identification of new quantum effects – interference and emergence – when these data were modeled using our quantum concept formalism. Fock space, a special Hilbert space also used in quantum field theory, turns out to constitute a natural environment for these data. For the combination of two concepts, the first sector of Fock space, mathematically describing interference, corresponds to emergent aspects of human thought, and the second sector, mathematically describing entanglement, corresponds to logical aspects of human thought (Aerts Reference Aerts2007; Reference Aerts2009; Reference Aerts, Aerts, Broekaert, D'Hooghe and Note2011; Aerts & D'Hooghe Reference Aerts and D'Hooghe2009; Aerts et al., in press). The quantum superposition in Fock space, representing both emergent and logical thought, models Hampton's (1988a; 1988b) data well in our approach. Our third argument to support and strengthen the authors' analysis is that the quantum formalism, and many detailed elements of its mathematical structure, for example, Fock space, has proved to be relevant for the structure of human thought itself.

We finish our commentary by presenting a graphic illustration of the interference of concepts as it appears in our quantum concept theory. Figure 1 represents the cognitive interference of the two concepts “fruits” and “vegetables” combined in the disjunction “fruits or vegetables.” Part “F,” Part “V,” and Part “F or V” illustrate the relative membership weights of the different exemplars with respect to “fruits,” “vegetables,” and “fruits or vegetables,” respectively, measured in Hampton (Reference Hampton1988a) and presented in Table 1. The illustration is built following standard quantum theory in a Hilbert space of complex wave functions in a plane. The exemplars are located at spots of the plane such that the squares of the absolute values of the quantum wave functions for “fruits,” “vegetables” and “fruits or vegetables” coincide with the relative membership weights measured.

Figure 1. Part “F,” Part “V,” and Part “F or V” are a graphical representation of the relative membership weights of the indicated exemplars with respect to “fruits,” “vegetables,” and “fruits or vegetables,” respectively. The light intensity at the spots where the exemplars are located is a measure of the relative membership weight at that spot, and hence the graphs can be interpreted as light sources passing through slits “F,” “V,” and “F and V.”

Table 1. Relative membership weights of exemplars with respect to fruits, vegetables and fruits or vegetables as measured by Hampton (Reference Hampton1988a)

The wave function for “fruits or vegetables” is the normalized sum – that is, the superposition – of the two wave functions for “fruits” and for “vegetables,” and hence the square of its absolute value includes an interference term. The light intensity at the spots where the exemplars are located is a measure of the relative membership weight at that spot, which means that the graphs can be seen as representations of light passing through slits, where Part “F” corresponds to slit “F” open, Part “V” to slit “V” open, and Part “F or V” to both slits “F” and “V” open. Hence, the graphs illustrate the cognitive interference of “fruits or vegetables” in analogy with the famous double-slit interference pattern of quantum theory (Feynman Reference Feynman1988). The interference pattern is clearly visible (Part “F or V” of Fig. 1), and very similar to well-known interference patterns of light passing through an elastic material under stress. Mathematical details can be found in Aerts et al. (in press).