Shepard made this statement in arguing that a universal law of generalization could be had only by formulating it around stimuli existing in a psychological, not a physical, space. Only then, according to Shepard, could principles of generalization be established that were invariant to context. Pothos & Busemeyer (P&B) are making a very similar argument. They contend that the probabilistic nature of cognition may adhere not to the assumptions of classical probability (CP), but instead to the properties of quantum probability (QP).

Perhaps this is the reconceptualization that is needed. Phenomena that were errors according to CP, such as violations of the conjunction rule, order effects, and violations of the sure-thing principle, are consistent within the structural constraints of QP. Moreover, a quantum account can offer a more parsimonious explanation of the phenomena. Take, for example, the Linda problem. The typical explanation of this problem has been the representativeness heuristic in which people substitute similarity for probability (Kahneman Reference Kahneman2003). In comparison, similarity is implicit in the QP calculation (reflected in the angle of the initial state vector relative to the bases).

The QP model of the Linda problem and other judgments of subjective probability (Busemeyer et al. Reference Busemeyer, Pothos, Franco and Trueblood2011) are intriguing because they describe how people actually assign probabilities. However, P&B use the same QP modeling framework to describe both the process of making subjective probabilities (e.g., the Linda problem) and the process of making a choice (e.g., Clinton vs. Gore honesty choice). This failure to distinguish between subjective probabilities and choice proportions presents a concern. To illustrate, consider another version of the Linda problem that researchers have used, in which people have to identify which of two statements is more probable (e.g., “Linda is a bank teller” or “Linda is a bank teller and is active in the feminist movement”) (e.g., Tversky & Kahneman Reference Tversky and Kahneman1983). When applied to choice proportions, QP is used to predict the objective probability of choosing either statement by projecting onto a basis vector. However, in modeling subjective probabilities, the model uses the same operator; therefore, it is as if people have direct access to the probability of entering into that state. Is the same cognitive system used for these different responses? To suppose these two models are of the same class seems to be an error. To do so would be to assume that subjective ratings of probability are the same as the probability of choosing a particular response.

The distinction between modeling the probability distributions over behaviors and modeling probability judgments is important. In the Linda problem, the probability judgment response is only one of many responses in which the conjunction fallacy is observed. The primary data, for example, were in terms of ranking eight possible hypotheses in order from least to most likely (Tversky & Kahneman Reference Tversky and Kahneman1983). We also know that the extent to which people commit the conjunction fallacy changes based on the format of the response, such as with bets (Tversky & Kahneman Reference Tversky and Kahneman1983), or with frequency formats (Mellers et al. Reference Mellers, Hertwig and Kahneman2001). It would seem that a more complete process level account of the conjunction fallacy would explain how these different response formats (or measurements) work, and would inform any model predicting the effect.

A more general comment regarding the thesis of the article is that quantum theory can provide a “better” probabilistic framework for cognition. The flexibility of QP theory may become a concern, as skeptics might argue that it lacks specific predictions. However, many predictions may be overlooked simply because they reflect common sense or because the theory is new enough that due scrutiny has not been afforded to uncover psychological inferences from Gleason's theorem or the Hilbert space. Take, for example, the prediction that projecting a state vector to a lower dimensional subspace will result in lost amplitude. This does appear to be consistent with the famous Korea/China asymmetry in similarity judgments (Tversky Reference Tversky1977), and certainly similar predictions can be made in other areas, such as comparing emotions of different dimensionality.

Perhaps the more difficult aspect is explanation. This seems tricky, as we are asked to take a different interpretation of a cognitive system under QP. For example, under a classical view, a cognitive system, at any given time point, is in a particular state. However, in QP we have superposition meaning a cognitive system is in no state at all! This idea would appear to change our ability to point to information as a causal mediating mechanism, as well as our understanding of the nature of mental events themselves. It also seems to change the very nature of what it means to explain cognition, as superposition seems almost uninterpretable. Hughes (Reference Hughes1989) suggests that an explanation in terms of QP is a structural explanation. That is, it shows how the stochastic nature of cognition and its probability functions can be modeled using Hilbert spaces.

Is a structural explanation sufficient for psychologists? This concern and the larger set of issues it raises may not come as a particular surprise, as QP theory was originally designed only to predict the probability of outcomes of physical events. The challenge of modeling subjective judgments and mental processes is obviously a new challenge for QP, and, therefore, it may simply be that more work is needed to attune the QP framework to address psychological, rather than physical, problems. Even so, we suspect that the success of QP in cognitive modeling will depend largely on its reinterpretation, reapplication, and resulting predictive power rather than its narrative explanation.