Pothos & Busemeyer (P&B) propose that many existing models fail to account for all aspects of cognitive processing, and discuss a number of phenomena for which the formal framework of classical probability theory (CPT) seems to fail. However, virtually all examples were based on hypothetical reasoning or experiments under artificial laboratory or Gallup poll conditions. Therefore, even if quantum probability theory (QPT) can account better for some of these phenomena, it is not clear that the much stronger claim that QPT offers unique insights regarding cognition and the nature of human rationality is supported at this stage. To support this stronger claim, P&B would have needed to establish that the cases discussed are paradigm cases of human cognition and not artifacts of the experimental setup. Further, to argue for QPT, P&B need to do more than observe that QPT predictions differ from CPT ones. Before arguing “that the quantum approach to cognition embodies all the characteristics of good theory” (sect. 6.4) P&B need to show in detail over a wide range of cases that specific numerical QPT predictions are correct.
One rather serious problem is the informal and vague presentation throughout. There seem to be two components/aspects to P&B's QP model, distinguishing it from CP. One aspect is the issue of and motivation for vectors space models, which could be real but, nevertheless, would have associated non-classical logical operators (disjunction, conjunction, negation, implication) that could account for non-commutativity effects. The second aspect is the issue of and motivation for complex vector spaces. P&B attempt to justify their preference for QPT by stressing the value of projection operators. But projection operators can be defined on real (in fact, on any) vector space. Real vector spaces are used in many areas such as machine learning and information retrieval (e.g., van Rijsbergen, Reference van Rijsbergen2004; Widdows Reference Widdows2004). The key concepts required for similarity matching include inner product, length, angle, and projection. But none of those are unique to complex vector spaces. However, P&B discuss these notions in the context of finite-dimensional complex vector spaces; their Hilbert spaces. As the arguments seem to support only real vector space models over classical set-theoretic models, P&B would need an independent argument showing that similarity matching for cognition requires complex numbers. P&B have not shown that any of the examples require quantum probability amplitudes, which involve complex numbers. For example, the concept of superposition does not require complex values but applies to linear systems quite generally and hence to vector spaces. Conflating notions that need to be kept separate makes it difficult to evaluate whether the application of quantum probability is justified.
Turning to the work discussed, it appears that especially the Linda case, on which much argumentative burden rests, is problematic for several reasons. First, the case is not based on an actual person but on one that had been “constructed to be representative of an active feminist and unrepresentative of a bank teller” (Tversky & Kahneman Reference Tversky, Kahneman and Shafir2004, p. 227). Reasoning under such artificial conditions seems hardly a prototypical case of rational decision making. Further, not all participants committed the conjunction fallacy and the authors of the study point out that “because individuals who have different knowledge or hold different beliefs must be allowed to assign different probabilities to the same event no single value can be correct for all people” (Tversky & Kahneman Reference Tversky, Kahneman and Shafir2004, p. 221). It is likely that some participants in such experiments rely, at least to some degree, on incorrect background information. This might explain the violation of the law of total probability, the conjunction fallacy. Undoubtedly, these experiments establish that models that assume fully rational agents do not apply to humans. But they do little to indicate whether QPT is a better model for human cognition than is CPT. Possibly non-QPT models can account for the observed anomalies if we assume that the reasoner relied to some degree on incorrect information and/or did not reason fully rationally. Similarly, the artificial conditions of the one-shot prisoner dilemma seem to interfere in some (but not all) individuals with rational decision making (Shafir & Tversky Reference Shafir, Tversky and Shafir2004). But more realistic reiterated prisoner dilemma tasks lead to different results (e.g., Axelrod & Hamilton Reference Axelrod and Hamilton1981; Nowak & Sigmund Reference Nowak and Sigmund1992). These cases seem to establish that rationality breaks down in some cases but not in others. This is very different from quantum physics, in which it is always the case that “knowledge of the position of a particle imposes uncertainty on its momentum” (sect. 1.1). Hence, the analogy between cognition and quantum physics might be much weaker than P&B suggest. They assume that any ad hoc models are inferior to a single, uniform mathematical framework (whether it be CPT or QPT). Given the complexity of human decision making, however, ad hoc models that make better predictions might be preferable over principled frameworks that make inferior predictions.
Another area of concern is the claim that “there are situations in which the distinctive features of QP provide a more accurate and elegant explanation for empirical data” (sect. 1.2). However, the specific examples discussed involve only the interaction between two parameters. In such a constrained setting, it may be true that for quantum probability theory “the relevant mathematics is simple and mostly based on geometry and linear algebra” (sect. 1.2). But it is not clear at all whether and how the proposed models can account for the multiply intertwined interactions between existing knowledge and any imaginable problem solving task. Take the discussion of the interference effects between “happiness” and “employment.” The superficial and sketchy discussion of the superposition effects of just these two variables seems to support the conclusion that emotional judgments are the results of constructive processes. However, anyone accepting the claim that QPT “fundamentally requires a constructive role for the process of disambiguating a superposition state” (sect. 2.1), has to ask whether such disambiguation is possible in realistic cases in which all factors that affect the emotional state of a person need to be considered. P&B seem to have no principled way of determining whether two attributes are incompatible. But such a distinction would be required before we can decide whether QPT is needed to replace CPT. The authors observe that classical optimality “assumes a measurable, objective reality and an omniscient observer” (sect. 5). This assumption is problematic. But this has been discussed in the philosophical literature at least since Hume (Reference Hume1751/1999; see also Cartwright Reference Cartwright1999), and current models do not attempt to make predictions for all aspects of the “real, noisy, confusing, ever-changing, chaotic world” (sect. 5). Many questions regarding the best model for cognition remain open and it seems premature to assert, “the QP cognitive program…is clearly an important direction for future research” (sect. 6.3, emphasis added).
Pothos & Busemeyer (P&B) propose that many existing models fail to account for all aspects of cognitive processing, and discuss a number of phenomena for which the formal framework of classical probability theory (CPT) seems to fail. However, virtually all examples were based on hypothetical reasoning or experiments under artificial laboratory or Gallup poll conditions. Therefore, even if quantum probability theory (QPT) can account better for some of these phenomena, it is not clear that the much stronger claim that QPT offers unique insights regarding cognition and the nature of human rationality is supported at this stage. To support this stronger claim, P&B would have needed to establish that the cases discussed are paradigm cases of human cognition and not artifacts of the experimental setup. Further, to argue for QPT, P&B need to do more than observe that QPT predictions differ from CPT ones. Before arguing “that the quantum approach to cognition embodies all the characteristics of good theory” (sect. 6.4) P&B need to show in detail over a wide range of cases that specific numerical QPT predictions are correct.
One rather serious problem is the informal and vague presentation throughout. There seem to be two components/aspects to P&B's QP model, distinguishing it from CP. One aspect is the issue of and motivation for vectors space models, which could be real but, nevertheless, would have associated non-classical logical operators (disjunction, conjunction, negation, implication) that could account for non-commutativity effects. The second aspect is the issue of and motivation for complex vector spaces. P&B attempt to justify their preference for QPT by stressing the value of projection operators. But projection operators can be defined on real (in fact, on any) vector space. Real vector spaces are used in many areas such as machine learning and information retrieval (e.g., van Rijsbergen, Reference van Rijsbergen2004; Widdows Reference Widdows2004). The key concepts required for similarity matching include inner product, length, angle, and projection. But none of those are unique to complex vector spaces. However, P&B discuss these notions in the context of finite-dimensional complex vector spaces; their Hilbert spaces. As the arguments seem to support only real vector space models over classical set-theoretic models, P&B would need an independent argument showing that similarity matching for cognition requires complex numbers. P&B have not shown that any of the examples require quantum probability amplitudes, which involve complex numbers. For example, the concept of superposition does not require complex values but applies to linear systems quite generally and hence to vector spaces. Conflating notions that need to be kept separate makes it difficult to evaluate whether the application of quantum probability is justified.
Turning to the work discussed, it appears that especially the Linda case, on which much argumentative burden rests, is problematic for several reasons. First, the case is not based on an actual person but on one that had been “constructed to be representative of an active feminist and unrepresentative of a bank teller” (Tversky & Kahneman Reference Tversky, Kahneman and Shafir2004, p. 227). Reasoning under such artificial conditions seems hardly a prototypical case of rational decision making. Further, not all participants committed the conjunction fallacy and the authors of the study point out that “because individuals who have different knowledge or hold different beliefs must be allowed to assign different probabilities to the same event no single value can be correct for all people” (Tversky & Kahneman Reference Tversky, Kahneman and Shafir2004, p. 221). It is likely that some participants in such experiments rely, at least to some degree, on incorrect background information. This might explain the violation of the law of total probability, the conjunction fallacy. Undoubtedly, these experiments establish that models that assume fully rational agents do not apply to humans. But they do little to indicate whether QPT is a better model for human cognition than is CPT. Possibly non-QPT models can account for the observed anomalies if we assume that the reasoner relied to some degree on incorrect information and/or did not reason fully rationally. Similarly, the artificial conditions of the one-shot prisoner dilemma seem to interfere in some (but not all) individuals with rational decision making (Shafir & Tversky Reference Shafir, Tversky and Shafir2004). But more realistic reiterated prisoner dilemma tasks lead to different results (e.g., Axelrod & Hamilton Reference Axelrod and Hamilton1981; Nowak & Sigmund Reference Nowak and Sigmund1992). These cases seem to establish that rationality breaks down in some cases but not in others. This is very different from quantum physics, in which it is always the case that “knowledge of the position of a particle imposes uncertainty on its momentum” (sect. 1.1). Hence, the analogy between cognition and quantum physics might be much weaker than P&B suggest. They assume that any ad hoc models are inferior to a single, uniform mathematical framework (whether it be CPT or QPT). Given the complexity of human decision making, however, ad hoc models that make better predictions might be preferable over principled frameworks that make inferior predictions.
Another area of concern is the claim that “there are situations in which the distinctive features of QP provide a more accurate and elegant explanation for empirical data” (sect. 1.2). However, the specific examples discussed involve only the interaction between two parameters. In such a constrained setting, it may be true that for quantum probability theory “the relevant mathematics is simple and mostly based on geometry and linear algebra” (sect. 1.2). But it is not clear at all whether and how the proposed models can account for the multiply intertwined interactions between existing knowledge and any imaginable problem solving task. Take the discussion of the interference effects between “happiness” and “employment.” The superficial and sketchy discussion of the superposition effects of just these two variables seems to support the conclusion that emotional judgments are the results of constructive processes. However, anyone accepting the claim that QPT “fundamentally requires a constructive role for the process of disambiguating a superposition state” (sect. 2.1), has to ask whether such disambiguation is possible in realistic cases in which all factors that affect the emotional state of a person need to be considered. P&B seem to have no principled way of determining whether two attributes are incompatible. But such a distinction would be required before we can decide whether QPT is needed to replace CPT. The authors observe that classical optimality “assumes a measurable, objective reality and an omniscient observer” (sect. 5). This assumption is problematic. But this has been discussed in the philosophical literature at least since Hume (Reference Hume1751/1999; see also Cartwright Reference Cartwright1999), and current models do not attempt to make predictions for all aspects of the “real, noisy, confusing, ever-changing, chaotic world” (sect. 5). Many questions regarding the best model for cognition remain open and it seems premature to assert, “the QP cognitive program…is clearly an important direction for future research” (sect. 6.3, emphasis added).
ACKNOWLEDGMENTS
I thank David Johnson and Robert Levine for their detailed comments. All remaining errors are mine.