Pothos & Busemeyer (P&B) are to be congratulated for their target article, which presents an exciting new paradigm for modeling cognition. We suggest that by including a comparative perspective grounded in scientific realism, quantum probability (QP) theory may have even more promise as a framework for describing cognition in humans and nonhumans alike. Moreover, from this view, QP and classical probability (CP) theory are viewed as complementary, not contradictory, frameworks.
Shepard (Reference Shepard1994) argued that organisms that evolved in a physical world would likely have internalized the constraints of that world. If QP theory provides the formal basis for the physical description of the world, then it should also provide the formal basis for the description of cognitive processes within organisms that evolved in that world. (It is interesting to consider why quantum mechanics is hard to understand, but we leave that for another day). However, P&B seem tempted to abandon scientific realism – prematurely, in our view. In the context of discussing the implications of QP for human rationality, they write: “The main problem with classical optimality is that it assumes a measurable, objective reality,” and later, “Quantum theory assumes no measurable objective reality” (sect 5). It is understandable that P&B draw clear distinctions between CP and QP models, but there is a risk of losing sight of their complementary aspect.
If, as most believe, CP describes the workings of chance in the real world, evolution would favor organisms that are able to make decisions in accordance with CP principles, particularly in situations in which recurrent choices are possible, because such decisions would produce the best outcome in the long run. There is considerable evidence for the utility of optimality models in describing nonhuman behavior, both in the laboratory and natural ecology (e.g., optimal foraging theory [Charnov Reference Charnov1976; Koselj et al. Reference Koselj, Schnitzler and Siemers2011; Stephens & Krebs Reference Stephens and Krebs1986]; ideal free distribution [Krivan et al. Reference Krivan, Cressman and Schneider2008; Sebastian-Gonzalez et al. Reference Sebastian-Gonzalez, Botella, Sempere and Sanchez-Zapata2010]; behavioral economics [Hursh Reference Hursh1984; Rachlin et al. Reference Rachlin, Green, Kagel, Battalio and Bower1976]). However, it is important to acknowledge that some theorists have proposed evolutionary accounts of apparently “irrational” human behavior. For example, De Fraja (Reference De Fraja2009) produced a plausible account of why conspicuous consumption might be evolutionarily adaptive, and Kenrick et al. (Reference Kenrick, Griskevicius, Sundie, Li, Li and Neuberg2009) argue for the modularity of utility across different domains (see also Robson & Samuelson Reference Robson and Samuelson2011).
Notably, there are situations in which nonhumans behave optimally and consistent with CP, whereas humans do not. A prominent example is the base-rate fallacy, associated with Tversky and Kahneman's (Reference Tversky, Kahneman and Fishbein1980) “taxi cab” problem. Here subjects are asked to estimate the probability that a taxi involved in a hit-and-run accident was blue or green. They are told that 85% of the cabs in the city are green, but that a witness identified the cab as blue, whose accuracy was later found to be 80% under comparable conditions. Typically humans estimate the probability of the cab being blue as greater than 50%, thus placing too much weight on the witness testimony; the correct answer (from Bayes' theorem) is 41%. By contrast, nonhumans (pigeons) did not neglect base rates and responded optimally when tested in an analogue task (Hartl & Fantino Reference Hartl and Fantino1996). Another example in which humans fail to respond optimally but nonhumans do not is probability matching (Fantino & Esfandiari Reference Fantino and Esfandiari2002).
These species differences can be accommodated within QP theory by assuming that nonhumans construct orthogonal representations in situations with multiple sources of information, leading to behavior consistent with CP and optimality, whereas humans do not. This corresponds to the distinction between “compatible” and “incompatible” questions in the target article. From a comparative perspective, the question is why and under what conditions would humans be more likely to construct nonorthogonal representations (“incompatible questions”).
Comparative research on visual categorization suggests that differences in attention-related processes may provide a clue. In the randomization procedure, subjects view circular discs with sine-wave gratings that vary in frequency and orientation, and assign each to one of two categories (Ashby & Gott Reference Ashby and Gott1988; Maddox et al. Reference Maddox, Ashby and Bohil2003). Two types of tasks are commonly used: one in which category exemplars are defined in terms of differences along one dimension, whereas the other dimension is irrelevant, and one in which both dimensions are relevant. The former is termed a “rule-based” (RB) task because accurate performance can be described by a verbal rule, whereas the latter is known as “information integration” (II) because correct responding depends on both dimensions. With training, humans can learn to respond correctly on the II task but when debriefed cannot describe their performance in terms of verbal rule, and typically say that they responded on the basis of a “gut feeling” (Ashby & Maddox Reference Ashby and Maddox2005).
The RB task is one in which selective attention facilitates performance, whereas the II task requires divided attention. Humans and nonhuman primates are able to learn the RB task faster and to a higher degree of asymptotic accuracy than the II task (Smith et al. Reference Smith, Beran, Crossley, Boomer and Ashby2010). In contrast, pigeons show no difference in learning rate or accuracy (Smith et al. Reference Smith, Ashby, Berg, Murphy, Spiering, Cook and Grace2011). Smith et al. (Reference Smith, Berg, Cook, Murphy, Crossley, Boomer, Spiering, Beran, Church, Ashby and Grace2012) have argued that these species differences suggest that the primate lineage may have evolved an increased capacity for selective attention. In terms of QP theory, selective attention may facilitate a shift in task representation. Pigeons may be constrained to use an orthogonal representation, in which both dimensions are given equal weight, leading to similar performance on both tasks, whereas the representation used by human and nonhuman primates is more flexible and allows for faster and more accurate learning in the RB task.
We agree that QP theory is an exciting new direction for modeling of human cognition, but as the previous examples suggest, it may benefit from a comparative perspective. An enduring controversy in comparative cognition has been the question of differences in intelligence across species (Macphail Reference Macphail1987), including whether human and nonhuman differences should be understood in qualitative or quantitative terms (Roth & Dicke Reference Roth and Dicke2005). We are intrigued by the possibility that QP theory may ultimately help to resolve this question.
Pothos & Busemeyer (P&B) are to be congratulated for their target article, which presents an exciting new paradigm for modeling cognition. We suggest that by including a comparative perspective grounded in scientific realism, quantum probability (QP) theory may have even more promise as a framework for describing cognition in humans and nonhumans alike. Moreover, from this view, QP and classical probability (CP) theory are viewed as complementary, not contradictory, frameworks.
Shepard (Reference Shepard1994) argued that organisms that evolved in a physical world would likely have internalized the constraints of that world. If QP theory provides the formal basis for the physical description of the world, then it should also provide the formal basis for the description of cognitive processes within organisms that evolved in that world. (It is interesting to consider why quantum mechanics is hard to understand, but we leave that for another day). However, P&B seem tempted to abandon scientific realism – prematurely, in our view. In the context of discussing the implications of QP for human rationality, they write: “The main problem with classical optimality is that it assumes a measurable, objective reality,” and later, “Quantum theory assumes no measurable objective reality” (sect 5). It is understandable that P&B draw clear distinctions between CP and QP models, but there is a risk of losing sight of their complementary aspect.
If, as most believe, CP describes the workings of chance in the real world, evolution would favor organisms that are able to make decisions in accordance with CP principles, particularly in situations in which recurrent choices are possible, because such decisions would produce the best outcome in the long run. There is considerable evidence for the utility of optimality models in describing nonhuman behavior, both in the laboratory and natural ecology (e.g., optimal foraging theory [Charnov Reference Charnov1976; Koselj et al. Reference Koselj, Schnitzler and Siemers2011; Stephens & Krebs Reference Stephens and Krebs1986]; ideal free distribution [Krivan et al. Reference Krivan, Cressman and Schneider2008; Sebastian-Gonzalez et al. Reference Sebastian-Gonzalez, Botella, Sempere and Sanchez-Zapata2010]; behavioral economics [Hursh Reference Hursh1984; Rachlin et al. Reference Rachlin, Green, Kagel, Battalio and Bower1976]). However, it is important to acknowledge that some theorists have proposed evolutionary accounts of apparently “irrational” human behavior. For example, De Fraja (Reference De Fraja2009) produced a plausible account of why conspicuous consumption might be evolutionarily adaptive, and Kenrick et al. (Reference Kenrick, Griskevicius, Sundie, Li, Li and Neuberg2009) argue for the modularity of utility across different domains (see also Robson & Samuelson Reference Robson and Samuelson2011).
Notably, there are situations in which nonhumans behave optimally and consistent with CP, whereas humans do not. A prominent example is the base-rate fallacy, associated with Tversky and Kahneman's (Reference Tversky, Kahneman and Fishbein1980) “taxi cab” problem. Here subjects are asked to estimate the probability that a taxi involved in a hit-and-run accident was blue or green. They are told that 85% of the cabs in the city are green, but that a witness identified the cab as blue, whose accuracy was later found to be 80% under comparable conditions. Typically humans estimate the probability of the cab being blue as greater than 50%, thus placing too much weight on the witness testimony; the correct answer (from Bayes' theorem) is 41%. By contrast, nonhumans (pigeons) did not neglect base rates and responded optimally when tested in an analogue task (Hartl & Fantino Reference Hartl and Fantino1996). Another example in which humans fail to respond optimally but nonhumans do not is probability matching (Fantino & Esfandiari Reference Fantino and Esfandiari2002).
These species differences can be accommodated within QP theory by assuming that nonhumans construct orthogonal representations in situations with multiple sources of information, leading to behavior consistent with CP and optimality, whereas humans do not. This corresponds to the distinction between “compatible” and “incompatible” questions in the target article. From a comparative perspective, the question is why and under what conditions would humans be more likely to construct nonorthogonal representations (“incompatible questions”).
Comparative research on visual categorization suggests that differences in attention-related processes may provide a clue. In the randomization procedure, subjects view circular discs with sine-wave gratings that vary in frequency and orientation, and assign each to one of two categories (Ashby & Gott Reference Ashby and Gott1988; Maddox et al. Reference Maddox, Ashby and Bohil2003). Two types of tasks are commonly used: one in which category exemplars are defined in terms of differences along one dimension, whereas the other dimension is irrelevant, and one in which both dimensions are relevant. The former is termed a “rule-based” (RB) task because accurate performance can be described by a verbal rule, whereas the latter is known as “information integration” (II) because correct responding depends on both dimensions. With training, humans can learn to respond correctly on the II task but when debriefed cannot describe their performance in terms of verbal rule, and typically say that they responded on the basis of a “gut feeling” (Ashby & Maddox Reference Ashby and Maddox2005).
The RB task is one in which selective attention facilitates performance, whereas the II task requires divided attention. Humans and nonhuman primates are able to learn the RB task faster and to a higher degree of asymptotic accuracy than the II task (Smith et al. Reference Smith, Beran, Crossley, Boomer and Ashby2010). In contrast, pigeons show no difference in learning rate or accuracy (Smith et al. Reference Smith, Ashby, Berg, Murphy, Spiering, Cook and Grace2011). Smith et al. (Reference Smith, Berg, Cook, Murphy, Crossley, Boomer, Spiering, Beran, Church, Ashby and Grace2012) have argued that these species differences suggest that the primate lineage may have evolved an increased capacity for selective attention. In terms of QP theory, selective attention may facilitate a shift in task representation. Pigeons may be constrained to use an orthogonal representation, in which both dimensions are given equal weight, leading to similar performance on both tasks, whereas the representation used by human and nonhuman primates is more flexible and allows for faster and more accurate learning in the RB task.
We agree that QP theory is an exciting new direction for modeling of human cognition, but as the previous examples suggest, it may benefit from a comparative perspective. An enduring controversy in comparative cognition has been the question of differences in intelligence across species (Macphail Reference Macphail1987), including whether human and nonhuman differences should be understood in qualitative or quantitative terms (Roth & Dicke Reference Roth and Dicke2005). We are intrigued by the possibility that QP theory may ultimately help to resolve this question.