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6. Modeling of cognitive effects

In cognitive and social science, the following opinion pool is known as the basic example of the order effect. This is the Clinton–Gore opinion pool (Moore, 2002). In this experiment, American citizens were asked one question at a time, e.g.,

$A=$ “Is Bill Clinton honest and trustworthy?”

$B=$ “Is Al Gore honest and trustworthy?”

Two sequential probability distributions were calculated on the basis of the experimental statistical data, $p_{A,B}$ and $p_{B,A}$ (first question$A$  and then question $B$ and vice verse).

6.1. Order effect for sequential questioning

The statistical data from this experiment demonstrated the question order effect QOE, dependence of sequential joint probability distribution for answers to the questions on their order $p_{(A,B)}\neq p_{(B,A)}$. We remark that in the CP-model these probability distributions coincide:

$p_{A,B}(\alpha ,\beta )=P(\omega \in \Omega :A(\omega )=\alpha ,B(\omega )=\beta )=p_{A,B}(\beta ,\alpha )$

where $\Omega$ is a sample space $P$ and  is a probability measure.

QOE stimulates application of the QP-calculus to cognition, see paper (Wang and Busemeyer, 2013). The authors of this paper stressed that noncommutative feature of joint probabilities can be modeled by using noncommutativity of incompatible quantum observables  $A,B$ represented by Hermitian operators ${\widehat {A}},{\widehat {B}}$ . Observable  $A$ represents the Clinton-question and observable $B$ represents Gore-question. In this model, QOE is identical incompatibility–noncommutativity of observables:

$[{\widehat {A}},{\widehat {B}}]\neq 0$

6.2. Response replicability effect for sequential questioning

The approach based on identification of the order effect with noncommutative representation of questions (Wang and Busemeyer, 2013) was criticized in paper (Khrennikov et al., 2014). To discuss this paper, we recall the notion of response replicability. Suppose that a person, say John, is asked some question $A$ and suppose that he replies, e.g, “yes”. If immediately after this, he is asked the same question again, then he replies “yes” with probability one. We call this property $A-A$ response replicability. In quantum physics,  $A-A$ response replicability is expressed by the projection postulate.The Clinton–Gore opinion poll as well as typical decision making experiments satisfy $A-A$ response replicability. Decision making has also another feature -  $A-A$response replicability. Suppose that after answering the $A$-question with say the “yes”-answer, John is asked another question $B$. He replied to it with some answer. And then he is asked $A$ again. In the aforementioned social opinion pool, John repeats her original answer to $A$, “yes” (with probability one).

This behavioral phenomenon we call $A-B-A$ response replicability. Combination of  $A-A$ with  $A-B-A$ and $B-A-B$ response replicability is called the response replicability effect RRE.

6.3. “QOE+RRE”: described by quantum instruments of non-projective type

In paper (Khrennikov et al., 2014), it was shown that by using the von Neumann calculus it is impossible to combine RRE with QOE. To generate QOE, Hermitian operators  ${\widehat {A}},{\widehat {B}}$ should be noncommutative, but the latter destroys$A-B-A$ response replicability of $A$. This was a rather unexpected result. It made even impression that, although the basic cognitive effects can be quantum-likely modeled separately, their combinations cannot be described by the quantum formalism.

However, recently it was shown that theory of quantum instruments provides a simple solution of the combination of QOE and RRE effects, see Ozawa and Khrennikov (2020a) for construction of such instruments. These instruments are of non-projective type. Thus, the essence of QOE is not in the structure of observables, but in the structure of the state transformation generated by measurements’ feedback. QOE is not about the joint measurement and incompatibility (noncommutativity) of observables, but about sequential measurement of observables and sequential (mental-)state update. Quantum instruments which are used in Ozawa and Khrennikov (2020a) to combine QOE and RRE correspond to measurement of observables represented by commuting operators ${\widehat {A}},{\widehat {B}}$. Moreover, it is possible to prove that (under natural mathematical restriction) QOE and RRE can be jointly modeled only with the aid of quantum instruments for commuting observables.

6.4. Mental realism

Since very beginning of quantum mechanics, noncommutativity of operators ${\widehat {A}},{\widehat {B}}$ representing observables $A,B$ was considered as the mathematical representation of their incompatibility. In philosophic terms, this situation is treated as impossibility of the realistic description. In cognitive science, this means that there exist mental states such that an individual cannot assign the definite values to both observables (e.g., questions). The mathematical description of QOE with observables represented by noncommutative operators (in the von Neumann’s scheme) in Wang and Busemeyer (2013) and Wang et al. (2014) made impression that this effect implies rejection of mental realism. The result of Ozawa and Khrennikov (2020a) demonstrates that, in spite of experimentally well documented QOE, the mental realism need not be rejected. QOE can be modeled within the realistic picture mathematically given by the joint probability distribution of observables $A$ and $B$, but with the noncommutative action of quantum instruments updating the mental state:

$[{\mathcal {J_{A}(x)}},{\mathcal {J_{B}(x)}}]={\mathcal {J_{A}(x)}}{\mathcal {J_{B}(x)}}-{\mathcal {J_{B}(x)}},{\mathcal {J_{A}(x)}}\neq 0$

$(20)$

This is the good place to remark that if, for some state $\rho ,[\Im _{A}(x),\Im _{B}(x)]\rho =0$, then QOE disappears, even if $[\Im _{A}(x),\Im _{B}(x)]\neq 0$. This can be considered as the right formulation of Wang–Bussemeyer statement on connection of QOE with noncommutativity. Instead of noncommutativity of operators  ${\widehat {A}}$ and  ${\widehat {B}}$ symbolically representing quantum obseravbles, one has to speak about noncommutativity of corresponding quantum instruments.

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