Lefebvre V.A., Webber J.A. Functions of Fast Reflexion in Bipolar Choice //Reflexive processes and control. Vol.1, N1. P.29-40.

Vladimir A. Lefebvre (USA), Jack Adams-Webber (Canada)

Vladimir A. Lefebvre University of California, Irvine, USA, professor

Jack Adams-Webber Brock University, Ontario, Canada, professor

inborn informational processor is built in into human psyche whose function is to automatically create these images together with their subjective domains. This processor generates a specific specter of human responses not controlled consciously and running extremely fast (one-two milliseconds). This type of reflexion, as distinct from the traditional concept is called fast reflexion (Lefebvre, 1987). In this paper we will decipher the mathematical laws governing the automatic functioning of this inborn processor and show how they reveal themselves in human behavior (Adams-Webber, 1996a). The result of this analysis will be a formal model of the subject with fast reflexion.

Ideally, global theoretical models ought to possess two properties: integrity and uniformity. Integrity means that the model must be able to reflect simultaneously the subjectís perception, behavior, and inner domain. Uniformity requires that different aspects of the subjectís activity must be described in terms of the same theoretical language. The general method for attaining these two properties is to represent the subject as a composition of mathematical functions. Various elements of this composition are interpreted

second phase, the program suddenly changed its tactics in such a way that the same mode of behavior became disadvantageous. In these experiments, it was discovered that the subjects would alter their behavior abruptly. No evidence of gradual retraining was observed (Lefebvre, 1967; 1969; 1972; 1977a; Baranov & Trudoliubov, 1969a,b; Lepsky, 1969).

An alternative approach was to construct simple formal models of subjects in the framework of which concepts reflecting the human subjective domain could acquire clear and unambiguous meaning. This approach led to the construction of several particular models of the subject with reflexion. In this paper we shall limit our consideration to one line of the development of such models associated with studying the bipolar choice (Lefebvre, 1977b; 1980; 1982; 1992; V.A.Lefebvre, V.D.Lefebvre, Adams-Webber, 1986; Krylov, 1994; Miller & Sulcoski, 1999). This theoretical model gave rise to many new methodological problems which have been addressed in an extensive literature (Adams-Webber, 1987; 1995; Batchelder, 1987; Kauffman, 1990; Lefebvre, 1987; 1995; Levitin, 1987; Popper, 1992; Rapoport, 1990; Schreider, 1994; 1998; Townsend, 1983; 1990; Wheeler, 1987; Zajonc, 1987).

Let us mention further that this human modeling mechanism is not a chain of verbal reasoning such as “I think that he thinks that I think,” etc. Such chains are purely linguistic structures. In contrast, we are concerned with the direct computational modeling of the self and other which proceeds automatically and independently of inner speech (Lefebvre, 1985, pp. 291-292).

This hypothesis was published sixteen years ago as more than a methodological declaration. It contained a detailed description of a possible mechanism for cognitive computations; however, experimental evidence in its favor began to appear only recently. For example, Hughes and Cutting (1999) demonstrated that children’s ability to reproduce other persons’ inner domains operates automatically and does not depend on verbal development.

The mechanism for the automated generation of images of the self and others described in this section lies outside the framework of the traditional understanding of reflexion as a human conscious constructive activity connected with his will and desire. While constructing one’s own image consciously a person may provide it with many features of his own choice. Unlike this ‘creative’ activity, the processor under consideration is inseparable from the human being itself. The subject in a normal state does not sense its presence. One cannot eliminate the control of this processor by an effort of will, which is similar to the impossibility of intentionally ceasing to understand words in one’s own native language. We have called the automatic process of generating the images fast reflexion, in order to avoid confusion with a conscious process of comprehending the self and others.

The model represents a subject facing the choice between two alternatives, A and B. One of them, A, plays the role of the positive pole, and the other that of the negative pole (Lefebvre, 1977b; 1980; 1995; 1997). In the simplest case, when the subject’s inner world does not contain the images of other subjects, his choice can be described by the function X₁ = f (x₁,x₂,x₃), where all variables take their values from the interval [0,1] (see Appendix). The value of x3 is the subject’s intention to choose the positive pole. The greater x₃, the more the “desire” to make this choice. The value of X₁ is interpreted as the probability with which the subject is ready to choose the positive pole, A, in reality. The value of x₁ is the world’s pressure toward the positive alternative A at the precise moment of choice. The value of x₂ is the world’s pressure toward A which is expected by the subject based on prior experience. We can consider the values of x₁ and (1 - x₁) as normalized utility-measures of the alternatives A and B at the moment of choice, and the values of x₂ and (1 - x₂) as normalized expected utility-measures.

A mathematical analysis of the function X₁ = f (x₁,x₂,x₃) of three variables demonstrates that it can be represented as a composition of one particular function of two variables, F(x,y), and that such a representation is unique; that is, X₁ = F (x₁, F (x₂,x₃)). Function X₂ = F (x₂,x₃) is interpreted as the image of the self possessed by the subject. Under this interpretation the subject has a functionally correct image of the self, since the same function, F (x,y), describes him from both his own perspective and external points of view. The first variable in this function, x, represents perceptual input, and the second variable, y, is a mental image of the self. Therefore, in considering the function F (x₂,x₃), which plays the role of an image of the self, we have to interpret the variable x₂ as an image of the input, and the variable x₃ as the subject’s mental representation of his image’s image of the self. To avoid confusion, we call this secondary image a model of the self. The variable x₃ now takes on an additional meaning: it represents not only the subject’s intention, but also his model of the self. As a result we obtain a formal analogue of the macro-structure of the subject’s inner domain. This analogue is a structure of composition F (x₁,F (x₂,x₃)). At the

same time, this structure describes the process involved in the cognitive computation of X₁ : first, X₂ = F (x₂,x₃) is computed, and then X₁ = F (x₁,X₂).

The subject’s intention may depend on factors distinct from x₁ and x₂, and, in principle, can take on any value from [0,1]. In cases in which the intention depends only on x₁ and x₂, we assume that the subject’s cognitive mechanism coordinates both the subjective intention, x₃, and an objective readiness, X₁, in such a way that X₁ = x₃ (Lefebvre, 1992). This equation corresponds

Theoretical models may assist in conducting thought experiments. With the help of such an experiment we will clarify the role which the subject’s image of the self plays in his activity. Let the subject face a moral choice. In his value system, alternative A is a good act, and alternative B a bad one. Suppose that on the basis of the previous experience the subject sees the world as pressing him toward choosing the positive pole, that is, x₂ = 1. Let the subject have the intention of choosing the positive pole, that is, x₃ = 1. With these data we find the value of the image of the self: X₂ = F (1,1) = 1 (see Appendix). Thus, the subject “sees” himself performing a good deed. Suppose that at the moment of choice, the world exerts pressure on the subject to choose the negative pole, which means x₁ = 0. Then the subject’s real choice will correspond to the value of the function X₁ = F (0,F(1,1)) = 0, i.e., in reality the subject carries out a bad action. The next time a similar situation arises, the subject will expect the world to press him toward the negative pole, which means that the value of x₂ is changed from 1 to 0. If the subject’s intention is still positive (x₃ = 1), the value of his image of the self becomes X₂ = F (0,1) = 0, and if the world keeps pressing toward a bad action (x₁ = 0), his real choice will correspond to the value X₁ = F (0,F(0,1)) = 1, that is, the subject will choose the positive pole. In the first case, the subject sees himself as “good” but commits a bad action (X₁ = 0); in the second case, the subject sees himself as “bad” but performs a good action (X₁ = 1). The image of the self plays the role the subject’s “conscience”: seeing oneself as “bad” prevents one from choosing the negative pole. It follows from the formal model that when the value of X₂= 0, then the value of X₁ є 1. Therefore, a possible function of the image of the self is to block some of the subject’s actions. If at x₁ = 0 the value of an image of the self suddenly changes from X₂ = 1 to X₂ = 0, we can describe this change in standard introspective language as follows: “the world presses the subject

A distinction between automated and deliberate choice is traditionally based on the notion that a person sometimes consciously plans and then performs an actions as planned, while other times the phase of conscious planning is absent (Bargh & Chartrand, 1999). The reflexive model allows us to make a clear distinction between these two cases: the variable x₃ represents the subject’s will (intention, desire); and the variable X₁ represents his behavior. Only behavior which does not depend on a subject’s will can be called “automatic”; that is, the value of X₁ does not depend on the value of x₃. Similarly, a subject’s behavior can be called “deliberate” only if the identity X₁ є x₃ holds, i.e., it is entirely determined by his will. Finally, mixed cases are possible in which intention x₃ influences X₁, but does not entirely determine it.

An automatic choice is possible only in such pairs x₁ = a and x₂ = b for which the value of the function X₁ = f (a,b,x₃) does not depend on the value of x₃ (see Appendix). As an illustration we will consider an experiment described in Wegner and Wheatley (1999). Their subjects were asked to attempt to read the unconscious muscle movement of a participant confederate whose fingers were placed on top of their own on “yes” and “no” response keys. Then a subject heard a trivial question of the type, “Is the capital of the USA Washington, D.C.?”. In reality, the confederate did not hear the questions and so his finger movements could not depend on their content. Nevertheless, the subjects pushed the correct button 87% of the times. In 63% of the cases they thought that they were acting according to their own will, and in 37% they believed that they felt a slight movement of the confederate’s fingers.

In commenting on this experiment the authors write: “They answered correctly, in other words, but did not have a strong sense of willfully having done so and instead thought the confederate had played a significant part. The pattern of findings across six experiments suggests that the correct answers are produced automatically” (Wegner & Wheatley, 1999, p. 457).

This conclusion corresponds to the mechanism which is represented in the formal model. Certainly, the majority of the subjects did not have any doubts that Washington, D.C. is the capital of the USA. The pressure of the external world was a social ‘demand characteristic’ directed toward the positive pole “tell the truth,” in the given case, “yes,” so x₁ = 1. After the first question the subjects expected the other questions to be of the same type: they anticipated pressure toward true answers; that is, x₂ = 1. As shown

in the Appendix, under such conditions choice is automatic, that is, it does not depend on intention x₃.

In this case the intention, x₃, predetermines entirely the subject’s readiness, X₁. In accordance with the model, this is possible only if both x₁ = 0 and x₂ = 0 simultaneously, that is, in the situation in which the world presses the subject to choose the negative pole (x₁ = 0) and the subject’s expectation is the same (x₂ = 0). Under this condition, X₁ є x₃, that is, any subject’s intention turns into his readiness, X₁. At first glance it seems that such a representation of deliberate or “free” choice is bound up with a particular problem, since real people even when they know the correct answers, may exhibit willfulness and answer incorrectly. To clarify this situation we return to Wegner and Weatney’s experiment (1999), in which the subjects heard questions of the type, “Is the capital of the USA Washington, D.C.?” As we demonstrated earlier, the correct answers to these questions are automatic. But many social psychologists who conduct surveys encounter subjects’ “rebellion,” especially during a series of monotonious trivial questions. Doesn’t this fact contradict the model’s predictions? In fact, the formal model explains the psychological mechanism underlying such rebellion. As we already have demonstrated, in experiments of the Wegner-Weatney type the subject takes on values x₁ = 1 and x₂ = 1. Suppose that after he answers a number of trivial questions, his “value system” suddenly changes: the positive pole, the choice of “tell the truth”, becomes negative and the negative pole, the choice of “lie”, becomes positive. The social imperative “tell the truth” did not disappear, but now it is directed toward the negative pole. A pressure toward the new positive pole is equal to zero. So, x₁ = 0 and x₂ = 0, and the subject’s choice becomes deliberate. Thus, the

An important impetus for the development of this model in the United States was a series of experiments based on bipolar dimensions of judgment represented by pairs of antonymous adjectives (such as strong-weak), within the framework of Kelly’s (1955) personal construct theory, summarized by Adams-Webber (1996b). In the early 1970s an unexpected finding was obtained repeatedly (Adams-Webber & Benjafield, 1973; Benjafield & Adams-Webber, 1976): experimental subjects, on average, assigned their personal acquaintances to the positive poles of dimensions with a relative frequency of 0.62. Benjafield and Adams-Webber (1976) conjectured that the theoretical value of this constant might be the golden section ((Ц5-1)/2=0.618...).

The first attempt to explain the underlying psychological mechanism and specify the conditions for the appearance of the golden section in such experiments was formulated with the help of the reflexive model (Lefebvre, 1985). In accordance with this model, the golden section appears under two conditions:

Let us clarify condition (2). It is not important that a given quality is positive or negative in its essence; the important factor is the subject’s evaluation of ascribing this quality to the object. For example, suppose someone bought a defective watch; in evaluating this purchase in terms of the construct ‘failure-success’ we have to consider ‘failure’ to be the positive pole and ‘success’ to be the negative pole. To test the model’s predictions, we need to analyze data from experiments in which both conditions specified above were satisfied. Several experiments conducted within the framework of the investigation of “mere exposure” (Zajonc, 1968) meet the criterion, especially those requiring subjects to choose between two patterns, one of which had been previously presented with an extremely short exposure time of 1-3 milliseconds (Kunst-Wilson & Zajonc, 1980). As shown in previous experiments, this time is inadequate for the subjects to memorize a pattern consciously, however, the “old” alternative, that is the one shown previously, was chosen by the subjects more often (Kunst-Wilson & Zajonc, 1980). We might reason that the prior exposure of patterns orients the pairs shown in the second phase of these experiments. Each “old” pattern assumes the role of the positive alternative, and its “new” counterpart that of the negative alternative. If we take this assumption into consideration, then in accordance with the reflexive model, the “old” alternative must be chosen not just “more often”, but with the relative frequency of 0.62. We analyzed all of the available data from similar experiments (Lefebvre, 1995) and obtained the following results:

The reflexive model yields numerical predictions also for cases in which, prior to the beginning of an experiment, the subjects develop the intention to evaluate objects only positively or only negatively. In the first case, x₃ = 1, and the model predicts that the subjects will make positive evaluations with the probability X₁ = (2/3); in the second case, x₃ = 0, the model predicts that the probability of positive evaluations be X₁ = (1/2).

Special experiments with bipolar constructs in which the subjects’ intentions were determined, have shown that with positive intentions the frequency of choosing positive adjectives was equal to 0.67; and with negative intentions this frequency was 0.5 (Adams-Webber, 1997; Adams-Webber & Rodney, 1983). Therefore, the reflexive model also passed this test.

It is important to keep in mind that reflexive models are not limited to the analysis of the subject’s fixed states. From the very beginning of the development of the reflexive approach, it has employed dynamic models (Lefebvre, Baranov, and Lepsky, 1969), which are now widely used in modeling various psychological phenomena (Barton, 1994; Kelso, 1995). A recent use of the dynamic model of the reflexive subject has led to the hypothesis that one of the fundamental functions of self-awareness consists of fighting against chaos, which appears in the series of sequential cognitive computations (Lefebvre, 1999, 2001).

1. The Main Equation

The function X₁ = f (x₁, x₂, x₃), which describes the subject’s readiness to choose the positive pole, is

X₁ = x₁ + (1 - x₁)(1 - x₂)x₃, ................... (1)

where x₁, x₂, x₃ О[0,1] (Lefebvre, 1991; 1992b).

2. A Theorem on Reflexion

The functional equation K (x₁,K (x₂, x₃)) = x₁ + (1 - x₁)(1 - x₂)x₃, where x₁, x₂, x₃ are any numbers from [0,1] and all values of K (x₂, x₃) belong to [0,1] has only one solution: K (x,y) = 1 - y + xy = F (x,y) (the proof is given in Lefebvre, 1992). It follows from this theorem that the subject can be represented as

X₁ = F (x₁,F (x₂,x₃))....................... (2)

The following function corresponds to the image of the self:

X₂ = F (x₂,x₃) = 1 - x₃ + x₂x₃...................................... (3)

3. The Correct Model of the Self

4. Automatic Choice

A choice is called automatic, if the values x₁ = a, x₂ = b are such that f (a,b,x₃) const, where x3 is any number from [0,1]. It follows from (1) that choice is automatic if at least one of the variables (x1, x2) is equal to 1.

5. Deliberate Choice

A choice is called deliberate, if the values x₁ = a, x₂ = b are such that function f (a,b,x₃) єx₃, where x₃ is any number from [0,1]. It follows from (1) that the choice is deliberate only if x₁ = x₂ = 0.

6. Golden Section and Other Constants

If the subject’s intention has not been determined in advance, the subject has a correct model of the self, X₁ = x₃; then (5) transforms into the equation X₁² + X₁ - 1 = 0, whose solution is X₁ = (Ц5-1)/2 = 0.618... . If the subject’s intention has been predetermined,

X₁ = (2/3) at x₃ = 1, and X₁ = (1/2) at x₃ = 0, as follows from (5).

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