Human Choice: Psycoloquy Call for Commentators

Stevan Harnad (harnad@ECS.SOTON.AC.UK)
Sat, 23 Sep 1995 18:41:34 +0100

Qualified professional biobehavioral and cognitive scientists are
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Stevan Harnad, Editor
PSYCOLOQUY (sci.psychology.journals.psycoloquy)
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psycoloquy.95.6.29.human-choice.1.lefebvre Saturday 23 September 1995
ISSN 1055-0143 (94 pars, 43 eqns, 3 figs, 1 tbl, 70 refs, 1225 lines)
PSYCOLOQUY is sponsored by the American Psychological Association (APA)
Copyright 1995 Vladimir Lefebvre


Vladimir A. Lefebvre
School of Social Sciences
University of California, Irvine
Irvine, California

ABSTRACT: We introduce a model of a subject facing a choice of an
alternative out of a set. The model ties together three aspects of
human activity: behavioral, mental, and neural-computational.
Parameters of this model cannot be estimated experimentally. Thus,
a problem arises of determining them by means of theoretical
considerations. Similar problems appeared in cosmology as well:
the values of the fundamental constants necessary for constructing
models of the Universe cannot be determined empirically. One
possible solution is to use the "anthropic principle," that is, an
abstract statement which allows narrowing the number of
combinations of values. We show that a similar methodological
gambit can be used in psychology. We formulate an abstract
statement and find the parameters of the model with its help. Here
we establish the relation of this model to other theories of choice
and to experimental psychology. Then we demonstrate that our model
is formally isomorphic to the process of gradual minimization of
work lost by a heat engine system. The existence of such an
isomorphism supports a hypothesis that mental phenomena are related
to macro-characteristics of an ensemble of neuron states.

KEYWORDS: choice; computation; decision theory; ethical cognition;
mathematical psychology; model building; parameter estimation;
probability; rationality.


1. In 1937, Paul Dirac, one of the founders of quantum mechanics,
noticed surprising coincidences between numbers characterizing the
micro-world and the Universe as a whole. Twenty years later, Robert X.
Dicke showed that this coincidence could be explained by assuming that
the conditions necessary for the appearance of an observer similar to a
human being correspond to a certain period in the evolution of the
Universe (Barrow & Tipler, 1986). Continuing this line of research
Brandon Carter suggested using a principle, which he called
"anthropic," for choosing constants in cosmological models. Its essence
is expressed in the statement:

...what we can expect to observe must be restricted by the
conditions necessary for our presence as observers.
(Carter, 1974, p.291)

2. This declaration "forbade" constructing models of Universe in which
an observer similar to a human being could not appear. Since the
existence of human beings depends on the existence of organic
molecules, only those relations between fundamental constants that
allow organic molecules to appear in the Universe are acceptable.
Therefore, the anthropic principle can be regarded as a special method
of reasoning, on which to base our selection of cosmological models
(Barrow & Tipler, 1986).

3. According to this methodological scheme we formulate a statement
which cannot be empirically falsified but which allows us to generate
hypotheses concerning possible values of parameters in mathematical
models. In this way we can progress from models of isolated phenomena
to more general theoretical constructions.

4. We consider a concrete model of human activity related to choice.
This model ties together three aspects of this activity: behavioral,
mental and neural-computational. We could not construct such a model,
however, if we followed a contemporary paradigm which requires that its
parameters must be estimated experimentally. Let us consider this
problem in more detail. We seek a function of variables x1, x2, and x3
by assuming that this function, X1=f(x1,x2,x3), is linear in each of
its arguments in the sense that if the values of any two are fixed, X1
is linear in the third. It follows from this assumption that X1 can be
represented as a polynomial

f(x1,x2,x3)=a0+a1x1+a2x2+a3x3+a4x1x2+a5x1x3+a6x2x3+a7x1x2x3. (1.1)

5. The right side of (1.1) contains eight parameters a0, a2,..., a7
whose values are unknown. Suppose that we have established
experimentally that /a0/<0.001 with probability 0.99. For predicting
behavior this information might be sufficient. But the properties of
f(x1,x2,x3), which must be known for establishing connections between
the different aspects of the model, may depend on the answer to a
cardinal question: whether a0 is equal to 0 or not? Experimental data,
in principle, cannot answer this question. Similar problems appear
concerning other parameters as well.

6. We will demonstrate later how the idea similar to the anthropic
principle in cosmology allowed us to find the exact values for the
parameters a0, a1,...,a7 in a particular psychological model and how
knowledge of an analytical form of a function allowed us to suggest a
concrete hypothesis about the nature of computational processes in the
human brain.

7. Our point of departure is the following statement, which we will
call the Principle of Freedom:

We have free will and under certain circumstances also
freedom of choice.

8. We will try to show that on this base a theory can be constructed
from which falsifiable models of psychological phenomena related to
choice can be deduced.

9. To give the terms "free will," "freedom of choice," and "certain
circumstances" an operational meaning, imagine an abstract subject who
is to choose one of six alternatives marked 1, 2, ..., 6 in the
following way. He must pick from a continuum of dies with different
centers of gravity, so that there is a one-to-one correspondence
between the set of dies and the set of all possible probability
distributions of choosing an alternative from the six. If there are no
obstacles, the subject (i) plans to use a certain die; (ii) picks it
from the continuum, and (iii) casts it, thereby choosing one of the six
alternatives. Note that the subject can make a completely determined
choice, namely, he can plan to use only those dies which always show
the same face up.

10. Imagine now that a state of the external world that may or may not
interfere with the process of actually taking from the continuum the
die the subject planned to use. For each plan there are such states of
the world.

11. The statement asserting the subject's freedom of will means that he
can plan to use any of the dies from the continuum independently from
the state of the world.

12. The statement that under certain circumstances the subject has
freedom of choice means that there are states of the world that do not
interfere with the process of using the die the subject planned to use.
In this way we distinguish between "free will" and "freedom of choice."

13. Complete absence of freedom of choice means that some state of the
world exactly predetermines the die to be taken. In this case the
subject's choice does not depend on his intention.

14. We will call a choice realistic if the subject plans to use only
those dice that can be taken under a given state of the world. A
subject who makes only realistic choices will be called a Realist.

15. This scheme reflects two different tendencies in psychology.
Miller, Galanter and Pribram (1960) emphasized the plan as a central
element of human activity. The distinction between a readiness (or
setting) to act, on the one hand, and the actual act, on the other,
evolved from gestalt-psychology and was developed by Uznadze (1968) and
his school (Prangishvili, 1967). In our scheme, the subject with a die
in his hands represents the state of readiness to act, and casting a
die the act itself.


16. A subject facing a choice from a set of n alternatives (n>=2) will
be represented by the function

a=G(b,c), a and c are elements of D, b is an element of S. (2.1)

17. The set D={(p1,p2,...,pn)}, where pi>=0, i=1,2,...,n, and
p1+p2+...+pn=1, is the set of all probability distributions on a given
set of n alternatives, and S is the set of all possible states of the
world. Variable a represents the probability distribution
characterizing the state of the subject's readiness to choose an
alternative. Variable b represents the world influence on a particular
choice. Variable c represents the probability distribution which the
subject plans to use for choosing an alternative.

18. The statement that the subject has a freedom of will corresponds to
the assumption that variables b and c are independent.

19. The statement that under certain circumstances the subject has
freedom of choice corresponds to the assumption that there exist an S1
subset of S such that for every b element of S1, the following identity

G(b,x)==x, (2.2)

where x is any distribution from D and "==" means identical.

20. The existence of states of the world that predetermine the
subject's choice behavior implies that there exists an S2 subset of S
such that for every b element of S2 there is exactly one y(b) element
of D that satisfies the identity

G(b,x)==y(b), (2.3)

where x is any distribution in D.

21. The Realist is represented by the Equation

G(b,x)=x, (2.4)

where x is element of D and b is element of S.

22. Let us put each element b which is element of S into correspondence
with the set D[b] using the following rule: if with a given b Equation
(2.4) does not have a solution, then D[b] is an empty set; otherwise,
D[b] consists of all the elements x which are elements of D for which
Equation (2.4) holds. Therefore we introduce a single-valued function

D[b]= g(b). (2.5)

23. Not all sets D[b] are empty, because when b is element of S1,
D[b]=D (follows from (2.2)), and when b is element of S2, D[b] is a
one-element set {y(b)} (follows from (2.3)).

24. Let us emphasize that on the right-hand side of (2.5), there is
only one variable b corresponding to the state of the world. By
comparing (2.5) with (2.1) we see that in the case when a choice is
realistic, that is, when a=c, we can exclude variable c from
consideration; its values are subjective phenomena and as such,
non-observable. Imagine that we want to predict a Realist decision if
we know the function g(b) and having observed the value of variable b.
If D[b] is empty set, this means that with a given b realistic choice
is impossible, and that the Realist refuses to choose. If D[b] contains
only one element, this means that the subject will use one particular
distribution, hence Equation (2.5) allows us to make a precise
prognosis. If D[b]=D, this means that the subject has freedom of choice
and can decide to use any of the distributions in D. Therefore, the
state in which the subject has the ability to make a free choice cannot
in principle correspond to any kind of concrete probability
distribution. In this case observation of the world does not allow us
to make a more precise prognosis. If D[b] consists of more than one
element and D-D[b] is not an empty set, we can predict that the Realist
will not use distributions belonging to D-D[b]. We cannot say, however,
which distribution from D[b] he will use; in this case we say that the
subject has partial freedom of choice.


25. We use the ideas described above to construct a concrete model of a
subject making binary choices (Lefebvre, 1992a) and a model of choice
among any number of alternatives (Lefebvre, 1994a). In the binary
choice model, we consider a subject choosing between a positive and a
negative pole. The subject is represented by the function

X1=f(x1,x2,x3), (3.1)

where X1,x1,x2,x3 are elements of [0,1]. The value of X1 is interpreted
as the readiness to choose a positive pole with probability X1, and the
value of x3 as the subject's plan or intention to choose a positive
pole with probability x3. Variables x1 and x2 represent the world
influence on the subject. Thus, variable b in Equation (2.1) above
corresponds to the pair (x1,x2) in Equation (3.1).

26. We assume that in every choice situation the world influences the
subject by two kinds of impulses. An impulse of the first kind
stimulates the subject to choose a positive pole, one of the second
kind a negative pole. The value of x1 is the frequency of the impulses
toward a positive pole (at the time of the subject makes his choice).
The value of x2 is the mean frequency of the impulses toward a positive
pole, sent to the subject in similar previous situations (including the
present one). Thus, the value of x2 reflects the subject's earlier
experience, in other words, the subject's "knowledge" about a class of
similar experiences.

27. Next we introduce


Function (3.1) is linear in each of its three arguments.

As mentioned in the Introduction, it follows from this assumption that
f(x1,x2,x3) can be represented as in a (1.1), i.e. containing eight
unknown parameters. We can find these parameters by formulating certain
limitations for function (3.1). Those limitations will correspond to
the requirements posed to the subject who possesses free will and the
ability for free choice. They are given by Equations (2.2) and (2.3).
First we formulate these limitations as axioms; then we will discuss
their meaning.

28. Equation (2.2) corresponds to


For any x3 element of [0,1], f(0,0,x3)==x3. (3.2)

The set S1, in this case, contains at least one element b=(0,0).

29. Expression (2.3) corresponds to the following two axioms:


For any x3 element of [0,1], f(0,1,x3)==0. (3.3)


For any x2,x3 elements of [0,1], f(1,x2,x3)==1. (3.4)

The set S2 in this case consists of at least an element (0,1)
and a set of elements (1,x2), where x2 takes on values from [0,1].

30. The restrictions corresponding to Equations (3.2), (3.3), and (3.4)
stem from moral philosophy, which has a rich history of logical
manipulation involving the categories of good and evil. In that field
freedom of choice is usually linked with responsibility: a person is
responsible for his actions if at the moment of its performance he was
free to perform it and conversely (Losskii, 1927). Thus expression
(3.2) corresponds to the statement: If the state of the world inclines
the subject to perform a negative action (x1=0), and the subject knows
this (x2=0), then the subject is responsible for the choice. Expression
(3.3) states: If the state of the world inclines the subject to perform
a negative action (x1=0), but the subject believes that the world
always presses toward good (x2=1), then the subject yields to the
world's pressure (X1=0). Finally, expression (3.4) corresponds to the
statement: A human being is not a source of evil; temptation can come
only from an external agent. Hence if the world urges the subject to
perform a positive action (x1=1), only the positive action can be
performed (X1=1). Such recourse to moral philosophy may create the
impression that I suppose a human being to be an innately moral being.
Karl Popper wrote in his note about my book (Lefebvre, 1992a):

No psychologist will like this. Everyone will say this means
thinking of man as an essentially moral being. (Popper, 1992)

31. Here is my answer: The problem of morality in a religious or any
other conventional sense is not important for constructing a model.
Relying on moral philosophy is a technique which helps to find the
values of initial conditions.

32. Using the Postulate of Simplicity, Axioms I, II, and III we can
write a system of equations for parameters a0, a1, ..., a7:

f(0,1,x3)==0, (3.5)

33. After solving this system we find that a0=0, a1=1, a2=0, a3=1,
a4=0, a5=-1, a6=-1, a7=1. Substituting these values into (1.1) we

X1=x1+(1-x1-x2+x1x2)x3. (3.6)

34. Heretofore, we have found a specific function from the class of
functions given by Equation (1.1) without experimental estimation of
the parameters a0, a2, ..., a7 but using an abstract principle of
freedom, instead (Lefebvre, 1992a).

35. To represent the Realist, (X1=x3), expression (3.6) can be written

x1/(x1+x2-x1x2), if x1+x2>0
X1 = { (3.7)
any number from [0,1], if x1+x2=0.

36. Equation (3.7) corresponds to expression (2.5), where b is an
element of the set of all pairs (x1,x2). If x1+x2>0, then the set
D[(x1,x2)] consists of a single distribution (X1,1-X1) given by
Equation (3.7), and if x1+x2=0, then D[(0,0)] coincides with the set of
all the distributions.

37. Recently Schreider (1994) showed that in another interpretation,
Equation (3.6) can be deduced from pure probabilistic considerations.
We can also deduce this equation from an earlier version of our model
(Lefebvre, 1977a; 1980; 1982). Based on this version a special method
of simulating human decision making has been developed (Shankin, 1994).
Experimental aspects related to the early version of the model were
discussed by Adams-Webber (1987) and Zajonc (1987), and its logical
and epistemological aspects were analyzed by Rapoport (1982; 1990),
Townsend (1983; 1990), Levitin (1987), McClain (1987), Wheeler (1987;
1990), Batchelder (1987), and Kauffman (1990).

38. Further we will show that the functions (3.6) and (3.7) can be
directly related to observable psychological phenomena (Lefebvre, 1990,
1992a, 1994a).


39. The BTL model is a distinctive starting point in the psychology of
choice. It helps to compare various probabilistic models of choice
behavior. In the case of binary choice, the BTL model is based on the
assumption that the probabilities of choosing alternatives belonging to
a certain set are proportional to their utilities. Thus, if utilities
of alternatives A and B are respectively v1 and v2, then the
probability that A will be chosen from the set {A,B} is given by

p(A,B)=v1/(v1+v2). (4.1)

40. Imagine now that the Realist, represented by function (3.7), is
making a choice between A and B. Without loss of generality, we can
assume v1>=v2. Let A be the positive pole, and B the negative pole.

41. We assume that the world's pressure toward the positive pole is
given by

x1=(v1-v2)/v1. (4.2)

We will explain the reasons for choosing this expression at the end of
section 8.

42. If the subject has had no prior experience in similar situations,
then in accordance with definition of x2, x1=x2. Substituting the
values of x1 and x2, given in (4.2), into (3.7) we find the relation
between the probability X1 that the subject is ready to choose
alternative A and utilities v1 and v2:

v1/(v1+v2), if v1>v2
X1 = { (4.3)
any number from [0,1], if v1=v2.

43. We see that on the basis of assumption (4.2) our model coincides
with the BTL model for the case of v1>v2. We can even say that the BTL
model is deduced from ours. However, for the case v1=v2 the model
predictions are different. According to the BTL model each alternative
will be chosen with probability 1/2, whereas our model implies that in
this case the subject has freedom of choice: he can select any
probability of choosing alternative A (Lefebvre, 1994a).


44. Almost forty years ago Stevens and Galanter (1957) found that the
results of magnitude estimation of stimuli intensities are related
non-linearly to their categorical estimation. In the first case,
subjects estimate intensities comparing them with a unit of
measurement. In the second case, subjects attribute the intensity of
each stimulus to a certain level. Such levels can be represented by a
set of numbers, where 1 corresponds to the weakest stimulus, and k to
the strongest. If we plot magnitude estimations against categorical
estimations, the graph will be an upward convex curve rather than a
straight line. Further, the convexity is greater when the mean values
of stimuli are shifted toward the weakest stimulus. The interrelation
between these two methods of estimation was debated for a long time
(Stevens & Galanter, 1957; Galanter & Messick, 1961; Parducci, 1965;
Marks, 1968; Haubensak, 1992; Parducci, 1992). However, no persuasive
theoretical explanation has been offered.

45. Lefebvre (1992a) attempted to explain this non-linear relation with
the help of the model described in this paper. A subject is represented
by equation

X1=x1/(x1+x2-x1x2), (5.1)

where x1+x2>0 (cf. (3.7)). Suppose the subject is able "to answer a
question" about his readiness to choose a positive pole. Let him do
this by marking a point X1 on the interval [0,1]. In this way we extend
our model by assuming that the final result of the subject's activity,
in addition to the actual choice, can also be the subject's "report"
about the value X1. We interpret such a report as a categorization.

46. Consider first a case of bipolar categorization. Let magnitude
estimations of stimuli found in earlier experiments be between Smax and
Smin. The subject's task in the given experiment is to categorize each
stimulus as "strong" or "weak." Suppose the attractiveness of
alternative "strong" is v1=Smax-Smin, while that of "weak" is
v2=Smax-S, where S is the intensity of the presented stimulus: the more
intensive is the stimulus, the less attractive is the alternative
"weak." Interpreting the alternative "strong" as the positive pole and
using Equation (4.2) we obtain

x1=(S-Smin)/(Smax-Smin). (5.2)

47. We will call this value a normalized magnitude estimation. In
accordance with the definition in section III, x2 takes on the mean
value of normalized magnitude estimations of stimuli preceding a given
one (including itself). The value of X1 is the subject's readiness to
choose "strong" with probability X1.

48. When a scale with a large number of categories is used, the subject
marks the value of X1 on the scale. For well-randomized long sequences
of stimuli presentation, we can assume x2 to be constant in a given
experiment. In this case expression (5.1) can be regarded as an
equation of a hyperbola with variables X1 and x1 and a constant
parameter x2. Larger convexity of the hyperbola corresponds to smaller
x2. Such hyperbolas appear to fit published experimental results on
categorization (sf. Lefebvre, 1992a). It is also noteworthy that values
of variables x1 and x2 can be estimated from experimental data. Thus,
the model contains no free parameters.


49. For many centuries the golden section [g=(SQRT5-1)/2=0.618...] was
regarded as especially attractive (cf., for example, Ghyka, 1946).
Attempts to refute or confirm this hypothesis began with Fechner
(1876); but the question of whether this phenomenon really exists
remained unanswered (cf. reviews in Valentine, 1962; Plug, 1980;
Lefebvre, 1992b). A new wave of interest in the golden section was
generated by the works of Adams-Webber and Benjafield, who found that
subjects evaluate their acquaintances positively in bipolar constructs
with a frequency close to 0.62. They hypothesized that the theoretical
value of this frequency is identical with the golden section
(Adams-Webber & Benjafield, 1973; Benjafield & Adams-Webber, 1976;
Adams-Webber, 1990).

50. Analyzing other subsequent experiments we found that the
frequencies of choice of the positive pole lie in the interval
0.60-0.64 (see review in Lefebvre, 1992b) and suggested that the golden
section is not limited to evaluations of other people but appears in
binary choice or categorical estimation of stimuli without measurable
qualities (Lefebvre, 1985; 1990). We also proposed a model similar to
the one described here from which an explanation of the phenomenon is
deduced (Lefebvre, 1985).

51. Suppose a subject represented by Equation (3.6) faces a choice
between two alternatives without objective measure in the context of
the given situation. Let the subject relate the first alternative to a
positive pole, the second to a negative pole, and let the
attractiveness of these alternatives be v1=1 and v2=x3. The latter
means that the greater the probability with which the subject is
planning to choose the first alternative, the more attractive becomes
for him the second alternative. Let also the attractiveness of the
positive alternative be always greater than that of negative one, that
is, v2=x3<1. In view of (4.2), we have x1=(v1-v2)/v1=1-x3. Next suppose
the subject had never faced a similar choice in the past. In view of
the definition of x2 (see section 3) we must set x1=x2. By substituting
the values x1=x2=1-x3 into Equation (3.6), we obtain

X1=1-x3+x3^3, where x3^3 means "x3 to the power of 3." (6.1)

52. Suppose that the subject is a Realist, that is, X1=x3. Then
Equation (6.1) is transformed into a cubic equation

x3^3-2x3+1=0. (6.2)

53. Positive roots of this equation are x3=1 and x3=(SQRT5-1)/2. The
first one does not satisfy the condition x3<1. Therefore, the subject
chooses a positive pole with the probability equal to the golden
section value.

54. Consider an experiment conducted by Kunst-Wilson and Zajonc (1980),
in which the ideas presented here played no part. The stimuli were 20
irregular octagons. During the first stage of the experiment, ten of
these octagons were presented one by one with a very short expositions,
1 millisecond, five times each. During the second stage, octagons were
presented in pairs: one which was presented earlier and one new one.
The subjects were not told that one of the octagons was among those
seen earlier; their task was to indicate which octagon they liked more.
The octagon seen earlier was preferred with frequency 0.60. Table 1
shows the frequencies of preference for objects seen earlier in a
number of similar experiments:

Table 1

Experimenter Frequency

Kunst-Wilson et al.(1980) 0.60
Seamon et al.(1983) 0.61
Mandler et al.(1987) 0.62
Bonanno et al.(1986) 0.66, 0.63, 0.62, 0.61, 0.63, 0.62

55. As can be seen, the frequencies cluster around 0.62. We found no
indications in the literature that this fact was noticed.

56. Our model suggests an explanation of this phenomenon. The first
stage of the experiments determined the polarization of the objects:
the object seen earlier became the positive pole; the new one the
negative. The stimuli had no measurable properties which might
determine the preference between the objects. Consequently the pressure
of the world was determined by the subject's intention, which led to a
preference for the positive pole with probability close to the golden
section ratio.


57. In 1908 a psychiatrist from Prague, H. Loewy, discovered a new
psychological phenomenon (see Reznik, 1969). One of his patients
complained that she had lost all her inner feelings. Her statements,
however, contrasted with the objective observations made by Loewy: her
tone, look, and style of behavior suggested that she had deep feelings
making her suffer. Loewy was confronted with a paradox: the patient
suffers because she is convinced that she does not.

58. According to Mayer-Gross,

The paradoxical occurrence of such complaints in patients who
obviously suffer from the alleged death of their feelings has
aroused great interest in all writers on the subject.
(Mayer-Gross, 1935, p.108)

59. Here are typical patients' reports:

"My emotions are gone, nothing affects me." "All the feeling is
gone, no remorse, no passion. I have only the feeling of being
alive, that my heart beats..." "My feeling is dead, I have no
interest for my husband and the baby." "Nothing impresses me, ...
I have lost all my love for the children, they seem miles away."
"I am unable to have any emotions, everything is detached from
me." "The trees in the garden seem unreal, I get no feeling, no
thrill, no joy ... ." "I cannot see the beauty of flowers -- the
feeling does not get past my eyes." (Mayer-Gross, 1935,
pp. 107-108)

60. Sometimes the patients themselves find that the meaning of their
statements contradicts their form:

"One of my patients remarked on her own surprise at this
paradox, viz. that she should weep in the very act of complaining
about loss of feeling." (Mayer-Gross, 1935, p.108)

61. One of the attempts to construct an explanatory scheme for
this phenomenon was made by Reznik (1969). He assumed that a person
in a normal state has the ability to sense his own feelings; that
is, in addition to the feelings as such, there is also a secondary
ability to "feel" the process of feeling, although this ability is
not consciously registered. In the case Loewy observed, the patient
had lost this secondary ability, so that she did not sense her own
feelings and believed that they were missing.

62. Although this phenomenon has been known to psychiatrists for a long
time, it did not attract attention of psychologists studying the
cognitive realm of human beings. Perhaps the inner mechanism of
"awareness" of one's own feelings is akin to the mechanism of awareness
of one's own visual perception (Gazzaniga, 1970; Griffin, 1976). In
that case, the loss of the ability to see one's own feeling is close to
the phenomenon of paradoxical vision or blindsight, the essence of
which is that subjects with damage to the visual cortex can move around
and not collide with objects but at the same time be sure that they
don't see these objects (Zihl, 1980; 1981; Zihl & von Cramon, 1985;
Paillard et al., 1983; Perenin & Jeannerod, 1978).

63. Another phenomenon which might be close to those already described
is the "feeling of knowing": sometimes subjects report that they have a
feeling that they will be able to retrieve unrecalled information under
specific conditions (Hart, 1965; see also Metcalfe & Shimamura, 1994).

64. On the other hand, although all those phenomena are usually
described in similar terms, such as awareness, metacognition, and
attention (Metcalfe & Shimamura, 1994), we cannot rule out the
possibility that by their neural physiological nature they belong to
different functional systems in the human brain (Gazzaniga, 1995).

65. We will show further that the theoretical representation of the
subject described in the previous sections allows us to develop a model
of the subject capable of becoming aware of his own feelings (Lefebvre,
1980; 1992a,b).

66. We consider a human being possessing a specialized functional
processor generating an "image of the self," which, in turn, generates
a second "image of the self." The first "image of the self" is related
to the subject's direct feelings; the second order "image of the self"
allows the subject to "see" himself feeling. The generation of the
first and second order of the "image of the self" goes on
automatically; it is not related to any voluntary constructive activity
(Lefebvre, 1967; 1977b). Under certain disruptive conditions, however,
a secondary image is not generated. This gives rise to the paradox
discovered by Loewy: a person has feelings but does not sense himself
having them. A connection between this scheme of the process of
automatic awareness and the model of the subject developed in the
previous sections is given by the following

67. STATEMENT: The function X1=x1+(1-x1-x2+x1x2)x3 of three variables
x1, x2, and x3 can be represented as composition

X1=F(x1,F(x2,x3)), (7.1)

of one function, F(x,y), of two variables x and y, where

F(x,y)=1-y+xy. (7.2)

This representation is unique. (The proof is given in Lefebvre,

68. We consider the function

X2=F(x2,x3) (7.3)

as a formal analogue for the subject's image of the self. In this
interpretation, the subject's self-description and the objective
description by an external observer are presented by the same function
F(x,y). We thus obtain a model of a subject "aware" of the self. In the
case in which the subject is a Realist (X1=x3), Equations (7.1) and
(7.3) transform into Equations (7.4) and (7.5) respectively:

X1=F(x1,F(x2,X1)), (7.4)

X2=F(x2,X1). (7.5)

69. It follows from these Equations that, if x1+x2>0, then

and if x1+x2=0, then

X2=1-X1, (7.7)

where X1 is an arbitrary number from [0,1].

70. Variables X1 and X2 can be represented as identities

X1==F(x1,X2), X2==F(x2,X1). (7.8)

71. In order to simplify the analysis, we represent X1 as a composition
of F1=F(x1,u) and u=X2, and X2 as a composition of F2=F(x2,v) and v=X1
by using a standard notation for function composition:

X1==F1oX2, X2==F2oX1. (7.9)

72. It follows from (7.9) that X1 can be put into correspondence with
the sequence of its representations as compositions of F1, F2, and Xj:

X[1]==F1 o X2,
X[2]==F1 o F2 o X1, (7.10)
.. . . . . . .

X[n]==F1 o F2 o F1 o ... o Fi o Xj,
where i,j=1,2; j=i.

73. A subject corresponding to composition X[1] has an image of the
self X2, but this image does not have an image of the self. A subject
corresponding to composition X[2] has an image of the self F2oX1, and
this image, in turn, has an image of the self X1. We identify an image
of the self, F2oX1, with the subject's inner feelings, and an image's
image of the self; X1, with the "sense" of one's own feelings (using
Reznik's term). Loewy's paradox appears when there is a disruption such
that a subject in the state X[1] cannot proceed to the state X[2]. The
subject has feelings connected with his image of the self, but his
image of the self does not have feelings, and the subject is incapable
of "seeing" the self with the feelings.

74. In the general case, we put an act of automatic awareness into
correspondence with the transformation X[k]->X[k+1] (Lefebvre, 1967;
1977b). Then composition X[n] will correspond to a state of the subject
appearing as a result of (n-1) acts of awareness starting with X[1].

75. When the subject is in the state X[n], every image of the self (in
the hierarchy of images) is presented together with the influence of
the world. Therefore, in addition to the hierarchy of self-images, the
subject has a hierarchy of world's images. These images are connected
with the subject's feelings of the world. So, the subject has feelings
of the two types and both are included into multiple awareness: some
are connected with self-reference, and the others with the reference to
the external world (Lefebvre, 1967; 1977b).

76. Every composition from (7.10) can be regarded as a formal operator
mapping a set of pairs of values (x1,x2) onto a set of pairs of vectors
(A,a), (See Fig. 1).

1 2 3 n-1 n
.................. .............
: : : : : : :
A = : X1 : X2 : X1 : ... : Xi : Xj :
.................. .............
X[n] == F1 o F2 o F1 o ... o Fi o Xj :
: : : : : : :
a = : x1 : x2 : x1 : ... : xi : xj :
.................. .............

Figure 1. Connection between composition X[n] and vectors of feelings A
and a. The upper lane corresponds to vector A which is a theoretical
analogue of the feelings linked to self-reference. The lower lane
corresponds to vector a which is a theoretical analogue of the feelings
linked to the reference of the external world.

77. The upper lane corresponds to vector A=(X1,X2,X1,...) whose
periodical components are the values of functions F1, F2, and Xj
(j=1,2) occurring in composition. This vector is a theoretical analogue
of the subject's feelings (and their multiple reflexion) linked to
self-reference. The lower lane corresponds to vector a=(x1,x2,x1,...)
whose components also constitute a periodical sequence. This vector is
an analogue of the subject's feelings (and their multiple reflexion)
linked with reference to the external world (Lefebvre, 1992b).

78. In the next section we will try to move further and show that the
model that we have constructed allows us to make some assumptions about
computational brain work.


79. One of the hypotheses concerning the general principles of the
functioning of real brain neural networks is that the ensemble of
neuron states is a similar statistical ensemble of physical particles
obeying the laws of statistical physics and thermodynamics (Cowan,
1967). The models of some types of such networks already exist
(Hopfield, 1982; Kirkpatrick et al., 1983; Hinton & Sejnowski, 1983;
Ackley et al., 1985; Churchland & Sejnowski, 1992).

80. In the framework of this "thermodynamic" representation, mental
phenomena can be connected with macro-characteristics of an ensemble of
neuron states, such as "heat," "temperature," and "entropy." If we
accept this hypothesis, we may expect that formal schemes of mental
processes reflect thermodynamic correlations between different
macro-characteristics of neuron ensembles. In the previous section we
constructed a formal model of one particular mental phenomenon, the
awareness of one's own feelings. No specific information taken from
statistical physics and thermodynamics was used to do this. Therefore,
if we show that our model can be easily and clearly translated into the
language of thermodynamics, this will serve as an argument in favor of
plausibility of both our model and the thermodynamic hypothesis as a
whole. Further we will make such a translation by demonstrating that
the process of multiple awareness described in the previous section is
isomorphic to the process of gradual minimization of work lost by a
heat engine system.

81. Let us consider an abstract heat machine consisting of a sequence
of heat engines and two tapes, on which the engines print their working
parameters (see Fig. 2).


................................. .............................

: : : : : : :
: M1 : M2 : M3 : ... : M{m-1} : Mm :
................................. .............................

................................. .............................

: W1 : W2 : W3 : : W{m-1} : Wm :
: : : : : : :
: Q1 Q2 : Q2 Q3 : Q3 Q4 : ... : Q{m-1} Qm : Qm Q{m+1} :
: -> -> : -> -> : -> -> : : -> -> : -> -> :
................................. .............................

T1 T2 T3 T4 T{m-1} Tm T{m+1}
................................. .............................

: : : : : : :
: R1 : R2 : R3 : ... : R{m-1} : Rm :
................................. .............................

Figure 2. A heat machine. Reservoirs correspond to vertical lines.
Each engine prints its efficiency Rm=(Qm-Q{m+1})/Qm on the bottom tape,
and the values Mm=Rm/[(T1-T2)/T1] on the upper tape.

82. In this sequence each succeeding engine performs work compensating
for the loss of available work by the preceding engine by receiving
from a reservoir the amount of heat equal to that yielded to this
reservoir by the preceding engine. The temperatures of reservoirs make
a decreasing geometrical progression:

T1/T2=T2/T3= ..., (8.1)

where T1>T2. Engine m performs work

Wm=Qm-Q{m+1}, where {m+1} is a subscript. (8.2)

83. We do not require an engine to be necessarily reversible, so

Wm<=Um, where "<=" means "lesser than or equal to," (8.3)

and where

Tm-T{m+1} T1-T2
Um = Qm --------- = Qm ----- (8.4)
Tm T1

is the work which would be performed by a reversible engine if it were
located between reservoirs m and (m+1). The loss of available work by
engine m is equal to

Q{m+1} Qm
dWm = Um - Wm = T2(------- - --) = T2dH, (8.5)
T2 T1

where dH is the change of entropy of the whole system caused by engine
m performing work Wm. Engine (m+1) receives from a reservoir (m+1) heat
Q{m+1} and performs work dWm. Each engine measures the work it performs
and registers it in two ways: as a portion of the heat received from
the reservoir, Rm, and as a portion of the work which would be
performed by a reversible engine under the same conditions, Mm. The
engine prints on the lower tape the value

Rm=Wm/Qm=(Qm-Q{m+1})/Qm, (8.6)

that is, the efficiency of engine m, and on the upper tape the value

Wm Rm
Mm = -------------- = ----------, (8.7)
Qm[(T1-T2)/T1] (T1-T2)/T1

that is, the ratio of the efficiency of engine m to the efficiency of
reversible engine.

84. We consider these printed quantities to be theoretical analogues
for subjective processes. We proved that the "text" printed by such a
heat machine is formally equivalent to the "text" generated by a
composition X[n] (Lefebvre, 1994b). The two following statements are
true concerning this sequence of heat engines.

85. STATEMENT 1. Sequences Rm and Mm are periodic and

R1 if m is odd
Rm = {
R2 if m is even ,
M1 if m is odd
Mm = {
M2 if m is even ,


R1=(Q1-Q2)/Q1, R2=[Q2-(T2/T1)Q1]/Q2, (8.9)

M1=R1/[(T1-T2)/T1], M2=R2/[(T1-T2)/T1]. (8.10)



We can see that equations (8.11) are equivalent to equations (7.6).

87. Consider an arbitrary heat machine and establish the correspondence

R1=x1, R2=x2. (8.12)

88. By using (8.9) we find that

Q1/Q2=1-x1, T2/T1=(1-x1)(1-x2). (8.13)


................................. ....................

: : : : : : :
: X1 : X2 : X1 : ... : Xi : Xj :
................................. ....................

................................. ....................

: : : : : : :
: -> -> : -> -> : -> -> : ... : -> -> : -> -> :
: : : : : : :
................................. ....................

................................. ....................

: : : : : : :
: x1 : x2 : x1 : ... : xi : xj :
................................. ....................

Figure 3. A heat machine which prints out vectors of feelings on

89. Let us choose an arbitrary pair (x1,x2), where neither x1 nor x2 is
equal to 1 and x1+x2>0, and two arbitrary pairs (Q1, Q2) and (T1, T2)
which satisfy equations (8.13). Construct now a finite machine in which
those four values are used (Fig. 3). In accordance with (7.6), (8.11),
and (8.12), composition X[n] in Fig. 1 and heat machine in Fig. 3, with
given values of x1 and x2, generate the same pair of vectors of
feelings A and a. Let us choose now x1=0 and x2=0. In this case T1=T2,
which follows from (8.13). Therefore, the subject's state in which he
has the ability to make free choice corresponds to the state of heat
equilibrium in which engines cannot perform work.

90. By establishing one-to-one correspondence between a sequence of
functions constituting the composition on Fig. 1 and a sequence of heat
engines constituting the heat machine on Fig. 3, we settle the
isomorphism between the analytical and the thermodynamic models.

91. Let us return to Equation (4.2). Its meaning becomes clear from the
thermodynamic model. Let utility v1 correspond to the heat Q1 received
by engine 1 from reservoir 1 and utility v2 to the heat Q2 yielded by
engine 1 to the reservoir 2. We showed in this section that x1
corresponds to the efficiency R1 of engine 1 (see (8.9) and (8.12)).
Equation (4.2) reflects this parallel.


92. We will conclude by adding to the formulation of the Anthropic
Principle in cosmology the formulation of the Principle of Freedom:

What we can expect to observe must be restricted by conditions
necessary for our presence as observers. We have free will and
under some circumstances also freedom of choice.

93. The first sentence helps us construct concrete models of Universe;
the second concrete models of the human being. However, the combined
statement is more than a sum of its two parts. We can now say

What we expect to observe must be restricted by conditions under
which we at least sometimes have a free choice.

94. This means that we are not only observers but also to a certain
degree creators of the reality we observe. This is the content of the
connections between psychology and cosmology on the level of abstract


I am sincerely grateful to Anatol Rapoport for numerous constructive
suggestions and for his invaluable help in expressing in English the
content of this paper. I am also grateful to Victorina Lefebvre without
whose constant help this work would never be completed.


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