**
Kurt Lewin’s Dynamical Psychology Revisited and Revised**

by

KULLERVO RAINIO

*
Professor emeritus,
Helsinki University*

Address: Suursuontie 29 as. 327, FIN-00630 Helsinki

e-mail: kullervo.rainio@pp.inet.fi

**
Abstract.**
The aim of this article is to show that the framework of the dynamic psychology
-created by Kurt Lewin – although it is 70 years old – can be a very good basis
for modern holistic theory of cognition and behavior. An essential new feature
in that “neo-Lewinian” theoretical framework – named here Discrete Process Model
(DPM) – is the definition of psychic force: it is treated as the *probability
*of a trial from a cognitive state to another. Thus, the strength of the
force has an exact measure: that probability. The cognitive course of events is
a stochastic process controlled by those probabilities. – Lewin’s brilliant idea
to assume certain valence fields which produce psychic forces is made
mathematically more exact in DPM. – The model makes possible to derive *
estimates* for the probability distributions of the behavior alternatives –
differently from the Lewinian theory, which is deterministic.

*
Keywords: *
choice* *behavior, cognitive trial, discrete model, graph, hodological,
holistic, psychic force, valence field.

1. Introduction. Background

**
**

It might seem strange to take into examination Kurt Lewin’s conceptual framework of dynamic psychology 70 years after it was published! – Lewin is such a classic figure, particularly in social psychology, that one may assume all essential to be said about his work. It has to be noticed, however, that his principal mathematical (topological) system which is described in detail in his work “The Conceptual Representation and the Measurement of Psychological Forces” in 1938 seems to be left without deep critics and is not developed further. Just in this work, however, Lewin’s genius and the depth and the originality of his thinking come out most strongly. I believe that that work remained rather unknown in the circumstances of war. In that time there was seemingly more use of Lewin’s fundamental analysis of group dynamics and social conflicts while the mathematical construction of psychological conceptual system appeared to be unfamiliar for the “real life”.

The basic
finding by Lewin was seemingly very simple, but revolutionized the scientific
thinking in psychology. He found that, in the description of behavior (and of
cognition), reality needs to be formed and organized in a new way. In
psychology, the world needs to be differentiated to such separate states which
have *meaning* to the subject, the psychological process being a locomotion
in such a space, i.e., transitions from a meaningful state to another meaningful
state. According to Lewin, only these states with meanings have *relevance*
in psychological description. (For example, when we walk, the steps – which are
physically clearly observable and measurable – are normally psychologically
entirely irrelevant; instead of that, the state of *walking* with its
meanings – the goal of it, the path to be used etc. – is relevant.)

Nowadays it may be difficult to realize what a remarkable jump in thinking Lewin’s view was: to change over from the concrete world of things to the abstract reality of meaningful states.

This deep insight
of a ”particular psychological world” was the basis on which Lewin constructed
his conceptual system of dynamical psychology trying to reach mathematical
exactness, taking on the use of particular the topological (“hodological”)
analysis; that suited well to the discrete space description what was needed.
Lewin was able to build *explaining *models for behavior situations and to
examine psychic occurrences in a way that was *relevant* to them.

Lewin’s
framework was, however, just a beginning of the new theory construction. There
were no possibilities to produce any useful *estimation* method in his
theory. – Why? In some way Lewin was in his thinking dependent on the *
deterministic* viewpoint – as we shall see. Quantum mechanics was not yet
broken its predominance in his time. The quantum theory, in principle, allows a
*non-deterministic* or stochastic view of mental states, as well as a
superposition of states. We say that this superposition of states in acted on
be a Hermitian operator, which, in the terms of quantum physics, acts according
to the principle of complementarity.
According to our view of complementarity, consciousness applies certain filters
(corresponding to mathematical operators) to create a reality of discrete
observables. But the primary process, in the unconscious, experiences the
unfiltered multiverse, the *unus mundus* of infinitely branching paths.
Before cognition can be understood, one needs to understand precognition. Freud
and Jung understood this. Much of modern psychology appears to have forgotten
it.

Kurt
Lewin’s sudden death 1947 was detrimental for the development of the new
dynamical psychology. At that time we had just started to create epoch-making
technical discoveries (computers) and constructing stochastic simulation methods
and new theoretical models. These strongly induced application of mathematics in
behavioral science. The stochastic learning models by William K. Estes (Estes,
1950) and
then by Robert Bush and Frederick Mosteller (Bush &
Mosteller, 1955)
gave impulses to stochastic theory construction in psychology. The quick
development of efficient computing gave opportunities to apply totally new
calculation methods. *Simulation *of complicated models which required
enormous amount of computations came possible in practice.

We can only guess what new, fruitful possibilities these methods could have opened to Lewin himself in further construction of dynamical psychology. But without Lewin’s genius the development of the basic mathematical concepts did not get sufficient attention in dynamical psychology. Instead of that, the mainstream psychological research was mostly directed to group dynamics. (I had an opportunity to perceive that trend when I was visiting the Group Dynamics Center in Ann Arbor 1956 and participating shortly a graph theoretical seminar led by Dorwin Cartwright and Frank Harary.)

It was rather late during my research work when I found the possibilities to further create the Lewinian system. – I had applied the stochastic learning theory first to sociological issues – to the birth and development of friendship relations in groups (Rainio, 1961, 1962, and 1966. See also Freeman – White – Romney, 1992, pp. 164-167 “Rainio’s Model of Social Interaction” and Wassermann, 1978, p. 811) -- and later to group problem-solving (Rainio, 1966 and 1970). -- About the year 1970 I realised that, in matter of fact, I was reconstructing the Lewinian system using new mathematical tools (Rainio, 1970 and 1983) .

I
realized that on the basis of Lewin’s insightful ideas one can create stochastic
(indeterministic) models which have – unlike Lewin’s model - *estimation power*.
A good demonstration of this appeared a model of group problem-solving
which gives estimates for behavior in Group Maze test
(Rainio,
1972, 1986, pp. 167-187, and 2000, pp. 50-85. See also Rainio, 2008, pp. 24-28
and 73-77). But why return again to investigations carried out many decades
ago?

The answer will be surprising: Lewin’s basic idea, discrete behavioural space (the use of topological description) contains seemingly much more than psychological aspects of interest. In its new form his conceptual system is, mathematically clearly analogous to the matrix-algebraic quantum mechanics created by Werner Heisenberg. This ultimate result is philosophically extremely interesting. We shall examine this claim briefly in the end of this article (See also: Rainio, 2008 and 2009).

2. Revision of the Lewinian conceptual system (Lewin, 1938)

**
2.1. Hodological
vs. graph representation**

The basic invention of
Lewin was to introduce a new geometrical framework, a ”hodological space”
(”hodos”, a Greek word meaning ”way”) to describe psychological occurrences.
Lewin realized clearly that what is relevant in describing the behavior is not
the perceived *physical changes* but the states of individual’s mind which
could be separated from each other on the basis of the psychological *meaning
*of the state.

________________________________________________________________________________

*
Fig. 1. Topological, matrix, and
graph representation of the mind (a life space) as a collection of the states A,
B, C, and D (fig. 1A1) , and with E (fig.1A2).*

________________________________________________________________________________

”... The purpose of hodological space”, writes Lewin, ”is to find a type of geometry which permits the use of the concept of direction in a manner which will correspond essentially with the meaning that direction has in psychology.” (Lewin, 1938, p. 23)

The hodological space
is a *discrete* presentation of the space. Unlike the Euclidean continuum
of points it is constituted of *regions *with boundaries between them, some
regions being ”neighboring” each other (*i.e.,* having common boundaries)
some not. The whole of all regions is called ”*life* *space*”.

The structure of the ”life space” is defined by the ”neighborhood relation” between the regions. It is conveniently represented in a matrix form - and illustrated by a graph form as exemplified in Fig.1. In fig. 1.A1 the regions A, B, C, and D constitute the life space. Should we add to the life space of A,B,C, and D a new region E as in 1.A2 requiring that E should be a neighbor to all the regions, then we necessarily need in the topological description a new meta-element called a ”tunnel” (T) to connect E and A. The matrix forms of the same life spaces are in Fig. 1.B where the names of rows and columns refer to a ”region” and the element 1 to the existing neighborhood between the row region and the column region. As can be seen from the matrix all regions are neighbors.

From the point of view of theory-building this change is, naturally, trivial.

**
2.2. Psychological forces**

The *psychological force*
is defined effecting in life space, as a cause of locomotion of the *point of
application* (the person) from one region to
another.

A force has: 1) direction, 2) strength, and 3) point of application.

According to Lewin,
the notation f^{P}_{A,B} means a psychological force effecting
in the region A, and having the direction from A to B, and P being as the point
of application (Lewin, 1938, p. 83, Fig. 2).

Lewin uses notation
½f_{A,B}½
for the *strength* of the force.

Usually the meaning of
the point of application seems clear from the context. It is a person (as a
whole) located in a region and according to Lewin the force influencing the
behavior of P can be expressed by notation f_{P,B} as well as f^{P}_{A,B}.
However, we shall see later that this is a foggy point in Lewin’s thinking.

Lewin recognizes 3 types of forces (see Fig. 2):

________________________________________________________________________________

*
Fig. 2. Life space of the
regions A and B and forces acting on person P in A.*

________________________________________________________________________________

**
**

1) *Driving force*,
notated as df^{P}_{A,B} if needed – or simply f_{A,B}.
It is a force applied to person P, located in the region A, driving him toward
the region B.

2) *Restraining
force*, notated as rf^{P}_{A,A} if needed, or simply f_{A,A}.
It forces the person P to stay on the same region A where he is already located.
Thus, it opposes the driving forces.

3) *Boundary force*,
notated as f_{B,-B} , is acting on the person in order to prevent the
person enter the region B when he is in the ”boundary region”. It ”corresponds
to the tendency to locomote from B to a region X outside B, for instance to A.
The force f_{B,-B} exists, therefore, at least at the common boundary of
A and B” (Lewin, 1938, p. 75). When Lewin uses this concept he seems to think
that there exists an obstacle between the regions. However, it seems to me
strange to see this obstacle as an active force, particularly so because the
boundary is *not* a region. - The concept of boundary force needs
clarification – as we shall see.

An essential concept is the *
resultant force, *which Lewin denotes by f*. The resultant force determines
the direction of locomotion. Therefore it is extremely important to show how the
resultant is derived from the totality of forces empowered in the person. Lewin
does not succeed in this. He gives a definition of ”totality of forces” in a
region A region A by the following formula:

Sf_{A,X}
º
f_{A,B} + f_{A,C} + ... + f_{A,N}

Lewin gives no clear
definition of how to get the resultant force as a sum of its components. He
writes about the direction of the resultant
force: ”The resultant of a group of forces can have the same direction as one
force of the group. ... For instance, it might be that
Sf_{A,X}
º
f_{A,B} + f_{A,C} + ... + f_{A,N} = f*_{A,B}
.” (Lewin, 1938, p. 85)

In connection to the
resultant force Lewin introduces a measure for the speed of locomotion, namely
the concept of *velocity*. If there exists a resultant force f*_{A,B},
directed from A to B and its strength is
½f*_{A,B}½>0,
then the velocity of locomotion is also > 0, and notated with v_{A,B} >
0.

As far as I can see,
Lewin thinks that the resultant force is the largest of the forces acting in the
same time in the same region. According to him, in a certain situation there is
one possible resultant force and the behavior could be exactly determined if
this could be found out. But Lewin did not have such a method. According to
him, a resultant force f*^{P}_{A,B} had appeared *ex-ante*
if a locomotion from A to B was observed. He shows no method to *estimate*
beforehand which direction the locomotion will take. This ruins the usefulness
of Lewin’s system as an experimentally testable theory. This weakness in the
system is understandable only on that ground that Lewin actually tries to create
a *deterministic* theory of behavior and, thus, as we now well understand,
was doomed to failure in his attempt. A radical change at this essential point
is necessary.

It is necessary to
leave out the concept of a resultant force. *The* *”psychological forces”
should be represented as probabilities *of cognitive trials*.*

To make the use of
probabilities possible we shall handle *also the time variable as discrete, in
addition to the discrete space.* The probability of an event will, thus,
always mean the probability* *per* a given unit of time* - one step
of the process taken to run in one unit of time measure.

The ”psychological
force” as a propensity to make a trial to change the prevailing state or
preserve it during one step in time, and the propensity measured with the
probability measure offers a more sound and flexible theory of dynamic
psychology. The greater the propensity the greater the probability and also the
psychological force involved. The strength and direction of a force are both
implicitly determined by the probabilities of locomotion to adjacent regions.
These probabilities capture the meanings of the Lewinian driving forces f_{A,X}
, and the probability to stay fully captures the meaning of the Lewinian force f_{A,A}.

The direction of
”locomotion”, *i.e.,* the trial alternative, is then determined with a
“lottery” *outcome* from a probabilistic choice using the vector of the
trial probabilities applicable to the situation concerned.

An example of this correction is shown in Fig. 3.

In this very important
revision the original Lewinian system loses its deterministic character and we
shall be dealing with a stochastic theory. This gives an opportunity to use the
revised theory also for *estimation* purposes.

________________________________________________________________________________

*
Fig. 3. Resultant force concept*

Lewin: Driving forces f^{P}_{A,B
}and f^{P}_{A,C }; restraining force rf^{P}_{A,A}
; resultant force f*^{P}_{A,B}.

Correction: Vector **p**_{A,X}

Pr[A,B] driving force

Pr[A,C] driving force

Pr[A,A] restraining force

(No resultant force.
Probabilistic choice using vector **p**_{A,X} determines the
locomotion.)

________________________________________________________________________________

**
2.3. Boundary of a region.
Barrier strength**

The concept ”strength of a barrier” by Lewin has nothing to do with the boundaries of topological (or hodological) representation. (See Fig. 4) The “barrier” has an illustrative character only.

________________________________________________________________________________

*
Fig. 4. Barrier strength*

Lewinian description; example:

Correction using probabilities of the non-success of trials:

Column vectors:

A B C D

A, p_{succ}
é1
ù
é.9ù
é0
ù
é0
ù

“, p_{Øsucc}
ë**0**
û
ë**.1**û
ë**1**
û
ë**1**
û
“strengths of barriers”

B, p_{succ}
é.8
ù
é1
ù
é.6
ù
é0
ù

“, p_{Øsucc}
ë**.2**
û
ë**0**
û
ë**.4**
û
ë**1**
û
“strengths of barriers”

C, p_{succ}
é0
ù
é.5ù
é1
ù
é.3
ù

“, p_{Øsucc}
ë**1**
û
ë**.5**û
ë**0**
û
ë**.7**
û
“strengths of barriers”

D, p_{succ}
é0
ù
é0
ù
é.6
ù
é1
ù

“, p_{Øsucc}
ë**1**
û
ë**1**
û
ë**.4**
û
ë**0**
û
“strengths of barriers”

The non-success probabilities of trials (bold numbers) indicate the “strengths of the barriers” (in the Lewinian terms). Thus, the non-success probability “1” means an “impassable barrier”.

________________________________________________________________________________

However, Lewin seems
to have touched here, intuitively, on an important topic, the resistance of the
psychic locomotion. But one can ask why he has not applied the idea more
systematically to *all* boundaries between the regions.

It seems convenient to
define - in analogy to the strength of a barrier - a particular concept of *
probability of the success of a trial* and apply it to *all* the
relations between the neighboring regions (graph nodes). Thus, the ”strength of
the barrier” will now have a new, mathematically exact meaning: it is defined by
the probabilities of success of
cognitive trial –
inversely: i.e., the strength of the barrier gets its minimum value when the
success probability is one and maximum when it is zero. (The success probability
of cognitive trial varies according to learning experiences.)

The transition from a behavioral state to another (subject’s real action) is also dependent of its success probability. The behavior success is, however, a property of environment (success of a move in game etc.) Thus, no learning is applied to the probabilities of the behavioural success.

**
2.4.**
**Point of application of forces. Group behavior**

Lewin emphasizes in his work the
need to see the individual as a *whole *in analyzing the psychological
situation. Thus,”the point of application of forces” in Lewin’s representation
is the *person*. Lewin shows this applying a special superscript symbol P
with the symbol of force f^{P}_{A,B} . Lewin uses also a
shortened expression f_{P,B} instead of f^{P}_{X,B}
where X indicates any of the regions. Lewin writes: “If we wish to indicate
specifically the point of application we will write: f^{P}_{A,C}
where A represents the region where the force exists, d_{A,C} the
direction of the force and P the point of application. We will not indicate
specifically in our formulas the point of application when the meaning is
clear.”
(Lewin, 1938, p. 74)

Lewin seems to forget it in his discussion of the ”inner personality”, particularly in his presentation of a ”relation between a force for locomotion acting on the person as a

whole and the tension in inner
systems”. (Lewin, 1938, p. 98). He uses in this context, *e.g.,* the
expression

indicating ”a force working on
the boundary b of S in the direction of the neighboring system S^{1}
” (Lewin, 1938, p. 99). Which is now the ”point of application of forces”? If
it is an ”inner system” of a person, is it meaningful to discuss the ”life
space” of such a partial system?

It seems to me that, in omitting the problem of point of application of forces, Lewin commits a mismatch. – Some ”group dynamists” are guilty of the same kind of a logical mismatch or an ”inventory error” when they take forces to act at the same time both on the group and on the members of that group.

For solving the formal
problem and for avoiding the confusion, the ”life spaces” of different subjects
should be kept logically separate from each other. But the subjects may yet
interact with each others, which means that the forces involved are then* *
given as *conditional *probabilities, the conditioned by the *
expectations* concerning the
behaviour of the
other group members. The point of application is always an individual.

**
2.5. Force field and its valence**

The origin of a force is,
according to Lewin, a specific characteristic of region (a mind state), which he
calls the *valence* of it. Lewin gives the definition:

”A region G which has a valence Va(G) is defined as a region within the life space of an individual P which attracts or repulses this individual. In other words:

Definition of positive valence:

If Va(G) > 0, then
½f_{P,G}½
> 0.”

”Definition of negative valence:

If Va(G) < 0 then
½f_{P,
- G}½
> 0.”

Lewin continues: ”The concept of valence ... does not imply any specific statement concerning the origin of the attractiveness or the repulsiveness of the valence. The valence might be due to a state of hunger, to emotional attachment, or to social constellation... The statement that a certain region of the life space has a positive or negative valence merely indicates that, for whatever reason, at the present time and for this specific individual a tendency exists to act in the direction toward this region or away from it.” (Lewin, 1938, p. 88)

Lewin makes an
assumption which is very essential in his framework: the valence induces a *
force field* which reaches more or less all regions in the life space
creating in them a force toward the goal region, G , if G has a positive
valence, or away from G, if the valence of G is negative. – ”As in physics, the
forces of a force field are only conditional ones; they are those forces which
*would* exist in a region if the individual should be located in this
region.” (Lewin, 1938, p. 90)

Lewin illustrates the force fields by figures indicating the forces by arrows[i]. The figure representing positive central force field is reproduced in Fig. 5. One can see that there is no exact theory but an illustration in question.

________________________________________________________________________________

*
Fig. 5. “Positive central force
field corresponding to a positive valence (Va>0)” (Lewin, fig. 33)*

“G, region of a positive valence
(Va(G)>0), located in C; P, person; the forces f_{A,C , }f_{H,C},
or f_{L,C} correspond to Va(G) in case P is located at A, H, or L,
respectively; f_{X,Y} = f_{X,G} .”

________________________________________________________________________________

Lewin’s
representation of force field is not exact. Later in this article (Chapter 4.3)
it will be shown that the Lewin’s invention
of valence is in a modified
probabilistic form a very suitable concept for the description of *intentional*
activity and goal-directed behavior.

Later, in Chapter 4.3, we shall handle the valences and the potencies of fields further and see how the trial probabilities (forces) are derived from them in an exact way.

**
2.6. Cognitive and behavioral
(“real”) world**

Lewin’s representation has a
confusing tendency to describe sometimes cognitive phenomena and sometimes the
so-called ”reality” by using the same terms and without formally making any
clear distinction between them. In order to create an *exact* theory, it
is, however, quite necessary to make this distinction, as clearly as possible.

In the “Discrete
Process Model” (DPM) as I should call the theory-model that will be described
later in this article more systematically, the psychic processes will be
presented as occurring, naturally, in the cognitive space. The ”regions” (nodes
in our graph representation) stand for *cognitive states* of the subject.
*The psychic forces act always and only on a cognitive state * - activating
cognitive trials in order to change or preserve the state.

The graph
representation (as well as the matrix representation of transition
probabilities) is used also for the ”real” world (*i.e.,* for the world
seen as well by the subject as by an observer and interpreted *according to
the psychological relevance*). Thus the “objective” states of an individual
are represented as nodes of a graph but this representation has no *a priori*
linkage to the cognitive world of the person in question.

The linkage is given
by a particular concept of *correspondence*, *i.e.,* in the form of a
correspondence probability matrix where the rows indicate cognitive states, the
columns real world states, and the elements the probabilities of the
correspondence between these states.

The correspondence probabilities may vary between 0 and 1. It is, however, plausible to assume that usually, for instance in game situations with clearly defined rules, the probabilities might well equal to 1 or 0.

Actually we have to take in our use two kind of correspondence matrices:

1) RCC - (reality-cognition) -correspondences, probabilities indicating which ”cognitive map” corresponds to a certain real situation.

2) CRC -
(cognition-reality) -correspondence, probabilities indicating which is the real
state to which the subject’s certain cognitive state refers, *e.g.,* which
behavior movement follows a certain decision to move.

The ** essential difference**
between the original Lewinian framework and the “neo-Lewinian” Discrete Process
Model (DPM) is that the latter offers the possibility to measure the variables
such as the strength of psychological force, the strength of barrier, and the
strength of valence in terms of probabilities. Thus, the DPM-framework may be
worth a more systematic, brief, summarizing description.

3. Discrete Process Model. Basic concepts

(Rainio, 2008)

In our theory, *both space and
time appear as discrete variables*.

At any point in time, the person is assumed to be in one and only one
state defined in the discrete space. In any *time-step* the person *
transits* from an actual (cognitive or behavioural) state to another one or
stays in the prevailing state. In principle, any transition between the states
is possible. It is assumed, however, that there exist conditional *
probabilities* of transitions from a given state. (Staying in the prevailing
state is one alternative among the transitions and occurs by a certain
probability, too.)

The transition principle is applied separately in cognitive and behavior occurrences.

**
The basic concepts and symbol
notations are as follows:**

*
Symbol: Verbal
description:*

*
Behavior*:

Be(i,a,t) Behavior
state: an individual **i** maintains a behavior state **a** at the moment
**t** – in other words: **i** is in the *real* behavior state **a**
in a space of states at **t, ***i.e.*, Be(i,a,t)
Î{Be(i,x,t)}.

*
Symbol: Verbal
description:*

By the ”real behavior state” we mean a psychologically relevant situation of an individual, available for observation, in principle.

Tr(i,a,b,t) Behavior
trial: it is a try to change a behavior state
maintained, *i.e.*, **i**,

being in the state **a**, tries to behave in the way **b** at the

time
point **t**.

Succ(i,a,b,t) Success of the trial, a shortened expression of Succ(Tr(i,a,b,t)).

Correspondingly, ¬Succ(i,a,b,t) means failure of the trial Tr(i,a,b,t).

*
*

*
Cognition*:

(Notice the sign ‘ indicating cognitive things.)

C(i,a’,t’)
Cognitive state: **i** imagines in the way **a’** at a cognitive
time point **t’** – in other words: **i** is in the state **a’** in the
space of cognitive states at the cognitive time point **t’. **And** **
C(i,a’,t’) Î
{ C(i,x’,t’)}.

Tr’(i,a’,b’,t’) Cognitive
trial: **i** whose cognitive state is **a’, **tries to make a cognitive

choice **b**’ at the cognitive time point t’.

Succ’(i,a’,b’,t’) Success of a cognitive trial Tr’(i,a’,b’,t’). ¬Succ’ means the failure.

NS’(i,a_{0}’,b’,t)
Cognitive success sum: number of consecutive successful cognitive trials

made
by **i** toward **b’** from the state **a _{0}’** at
real time point

AT(i) Action threshold: an integer indicating the number of consecutive

successful cognitive trials (NS’) needed to make a decision and to start an

action (a “real” trial) according to it.

*
*

*
Cognition and reality:*

RCCor(i,a,M’,t)
Reality-cognition correspondence: **i**, being in a behavioral state **a**
at the

time
point **t**, creates in his cognition a *cognitive map* **M’ **(Life
Space) which

consists of the relative
cognitive states in
the actual situation **a**.

The cognitive map resembles, for
its structure, the graph of the real states but need not to be identical with
it. In a problem-solving situation, where **i** commits himself to the rules
of the problem, the cognitive states of M’ may well be in one-to-one relation to
the real states.

CRCor(i,a’,a,t)
Cognition-reality correspondence: in **i**’s cognition, the cognitive

alternative **a’** corresponds to the behavior alternative **a** at the
moment **t**.

The correspondence between
individual’s cognition and the behavior space is given by probabilities **p**(CRCor(i,x’,x,t)).

(In laboratory game situations it is usually plausible to assume the simplified one-to-one relation, i.e., there is always for every x’ one and only one x corresponding to it.)

*
*

*
Cognitive and real time:*

Real time points are denoted by t, t+1, t+2 etc. Analogously, t’, t’+1, t’+2

etc. indicate points in cognitive time. If the cognitive time-step is denoted

by Dt’ and behavioral time-step by Dt then Dt’ £ Dt.

Cognitive information in the mind (I), according to Germine (1993), can, in principle, be determined by the frequency of state changes (P), the probability of the state (R) from 0 to 1 (relative entropy), the total of possible microstates (g), and Boltzman’s Constant (k) according to the equation:

I = (1-R) k Pt ln g

In this model, a superposition of states could be said to exist, each with a certain probability, with a higher information content associated with realization of a lower probability state. The superposition of states would then be the sum of the fields. The function g can be viewed abstractly as the area of the life space in Lewinian terms. The transition from one state to the next may be viewed is collapse of the wavefunction, which would presumable be the entirety of the life space at a given time, into a single region of the life space. The movement involved would then involve a path in hodological space.

4. The cognitive process and behavior according to the DPM

**
4.1. Cognitive and behavior
transitions **

Lewin described the
psychological process as *locomotion* of the person from a region to
another in topological space. The direction of locomotion was determined by the
resultant force. Lewin did not make any distinction between the cognitive and
behavior locomotion. DPM makes this distinction. According to it, there occur *
cognitive transitions* from a cognitive (mental) state to another and *
behavior transitions* from a behavior (observable) state to another. The
direction of a transition is determined by *transition probability vector*
and by a *probabilistic choice* using it. (In simulation of the process,
Monte-Carlo –method is applied.)

The main features of the cognitive process and of the first behavior choice are given in the following “flow chart” list (Table 1). (The topological structure – the graph – of the Life Space and the corresponding “cognitive map” – cognitive Life Space – are assumed to be set in the beginning.)

Table 1 shows the cognitive steps needed for one behavior transition. (For the sake of simplification, action threshold AT(i) is assumed to be 1.)

________________________________________________________________________________

*
Table 1. Cognitive process and
the first behavior choice. “Flow-chart”.*

1) Start. – Give to the
state-variable x the value x_{0} and to the time-variable t the value t_{0}
.

Thus, the behavior
state is Be(i, x_{0} ,t_{0 }).

2) Give to the cognitive
time-variable t’ the value t_{0}’ and to the cognitive state-variable x’
the value x_{0} ’.

Thus, the cognitive
state is C(i, x_{0}’,t_{0}’).

3) Determine the *cognitive*
trial Tr’(i,x_{0}’,y_{k}’,t’), applying the Monte-Carlo-method
to the probability vector **p**(y’½x_{0}’)
and denote the outcome by y_{i}’. – Add 1 to t’.

4) Determine the *success*
of the cognitive trial, applying the Monte-Carlo-method to the probability
vector **p**(succ’(Tr’(i,x_{0}’,y_{i}’,t’))), the outcome
being succ’(Tr’(i,x_{0}’,y_{i}’,t’) or ¬succ’(Tr’(i,x_{0}’,y_{i}’,t’).

5) Test the outcome of the success of the cognitive trial.

6) If ¬succ’ then go to 3

7) If succ’ then set y_{i}
as an argument of the *real* trial Tr(i,x_{0},y_{i},t)
– x_{0 }corresponding to x_{0}’ and y_{i} to y_{i}’.

8) Determine the success of real
trial Tr(i,x_{0},y_{i},t), outcome being succ (Tr(i,x_{0},y_{i},t)
or ¬succ Tr(i,x_{0},y_{i},t). Carry out *learning*.

9) If ¬succ then go to 11.

10) If succ then set y_{i}
in the *real behavior* Be(i,y_{i},t+1) and go to end (12).

11) Let t_{n} indicate
the end of time available. – Test t. If t < t_{n} then set x’, y’, and
t_{0}’ and go to 3 but if t = t_{n} then no action occurs and go
to end (12).

12) End.

________________________________________________________________________________

**
4.2. Learning**

DPM offers an easy way to mathematically take the learning phenomenon into account. We may well assume learning reinforcement so that 1) the success of trial affects the cognitive trial probabilities rewardingly, the failure having a punishing effect and 2) the reached real state has the quality of producing reward or punishment concerning both the trial probabilities and the success probabilities. We can apply some stochastic learning model, e.g., the Bush-Mosteller’s Two Operator Learning Model (Bush and Mosteller, 1955).

An
example: Let’s assume a cognitive trial probability vector **F**:

**F**

A’ B’ C’ D’ S

A’ (.2 .5 .2 .1) 1

If the Monte Carlo –lottery has given the outcome C’ and, thus, the person makes the trial from A’ to C’ and, further, it is assumed that this C’ state happens to be rewarding and the trial successful; then the rewarding Bush-Mosteller operator is applied. This means that the cognitive trial probability toward the state C’ increases according to the equation:

p_{t+1}
= p_{t} + a(1
– p_{t} )

Suppose
that the learning coefficient
a
= .5; the equation gives the new trial probability p_{t+1} toward the
state C’. It is .2 + .5(1-.2) = .6. The other elements of the vector

need to be decreased so that the sum equals 1. Thus, the new trial probability vector is:

(.1 .25 .6 .05) S=1

The same
learning principle is applied to the probabilities of the *success* of the
cognitive trials, too.

**
4.3. Valence, field
and potency of field**

One of the great
inventions of Kurt Lewin was to borrow the concepts of *force* *field*
from the physics and apply it with a concept *valence* to psychological
life space. The fields organize the life space. The valence gives to a field
(Lewin: region) its *goal* character.

However, as we have seen in Chapter 2.5, Lewin used the concepts of field and of valence mostly in an illustrative way. The new, more specified definitions of those concepts will be given in the following.

A *
field*, in DPM, is identified or defined by a subgroup of states within the
LS. The field partitions the life space LS into two subspaces: the states
belonging to the field (F) and the states outside F denoted as (ØF).

All fields in DPM
description are *cognitive*. – There may exist none, one or more states as
elements of a field. A state may be an element in many fields; thus, the fields
may be distinct or overlapping.

The whole *”life
space”* itself is a field. This field (F_{LS} or simply LS) includes,
then, all the *relevant* cognitive states as its elements (ØLS
being an empty subspace).

An example of 3 fields in LS is given in Fig. 6.

________________________________________________________________________________

*
Fig. 6. Three fields in LS*

LS = {a’, b’, c’, d’, e’} ØLS = {empty}

F1 = {b’, c’, d’} ØF1 = {a’, e’}

F2 = {b’, c’} ØF2 = {a’, d’, e’}

F3 = {d’, e’} ØF3 = {a’, b’, c’}

Notice that F2 is included in F1 but F3 not, and that fields F2 and F3

are distinct but F1 and F3 overlapping – the intersection being d’.

________________________________________________________________________________

*
Definition of valence and its
measure:*

Lewin gives actually
no exact *measure* for the valence: according to him there are only
positive and negative categories of valences. The new measure of psychological
force in terms of the trial probability gives an opportunity to define the
measure of valence.

If there exists a field F with a valence in the life space LS, the field F partitions the states of LS into two groups:

F = {s_{1}, s_{2},
… , s_{n} } , the field including states from s_{1} to s_{n}
in LS, and

ØF = all the other states, i.e., states of LS not belonging to the field F.

We define the *
measure of valence of the field *as follows* *:

There are 4 “field forces” induced by the field; they form the following probability matrix:

F ØF S

F
ép_{F,F}
p_{F,}_{ØF
} ù
1

ØF
ëp_{ØF,F
}p_{Ø,F,ØF
}û
1

An example in a matrix form and in an illustrative graph form is given in Fig. 7.

In the example in Fig.
7 there are only two states, one inside F and the other outside it. Thus, p_{F,F}
needs to be equal to p_{b,b} , p_{F,}_{ØF}
to p_{b,a }etc. What the “field forces”* generally* mean in terms
of cognitive trial probabilities will be presented later.

________________________________________________________________________________

*
Fig.7 . Life Space including one
field F and two states a and b. Forces produced by the field F*

“Field forces”

The “field force p_{F,F }
is acting as a “restraining force” p_{b,b} , the strength of which is
assumed to be .9, while p_{F,}_{ØF
}(strength .1) is
acting as a “driving force” p_{b,a} from b to a, _{ } p_{ØF,F
}(strength .8) as a
“driving force” p_{a,b} from a to b, and p_{Ø,F,ØF
}(strength .2) as a
“restraining force” p_{a,a}.

________________________________________________________________________________

According to Lewin, if
the field G has a positive valence, there must exist a force
½f_{P,G}½>0;
if the field G has a negative valence, there will exist, correspondingly, a
force ½f_{P,-G}½
> 0. (Lewin, 1938, p. 88, statements 20 and 20a)

Both definitions of the valence are too narrow. Using the “field force” probabilities above, we can define the valences more completely:

Va(F) = [(p_{F,F}
– p_{F,}_{ØF})
+ (p_{ØF,F}
- p_{ØF,ØF})]/2

The equation gives the
maximum positive value 1, if p_{F,F} = 1 and p_{ØF,F}
= 1 and the maximum negative value -1, if they both are 0. – In the example in
Fig. 7 , the measure of valence is: [(0.9 - 0.1)+(0.8 - 0.2)]/2 = 0.7.

*
The field’s effect on the
cognitive trial probabilities. Potency of field*

The fields produce the psychological forces acting on states. If there exist no fields inside the Life Space, the LS is homogeneous, i.e., all forces have equal strength (probability) and each trial probability is 1/n , n indicating the number of states in LS.

If there are one or more fields in LS, we have to combine the effects of them for deriving the strengths of the forces, the trial probabilities. This will be done by giving certain weights to the fields in the following way:

We use the term *
potency of a field* to indicate *the weights used in combining (summing up)
the effects of the fields in calculation of the cognitive trial probabilities*.
Thus, the sum of the potencies has to be 1. –
The rule of the
calculation is following:

Let’s
matrix **p**_{F} indicate the 4 field forces produced by a field F.

**p**_{F}

F ¬F

F
ép_{F,F}
p_{F,}_{ØF
} ù

¬F
ëp_{ØF,F}
p_{ØF,ØF
}û

Suppose that there are
n_{F} states *inside *the field F and n_{ØF
}*outside
*it. It seems plausible to
assume that the force p_{F,F } is distributed equally to the states
inside the F, the force p_{F,}_{ØF
}equally to the
states outside F, the force p_{ØF,F}
equally to the states inside F, and the force p_{ØF,ØF
} equally to the
states outside F.

The general equation for this calculation is given in Rainio 2000, Chapter 2.6. We examine here (Table 2) only a numerical example calculating the trial probabilities according to Fig. 6. We make a simplification and assume that the valences of all the fields are = +1. The potencies are given in Table 2, too.

After calculating the component vectors we shall form a weighted sum of them using potencies of the fields as weights. These weighted sums are, thus, the trial probabilities.

________________________________________________________________________________

*
Table 2. Computing the trial
probabilities produced by fields*

(See the example in Fig. 6)

Fields n_{F}
Vectors of probabilities Potencies

a’ b’ c’ d’ e’

LS
5 **a’** (.2 .2 .2 .2 .2) .1

F1
3 **a’** (0 .33 .33 .33 0 ) .3

F2
2 **a’** (0 .5 .5 0 0 ) .4

F3
2 **a’** (0 0 0 .5 .5) .2

Components weighted by potencies:

LS **a’** (.02 .02 .02 .02 .02)

F1 **a’** (0 .1 .1 .1 0 )

F2 **a’** (0 .2 .2 0 0 )

F3 **a’** (0 0 0 .1 .1)

Sums, the trial

probabilities

(“driving forces”): **a’**
(.02 .32 .32 .22 .12) S
@
1

(The trial probability vectors
**b’**, **c’**, **d’**, and **e’** are equal to **a’**.)

________________________________________________________________________________

Note: The
high potencies of fields F_{1} , F_{2} , and F_{3} (in
the example in Table 2) mean psychologically *high differentiation *of
cognition. If they were low and Pot(LS), correspondingly, high, the
cognition would be undifferentiated, diffuse. The extreme case while Pot(LS) = 1
and, thus, all trial probability vectors (“forces”) (.2, .2, .2, .2, .2) ,
would indicate the most primitive cognition, “random walk” through the states. –
(The “picking up” the *relevant* states, giving form to the actual life
space, means, naturally, the first and most fundamental differentiating act of
subject.)

Lewin talks about some concrete
things which can produce valence fields, e.g., food, social acceptance,
producing something valuable, electric shock (negative field) etc. – We
emphasize, however, the cognitive aspect: cognitively, the *goals* (or
cognitive states having valence) mean that those states are bringing positive *
expectations*. (Correspondingly, the state with negative valence creates
negative expectations which lead to the avoidance of it.) Not until the goal is
*reached *(i.e., the corresponding behavior trial occurs and succeeds), a
learning reinforcement takes place. – The goal needs not to be a “real thing”;
it can well be a cognitive state, e.g., during a thinking process a state where
the individual experiences that the problem is solved. This can produce –
without any “outer behavior” – a reinforcement effect, too.

The expectations play an
essential role in *group dynamics*. According to DPM, the group is defined
as a set of individuals (members) who have *at least one expectation*
concerning the behavior of some other member. This makes every individual in
that set of individuals somehow dependent on the others. In human groups the
expectations are strengthened by *communication*. The expectations are
definite, if the person *believes* completely the message he receives,
otherwise he does not take the message into account; thus, a probability vector
is used for determining the believing. (Seemingly, there are cultural
differences: if in a group – or in a culture – the probabilities to believe the
messages are high, the group behavior is more *articulated*
(differentiated) than in groups with low belief probabilities.)

The analysis of group behavior is not examined here further. (A detailed description of group behavior – in terms of DPM – is available in Rainio 1972 and 2000.)

5. Discussion

Although very
little ** empirical work** has been done using Discrete Process
Model for estimation, some laboratory experiments show that the model can well
be operationalized at least in well controlled laboratory game situations; in
those situations, namely, the behavior choices and communication acts are
observable. A particular Group Maze experiment has been constructed and
simulated using DPM. The goodness-of-fit of the simulations has been rather high
(Rainio, 1972 and 2000).

*
Physical quantum
mechanics and DPM:*

The mathematical
structure of DPM is *general*; it covers different kinds of systems. Thus,
it is actually not very surprising that it is applicable to the description of
physical quantum evolution, too.

The
probability of cognitive trial corresponds – in quantum mechanics – to the
probability of *transition* of a system from one quantum state to another.
(But no “success” of transition is assumed.) The vector of the transition
probabilities at a certain point in time indicates a system’s *superposition
state*. This corresponds to Feynman’s sum over all paths. The system’s
evolution process is manifested as “jumps” from a superposition state to
another. If some vector, however, allows only the staying in the actual state
(i.e., the transition probability from the state *i *to itself, p_{i,i
} , is 1 and the others 0), then the process ends in this stable state.
Such a state is not a superposition state but a so-called “definite” state and
it is *observable, *in principle – i.e., the system exists in that state.

The application of DPM both to physical quantum mechanics and to cognition is represented in detail in Rainio’s several works (Rainio, 2008 and 2009). Notice that, independently of DPM and without knowing it, McCall, Whitaker, and George have also used a discrete quantum evolution model which is based on the transition probabilities (McCall, Whitaker, and George, 2001).

It is **
philosophically** interesting and important that the DPM approach – being
extremely abstract and general – seems to be able to combine the quantum
mechanical theory and the dynamical psychology to the same conceptual framework.
(One has to notice, however, that the dynamical psychology is not identical with

It seems obvious that Kurt Lewin’s brilliant intuitive insights concealed fundamental ideas which lead to new understanding not only in dynamical psychology but also in a much greater domain of philosophy.

Acknowledgement

I am very grateful to many members of the Finnish Society of Natural Philosophy for their interest in my ideas and I want to thank particularly Pentti Malaska – emeritus professor in mathematics of economy and honorary doctor in technology – who has been patiently great help to me.

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