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The mathematico-symbolic formulation of teleonic principles.


Teleonics has its roots in computer simulation studies of calcium homeostasis (Jaros et al., 1979, 1980). Thus, its origins lie in the mathematical, computer-modelling field applied mainly to the analysis of complex physiological problems (Jaros et al., 1987, 1988). However, it is clear that the method could be applied to complex fields outside the realm of physiology and was thus formulated as a general systemic approach under the name 'biomatrix approach' (Jaros and Cloete, 1987). The goal-oriented process systems that form its basis were later given the name 'teleon' (Jaros and Cloete, 1990) and the method has frequently been referred to as teleonics (Jaros and Peeno, 1996; Jaros, 1998, 1999; Jaros and Bunn, 1998; Jaros and Cloete, 1993). An extensive study of its principles has been completed recently (Cloete, 1999). The approach has been used in many different fields, such as healthcare (Dostal et al., 1998), development (Dodds and Jaros, 1995), psychology (Edwards and Jaros, 1995), education (Dostal and Jaros, 1996) and organizational business development (Jaros and Dostal, 1999).

Recently, we have been attempting to apply the principles of the methodologies to the analysis and design of information systems. Although the general descriptive method has been very successful in analysing soft systems it has become essential to return to a more mathematical approach for teleonics to be rigorously applied to program design. The present paper focuses on formulating the main principles of teleonics (biomatrix approach) in a mathematico-symbolic way.

The body of the paper is divided into three main sections. In the first section some of the assumptions and definitions that form the background for the development of teleonics are introduced. In the next section the fundamental elements of a teleonic 'universe', such as teleos, ethos, teleon, doublet and biomatrix, are dealt with. In the last section some more advanced teleonic concepts, relations and operations such as strength and adaptivity of teleons, telentropy, etc. are discussed.


In this section, we present some basic assumptions that have been used in the development of teleonic principles. Some of these assumptions may be at variance with those generally accepted in systems sciences and in classical sciences.


According to the generally accepted scientific paradigm, the substance of the universe is matter. Energy and information are simply 'attributes' of matter, in spite of the fact that matter and energy were proven to be interchangeable almost 100 years ago.

We propose that the substance of the universe is mei ([mu]), viz. an inseparable, interchangeable, transformable, time and space variable amalgam of Matter, Energy and Information.

(1) Matter = [mu]|matter, Energy = [mu]|energy, Information = [mu]|information

Note that even in a substance of 'purely' material nature, energetic and informational aspects are always present, albeit to a generally negligible extent. For example, the cohesive forces that keep the constituents of matter together represent the energy and the factors that determine the intra-atomic and surface configurations represent the information components. Even a 'purely' informational event such as thought requires matter (neuronal substrate, ions and carriers) and energy (to move ions between compartments).

Systems in Teleonics

Spatial Structures

A system, or a spatial structure ([S.sub.s]) is a set of entities (O) and the relationships (R) between these entities within a defined boundary (Klir, 1991):

(2) [S.sub.s] := (O, [R.sub.o])

This general definition can be supported with the explicit handling of time variance, i.e. with the introduction of the so-called emergent properties ([O.sup.e]), which cannot be fully predicted during the analysis or design of the system from either the component entities or from their relationships:

(3) [S'.sub.s] := (O [union] [O.sup.e], [R.sup.e.sub.o])

where [O.sup.e] denotes emergent entities and [R.sup.e.sub.o] the emergent relations, both among the already existing entities and the emergent entities.

However, with the extended concept of substance the concept of a discrete entity (o [member of] O) also changes. It is a part of the total substance mei, separated from the rest of the world by well-defined, universally (or at least mutually) accepted spatial boundaries ([B.sub.u]). Since mei contains matter, energy and information, discrete entities may also possess material, energetic or informational characteristics. Physical entities can be described in terms of physical space and time; examples of informational entities are words and ideas, and that of an energetic entity is radiation. Boundaries of such discrete entities can thus be of a compound nature, consisting of all the elements of mei, viz. matter, energy and information. Thus, a spoken word can be considered as a distinct entity, even though it is mainly informational in nature.


A process (p [member of] P) can be seen as a simple flow or sequence of atomic activities called actions ([[pi].sub.i]):

(4) p :=< [[pi].sub.1], [[pi].sub.2], ..., [[pi].sub.m] >

where actions represent the basic units of changes in mei.

Temporal Structures

Processes can also be regarded as thread-like entities. It is also possible to build up systems from processes if agreement can be found to demarcate them. In this case one could introduce the definition of a different kind of system, or temporal structure ([S.sub.t]):

(5) [S.sub.t] := (P [union] [P.sup.e], [R.sup.e.sub.p])

where P denotes the processes and [P.sup.e] the emergent processes in the system, while [R.sub.p] denotes the relationships between these processes.

It is thus conceivable that a system can be composed of processes, which in turn are composed of discrete entities. This is one of the major distinctions of the teleonic approach from the classical systems approach.


Governance ([GAMMA] [subset] P) is a special subset of processes. It has the role to keep systems on track. There may be one or usually more parallel governing feedback loops (gl) in a system, each characterized by its time-constant ([t.sup.c]) and its gain ([omega]).

(6) Loop := {[gl.sub.1], [gl.sub.2], [gl.sub.3], ..., [gl.sub.n]}

(7) [GAMMA] := (Loop, [R.sub.loop])

where [R.sub.loop] stands for the relations between the feedback and feedforward loops in the teleon.

Usually there are several parallel feedback loops in a system. The time constants of the governance loops play an important role in the functioning of the system. The loop with the smallest time constant dominates the governance of processes. However, saturation or failure of the fastest loop may allow the next fastest loops to assume governance.

If the effects of an event on a system are shorter than the smallest time constant of the system, then the latter is unprotected against these effects. Whether a feedback loop is successful in controlling the output of a system depends on the available resources and the structural as well as temporal integrity of the feedback loop.

Levels of the Universe

It has been proposed that the substance of the universe is arranged in concentric layers (Koestler, 1978; Laszlo, 1972; Miller, 1978). Assume that there are N = n + m + 1 levels ([[LAMBDA].sub.i]) in the examined (part of) universe; the following relationship can be established between them:


For example:


If one chooses a point of reference within this arrangement at level A0, the levels outside (above) this level are referred to as the external environment ([Z.sub.ext] := [U.sup.n.sub.i=1] [[LAMBDA].sub.i]), while levels inside (below) this level form the internal environment ([] := [U.sup.-1.sub.i=-m] [[LAMBDA].sub.i]).


In the following paragraphs we introduce the four basic building blocks of the teleonic universe (biomatrix): ethos, teleos, teleon and doublet.


The ethos (E) of a system is a collection of all the laws, rules, regulations, constraints and values according to which a system is conceived. For example, in natural systems the laws of nature constitute the ethos, while in a human or social system the ethos represents the written and unwritten laws of a community, people or system. Depending on the system under consideration, there are different ethoses on the different holarchic levels of the universe.


Teleos ([tau]) denotes the end state towards which a process or system of processes is progressing. This creates the impression that the teleos is attracting the process or is attempting to maintain it dynamically. One can thus consider teleos as being the driving force behind living processes. Concepts such as aim, goal, purpose, meaning, end-point, set-point, function and attractor could all be included in the notion of teleos. In a system, the set of teleoses can be denoted by T. Teleos can be regarded as a generalization (de-anthropolorization) or metalevel interpretation of the idea of everyday 'goal' in human processes (Francois, 1997). The following are examples of teleos:

* preferred end-point, set-point of a process that has come into being during evolution;

* goal, in a general sense usually expressed as the preferred mei state in time and space; and

* meaning or purpose as interpreted or expressed by an observer.


Teleon (x) is a self-governed, goal-related process system, in which the informationally determined boundaries of the teleon are set according to the teleos. Teleons' goal-relatedness means that process systems have definite preferred endpoints where they are heading. Self-governance is achieved through internal morphogenetic and morphostatic feedback/feedforward. If the teleos of the teleon is to maintain a preferred state of mei then morphostatic feedback loops are employed, while for a controlled change of the mei state morphogenetic and morpholytic feedbacks are used.

X denotes the set of teleons in a system, while the set of teleons on the ith level in the biomatrix is represented by [X.sub.i].


A doublet (D) at the referential level, denoted by [d.sub.i] [member of] D or [d.sub.i]([alpha]) [member of] D, is an autopoietic system composed of teleons of that level that are organized around the [alpha] [member of] E attractor:

(9) [d.sub.i] ([alpha]) := ([alpha], [X.sub.[alpha]], [R.sub.x]

Grouping of Teleons

Originally, doublets represented a bipolar field of teleons. One of the two poles has its goals in the inner environment, while the other is 'aimed at' the external environment.

Endopole ([X.sub.endo](d)) contains all the endoteleons that constitute the doublet:


where [upsilon] denotes endoteleons around [alpha]. An endoteleon is a special teleon whose teleos is situated in the internal environment of the double.

Exopole ([X.sub.exo](d)) contains all the exoteleons that constitute the d doublet:


where [xi] denotes the exoteleons around [alpha]. An exoteleon is a special teleon whose teleos is situated in the external environment.

Further studies discovered other dualities and a third field, the centro-field, was added to the model. The centro-field consists of centroteleons ([sigma]) around [alpha], which originate from the same referential level of the doublet and whose teleos is also situated on this same level:


The other significant dual fled is the pair of the exo-tapping-teleons and endo-tapping-teleons. The teleoses of these kinds of teleons support the 'Link_up' or 'Tapping of' type of goals, but they reach out towards other levels during their functioning:




where [alpha] is the attractor of doublet d.

(16) [X.sub.tapping](d) := [X.sup.t.sub.endo](d) [union] [X.sup.t.sub.exo](d) [union] [X.sup.t.sub.centro](d)

We denote the set of teleons belonging to doublet d [member of] D with [X.sub.d]. Thus the teleons of the doublet are defined as follows:

(17) [X.sub.d] := [X.sub.exo] [union] [X.sub.endo] [union] [X.sub.centro] [union] [X.sub.tapping]

Relations of Doublets

Doublets can contain ([subset]) other doublets from different levels, arranged like Russian dolls:


In many doublets, we might find a whole series of concentric doublets, around [[LAMBDA].sub.0] as the reference level.

Holarchy of Doublets in the Biomatrix

The term of holarchy was suggested by Koestler (1978) for hierarchies that are made up of systems that consist of subsystems and participate in supersystems. This has partially inspired

the application of this term in teleonics, since it also follows this type of 'symmetrical' paradigm. The aforementioned poles of the doublets on any level of abstraction can potentially reach out to every corner of the world under observation. This also emphasizes the holographic nature of the net of doublets and teleons; namely, a small segment of the whole net has similar structural features to the whole web.


Doublets (D) and teleons (X) form a multidimensional web, in which knots represent doublets, and teleons are the threads:

(19) BMX := (D,X)


The Creation and Maintenance of Teleons

We suggest that the planning (including understanding) and executing of teleons proceed on two distinctive planes, viz. the specification plane and the manifestation plane These planes are not to be confused with the holarchic levels described previously. The specification plane (SP) is a conceptual plane, where the analysis of natural teleons and the design of man-made teleons are perceived and conceptualized. The manifestation plane (MP) is the plane where the things actually 'happen', where plans are executed and the work is done. Ideally, the two planes should simply mirror one another, viz. real world would be conceptualized correctly, and things would be happening exactly as we imagine them to happen. In the real world, such an ideal is very seldom achieved, leaving discrepancies between the desired and actual state of affairs. However, it is these discrepancies that help us to make corrective actions that would take us closer to the desired outcomes. There are special relationships between the two planes, which should be kept in mind at all times. In the following discussion we shall be looking at the two planes separately first, before putting them together and having a glance at the interactions between them.

The Specification Plane (SP)

The specification plane (see Figure 1) can be thought of as having four quadrants, viz. ethos (E), teleos (T), processes (P) and entities (O), as discussed in previous sections.


Specification of Teleons

The teleon (x [member of] X) or, using a longer term, the specification of the teleon on the SP is therefore given by the following quadruple:

(20) x := (E, [tau], [S.sub.t], [S.sub.s])

This means that in the general 'atmosphere' of the ethos E, the active entities (agents) and passive entities (resources) of the [S.sub.s] play their role in the core of the specification, namely in the goal-oriented process systems ([tau], [S.sub.t]).

Connection between the Quadrants of the Plane. The communication between the quadrants of the SP is generally clockwise, starting from E, through T and [S.sub.t] to [S.sub.s]. It is often described by the three relations Influence, Attract and Employ.

(21) Influence (E) [subset or equal to] T [union] [S.sub.t] [union] [S.sub.s]

where T is the set of teleoses that are to be reached by the system, from which the teleos ([tau]) of the particular teleon is chosen. The ethos (E) also influences the execution of actions (P) as well as the relations of agents and resources (O):

(22) Attract([tau]) [subset or equal to] [2.sup.P]

where P is the set of processes in the definition of the temporal structure ([S.sub.t]) of the teleon, and [tau] denotes the teleos of the teleon:

(23) Employ(P) [subset equal to] [2.sup.O]

where O is the set of discrete entities used in the definition of the physical structure of the teleon ([S.sub.s]).

Different sets of agents from O execute the processes of Attract ([tau]) with the use of diverse resources:

(24) [for all]p [member of] Attract([tau]) : Employ(p) [subset or equal to] [2.sup.O]

There are agents (o [member of] O) that are responsible for the teleons to reach their teleos:

(25) InCharge(o) [subset or equal to] [2.sup.X]

Manifestation Plane (MP)

The concrete rules and policies (R) in the MP (see Figure 2) are identified and selected from the ethos of the SP. [O.sub.m] denotes the set of available resources and agents. The processes ([P.sub.m]) in the MP are the results of the interaction of the available agents and resources. Goals (G) are the time- and space-conscious projections of teleoses on the MP. The result of this projection can be one well-defined or poorly defined goal, or even several goals. It very much depends on the actual problem.


Relationship between Quadrants. The communication between the quadrants on the MP is anticlockwise, through the following relations (see Figure 2):

(26) Participate (o) [subset] [2.sup.Pm]

(27) Execute (a) [subset] [2.sup.Pm]

(28) Contribute (p) [subset] G

(29) Determine (r) [subset] G [union] [O.sub.m] [union] [P.sub.m]

Rules determine goals, participants and applied processes in the teleons of our model. Resources participate in processes. An agent executes his/her own processes. Objects usually participate in more than one process, and agents are often forced--e.g. in case of emergency--to execute more than one process at the same time.

The participants of the process p contribute with their activities to the attainment of the present goal, which might or might not correspond to the desired teleos specified on SP:

(30) [for all]p [member of] Participate(s) : Contribute(p) [subset or equal to] T

Working Teleons

Thus we define the working teleon (x') on the manifestation plane as a quadruple:

(31) x' := (R, G, [P.sub.m], [O.sub.m])

Here, we also want to accentuate the core of the working teleon: (G, [P.sub.m]).

The Relationship between SP and MP

The actual goals (g [member of] G) of the teleon depend on the teleos ([tau]) and the present ethos (E), plus on the inner ([]) and outer ([Z.sub.ext]) environments. A further set of relationships (X) can be introduced to describe the connection between the SP and MP (see Figure 3), in which Take_aim is the one that denotes the relations describing the teleos of the teleon and its time- and space-dependent goals, viz.


(32) Take_aim([tau], E, [], [Z.sub.ext]) [subset] G

The O and P sets of the relatively stable [S.sub.s] and [S.sub.t] structures are the domains of the necessary roles in a teleon to reach its teleos. Roles include both the application of discrete entities and the selection between alternative processes, if available. For the description of the above role assignments we propose the introduction of the following additional relationships:

(33) Select_process(P, G, E, [], [Z.sub.ext]) [subset] [P.sub.m]

(34) Play_role(O, G, E, [], [Z.sub.ext]) [subset] [O.sub.m]

Emergent entities ([a.sub.i], [o.sub.i] [member of] [O.sub.m] or [p.sub.i] [member of] [P.sub.m]) on the manifestation level are linked to the [S.sub.s] and [S.sub.t] structures of the specification level. New entities can take the roles of other 'older' entities and new processes can be integrated into the [S.sub.t] of the system. Naturally, these processes have their needs for agents and resources, so they do make changes in the [S.sub.s] as well.


Strength of Teleons

The strength of a teleon signifies its ability at a point in its path (in time and space) to persevere towards the reaching of its teleos in spite of internal and external influences:

(35) STG(x): X [right arrow] [0,1]

This ability of a teleon varies according to circumstances and depends, among others, on the following factors:

* internal governance;

* absence of external control interference;

* internal flexibility;

* redundancy;

* maintenance of alternatives;

* motivation;

* ability of participants (if any);

* fitness of participants;

* material resources;

* minimum inertia;

* availability of time;

* authority of the executor or that of the person in charge of the teleon

STG (x) = 1 in the case of the strongest teleon in the system.

Adaptivity of Teleons

The adaptivity of a teleon is also an important issue. It mirrors the teleon's ability to reach its teleos in a changing environment, which can make the teleon mobilize its resources or change its current goal for reaching its final teleos. Redundancy and maintenance of alternative routines are important in this connection.

Performance Function

The normalized performance function of an x teleon describes the difference between the actual mei state in the teleon and the mei state determined by the current goal of the teleon (Klir, 1991):

(36) Perf (x) : X [right arrow] [0,1]


Telentropy is reciprocally proportional to the chances of the teleon reaching its teleos:

(37) TS(x) : X [right arrow] [0,1]

Telentropy equals 0 if the teleon has reached its teleos. It is important to mention that a complication just before the end of the completion of a task (at a high value of the performance function) could boost the telentropy of the teleon.


1. The Perf (x) performance function is not equal to the TS(x) telentropy function of the x teleon.

2. Telentropy originates not only from the unforeseen events of the environments, but can also spring from the incorrectly defined teleos. It can change very drastically during the functioning of teleons.

3. Telentropy can be transferred between teleons ([x.sub.i], [x.sub.j] [member of] X : [x.sub.i] [??] [x.sub.j]) that are closely associated with each other. The telentropy of a teleon and that of the whole system characterize very well the current state of the system, or in other words how healthy the system is.

4. Organizational teleons ([x.sub.o] [member of] [X.sub.o]), which manage subteleons and their procedures with special teleons called safety teleons ([x.sub.s] [member of] [X.sub.s]), have an essential role in the control of telentropy.


Ordering of Teleoses

1. According to the ethos, one can differentiate between agonist (positive), neutral and antagonist (negative) teleoses, according to whether they contribute, are indifferent to or oppose the preferable values of the ethos, respectively.

2. An importance ordering can be defined on the teleoses according to the universal ethos of the system ([??]). The importance ordering is a partial ordering, so the set of teleoses is partially ordered with it (T, [??]).

3. A temporal ordering ([??]) can be also defined on the teleoses according to the time of their accomplishment.

Both ways of ordering are essential in the analysis and design of systems. The temporal ordering determines the order of process execution, while the importance ordering differentiates between the parallel and alternative processes. These kinds of ordering will be applied to further concepts related to the teleos. Ethos has a key role when the functioning of systems is examined. It is especially important when questions such as 'Is this system healthy?' are being answered.

4. A particular teleos can be evaluated in several different ethoses.

Ordering of Teleons

Importance Ordering

Teleons should be ranked according to the importance of their teleoses within the prevailing ethos of the system that can be observed or constructed by the observer. Naturally, different observers evaluate the ethos in question differently, which means that the importance of a mutually accepted ethos for the functioning of the overall systems is always highly welcomed.

If within the framework of a prevalent ethos teleos [[tau].sub.1] is considered to be more important than teleon [[tau].sub.2], we write:


Thus the set of teleons (X, [??]) is also partially ordered with the [??] relation.

Strength Ordering

Teleons can be ordered according their strength as well:


The set of teleons (X, [??]) is also partially ordered with this relation.

Other orderings are also possible, based on any measurable attribute of the teleon (number of teleons, number of feedback loops, telentropy, performance, etc.).

This is part of our ethos that in a 'healthy' system the ordering according to strength (X, [??]) should preferably match the importance ordering of the teleons according to their importance (X, [??]).

Connection between the Different Kinds of Ordering

Examining a given system at the moment ([t.sub.0]) and later ([t.sub.1]), one will first elaborate a certain mental picture about the presumed 'normal', 'healthy' functioning of the system's teleons if otherwise not stated. Later the observer will compare the current state ([t.sub.1]) to the standard level examined before, which is mirrored in the mental picture of the observer. The four basic steps of this kind of comparison are shown in Figure 4:


* Step 1. On the basis of observations at a given [t.sub.o] point in time, the observer ranks the strength of the teleons ([??]).

* Step 2. He has an idea about the importance of the teleoses ([??]), based on the prevalent ethos ([epsilon]).

* Step 3. In an assumably pathological situation ([t.sub.1]) the ranking of the strengths ([??]) of teleons will change, which might or might not be observable from the general behaviour of the system.

* Step 4. The observer or therapist tries to re-establish the original ([??]) ranking according to the importance ([??]) of the teleos.

Teleonic Operations

Intersection of Teleons, Independent Teleons

Assume that there are two teleons: [x.sub.1] and [x.sub.2] Their intersection is defined by the intersection of the sets of their components. Two teleons are independent from one another ([x.sub.1] [intersection] [x.sub.2] = [empty set]) if none of their components have common parts.

Relationship of Ethoses. Let us call the common ethos of two teleons

(40) [E.sub.1,2] := [E.sub.1] [intersection] [E.sub.2]

If the ethoses have common parts their intersection is not empty, [E.sub.1,2] [not equal to] 0. The more the two ethoses suit each other the easier it is to determine the importance ordering of the teleoses in the system. It also lowers the risk of value conflicts in the model. The ideal situation would be if [E.sub.1,2] = [E.sub.1] = [E.sub.2] could be managed at a reasonable cost.

Relationship of Teleoses. Teleoses can also have common parts, if

(41) [[tau].sub.1,2] := [[tau].sub.1] [intersection] [[tau].sub.2] [not equal to] [empty set]

If [[tau].sub.1] [subset] [[tau].sub.2] then [x.sub.1] is the subteleon of teleon [x.sub.2].

If [[tau].sub.1] = [[tau].sub.2], i.e. the two teleoses are the same, the corresponding teleons are alternative subteleons of a common suprateleon. They have a telentropy-lowering effect.

Shared Processes. The shared processes of teleon [x.sub.1] and [x.sub.2] are defined as the intersection of the set of their processes:

(42) [P.sub.1,2] := [P.sub.1] [intersection] [P.sub.2]

[P.sub.1,2] [not equal to] [empty set] indicates that two teleons do have shared processes, common activities.

In these cases, the chance of telentropy transfer between the two teleons increases proportionately with the number of shared processes. Even if [P.sub.1,2] [not equal to] [empty set] and at the same time [[epsilon].sub.1] [intersection] [[epsilon].sub.2] [not equal to] [empty set] and [[tau].sub.1] [intersection] [[tau].sub.2] [not equal to] [empty set] are true as well, there is no change in the telentropy transfer situation. For example: eating as a very common activity can be part of at least four different teleons: feeding the cells of the human body, enjoying a good meal, joining a party, socializing.

Shared Entities. The shared entities of teleons [x.sub.1] and [x.sub.2] are defined as the intersection of the set of their entities:

(43) [S.sub.1,2] := [S.sub.1] [intersection] [S.sub.2]

[S.sub.1,2] [not equal to] [empty set] indicates that two teleons share some common entities, agents or resources. For how shared entities influence the telentropy transfer see the section on 'Shared processes'.

In self-regulating systems, if two or more teleons assert the same resources at the same time, there is a conflict situation, where the 'winner' is the 'stronger' teleon and not necessarily the more important one.

For example: several different teleons in the human body share the same physical structure: the 'blood-vascular system'. This shared medium enables its users to easily transfer telentropy among each other with rather unexpected effects.

Difference of Teleons

Assume that there are two teleons: [x.sub.1], [x.sub.2] [member of] X. The difference ([x.sub.1/2] := [x.sub.1] / [x.sub.2]) of [x.sub.1] and [x.sub.2] is defined by the difference of the sets of their components. The [x.sub.1/2] teleon works only in accordance with the ethos of [x.sub.1]; its teleos is the same as the teleos of [x.sub.1]; and for reaching its goal it only has the entities and processes of teleon [x.sub.1], which do not belong to [x.sub.2].

Teleon [x.sub.1] is independent from [x.sub.2] if [x.sub.1]/[x.sub.2] = [x.sub.1].

Analysis of Complex, Interrelated Teleons

The dynamic analysis of a complex model is conducted in three major steps:

* Assume that we analyse the jth teleon [x.sub.j].

* Examine the parts of the teleon that are not related to any other teleon: [x.sub.j]\([union][x.sub.i]) and i [not equal to] j.

* If [x.sub.j] [intersection] ([union][x.sub.i]) [not equal to] 0, i.e. the jth teleon has shared attributes with other teleons, one should analyse them (processes, agents or resources) with special care.

* And finally merge the results of the previous two studies.

Operations on Teleons

Sequential Connection

Assume that [x.sub.1], [x.sub.2] [member of] X are two teleons; denote their sequential connection with [x.sub.1] [direct sum] [x.sub.2], i.e. [x.sub.2] starts to work after [x.sub.1] no longer functions. Its properties are:
Non-commutative:   [x.sub.1] [direct sum] [x.sub.2] [not equal to]
                   [x.sub.2] [direct sum] [x.sub.1]

Associative:       [x.sub.1] [direct sum] ([x.sub.2] [direct sum]
                   [x.sub.3]) = ([x.sub.1] [direct sum] [x.sub.2])
                   [direct sum] [x.sub.3]

Transitive:        [x.sub.1] [direct sum] [x.sub.2] [conjunction]
                   [x.sub.2] [direct sum] [x.sub.3] [??] [x.sub.1]
                   [direct sum] [x.sub.3]

Consequential Connection

Assume that [x.sub.1], [x.sub.2] [member of] X; we speak about the consequential connection of the two teleons if [x.sub.1] triggers the other teleon [x.sub.2]. Let us denote this type of connection by [x.sub.1] [??] [x.sub.2]. Its properties are:

Transitive: [x.sub.1] [??] [x.sub.2] [conjunction] [x.sub.2] [??] [x.sub.3] [??] [x.sub.1] [??] [x.sub.3]

Parallel Teleons

Assume that [x.sub.1], [x.sub.2] [member of] X; denote the two teleons that operate parallel with [x.sub.1] [parallel] [x.sub.2].

The operational precedence of parallel teleons is greater than that of the sequential teleons. Parallel teleons have the following properties:
Commutative:   [x.sub.1] [parallel] [x.sub.2] = [x.sub.2] [parallel]

Associative:   [x.sub.1] [parallel] ([x.sub.2] [parallel] [x.sub.3])
               = ([x.sub.1] [parallel] [x.sub.3]) [parallel]

Transitive:    [x.sub.1] [parallel] [x.sub.2] [conjunction] [x.sub.2]
               [parallel] [x.sub.3] [??] [x.sub.1] [parallel]

Alternative Teleons

Assume that [x.sub.1], [x.sub.2] [member of] X are teleons and denote by [[x.sub.1], [x.sub.2]] the fact that one can choose either [x.sub.1] or [x.sub.2] for accomplishing his goal.

The support and maintenance of alternative teleons have an accentuated importance both within a teleon and in any teleonic model. Generating alternatives play an important role at design time, but if they are neglected after the initiation of the system they will weaken and disappear as available alternatives.

Map of Teleons

Assume that x [member of] [X.sub.k] and [x.sub.i] [member of] [X.sub.j] ([for all]i [member of] [1,12] and j < k and i, j, k [member of] N) are the subteleons of x:


A general map of teleon x is shown on Figure 5. Parallel teleons are accentuated with loops around them.



In this paper we have introduced a mathematico-symbolic notation for some of the principles of teleonics, which are summarized in Table 1. We believe that their application to real-life problems will be simpler and more readily definable for those who prefer the short-hand notation of mathematics. The danger is that this might cause problems for those who do not belong to the privileged group. However, for these teleonics has been repeatedly presented in a descriptive manner in several papers listed in the References.

Present research also focuses on the quantitative and qualitative examination of telentropy. Further studies will target the more detailed modelling of telentropy, its transfer through out the biomatrix, and the relationship of telentropy and teleonic strength. The understanding and modelling of the multilevel teleon-doublet duality also needs further detailed study.

There is also an ongoing effort to utilize the pervading modelling language UML (Unified Modeling Language) for a standardized graphical description of teleonic models. UML is an extensively used tool in the contemporary IT community, and has also been adapted to business process modelling. It works with diverse views and diagrams, which makes it very useful for teleonics. Besides, UML has built-in extension mechanisms, which enable us to enrich the present set of UML diagrams with some of the important and sophisticated concepts of teleonics.
Table 1. Summary of notations

                   Basic 'systemic'                    Teleonic
                   components                          elements

                   Resources                           Teleons: X
                     entities): Res
Elements of        Actors (physical                    Doublets: D
the universe         entities): A
                   Entities O =                        Biomatrix: BMX
                   Processes: P
                   Ethos elements:
                   Teleoses: T

                   Operations                          Attributes

Basic teleonic     Sequential:                         Strength (x)
operations           [x.sub.1]                           [subset or
                     [direct sum]                        equal to]
                     [x.sub.2]                           [0, 1]
                   Parallel:                           Performance (x)
                     [x.sub.1]                           [subset or
                     [parallel]                          equal to]
                     [x.sub.2]                           [0, 1]
                   Alternatives:                       Telentropy (x)
                     [[x.sub.1.],                        [subset or
                     [x.sub.2]]                          equal to]
                                                         [0, 1]

                   Intra-teleon                        Mixed relations

Specification      Manifestation                       Executor (x, a)
  plane              plane
(Figure 1)         (Figure 2)                          InCharge (x, a)
Influence (e,      Determine (r, g)                    Owner (x, a)
Influence (e, o)   Determine (r,
Influence (e, p)   Determine (r,                       Invite
                     [o.sub.m])                          ([alpha], x)
Attract ([tau],    Participate (r,
  {[p.sub.i]})       {[p.sub.m]})
Employ (p,         Contribute (p,
  {[o.sub.i]})       {[g.sub.i]})

Orderings          Execute
                     (a, [p.sub.m])

[MATHEMATICAL      Interplane
  EXPRESSION NOT   relations
  REPRODUCIBLE     (Figure 4)
  IN ASCII.]       Play_role
                     g, o)
                     ([p.sub.m] g,
                   Take_aim ([tau],
                     Z, {r}, g)

Inter-teleon                          Others
relations                             teleonic

Consequential:                        Works_at (x,
  [x.sub.1] [??]                        [[LAMBDA]
  [x.sub.2]                             .sub.i])
                                      Centered (d,
Subteleon                               [[LAMBDA]
  ([x.sub.1],                           .sub.i])
Superteleon                           Subdoublet
  ([x.sub.1],                           ([d.sub.1],
  [x.sub.2])                            [d.sub.2])
Independent                           Superdoublet
  ([x.sub.1],                           ([d.sub.1],
  [x.sub.2])                            [d.sub.2])
                                      Link_up (d, Z,



a [member of] A, r [member of] R, o [member of] O, p [member of] P,
e [member of] E, [tau] [member of] [TAU], g [member of] G,
r [member of] R, x [member of] X, d [member of] D, and
[[LAMBDA].sub.i], ith level in the biomatrix, Z stands for either
internal or external environment, {[p.sub.i]} [subset or equal to] P
is a subset of processes, {[o.sub.i]}, [subset or equal to] O is a
subset of entities, {[g.sub.i]} [subset or equal to] G is subset of

Figure 4. Ordering of teleons

[t.sub.0]                                    [t.sub.1]

1. [(X, [??]).sub.0]   [not equal to]   [(X, [??]).sub.1]   3.


2. [(X, [??]).sub.0]         =          [(X, [??]).sub.1]   4.


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Received 3 October 2000

Accepted 28 November 2003

Gabor Horvath (1) * and Gyorgy Jaros (2)

(1) 1117 Budapest, Pazmany Peter Setany 1/C, Hungary

(2) Central Clinical School, University of Sydney, Australia

* Correspondence to: Gabor Horvath, 1117 Budapest, Pazmany Peter Setany 1/C, Hungary. E-mail:
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Title Annotation:Research Paper
Author:Horvath, Gabor; Jaros, Gyorgy
Publication:Systems Research and Behavioral Science
Date:Jan 1, 2005
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