Patterns of Relational Paradigm

Twofold Operators-Hierarchy/Network











  • The network was formally excluded from Western logic by Bertrand Russell's The Theory of Logical Types.

  • The theory states; "No class can be a member of itself."

  • It excludes self-reflective statements like; "This statement is false." "He who knows, keeps quiet. he who doesn't know, speaks."

  • Self-reflexive statements or networks result in a paradoxical or oscillating condition.



  • The theory was a basic concept in Russell and Alfred Whitehead's Principia Mathematica.

  • Kurt Gödel used these concepts to prove Göbel's A, which stated that any system that is consistent cannot be complete.

  • But consistency was based on exclusion of the network.

  • Allow The Network and Göbels's proof no longer applies.




Patterns of Relational Paradigm


Twofold Operators







Summary of Emergent Axioms

Threefold Operators

Fourfold Operators

Fifth Business

Relational Organization

Relational Structure & Root Language

          The hierarchy and the network are very simple concepts that have caused great problems in existing world views. The hierarchy is like a tree; it begins from one central trunk and branches into many limbs, none of which connect in a circular manner.

         The network in some fashion always forms a circular connection between the beginning and the end; it is self-referencing in that it refers back to itself. Self-reflexive, recursive, circular, feedback, uroboris are some of the terms which imply a network rather than a hierarchy.

          The network or self-referencing concept was formally excluded from Western logic by Bertrand Russell’s Theory of Logical Types. This theory states that no class can be a member of itself—a basic requirement of the network.

          Russell’s theory forbids self-reflexive statements like, “This statement is false.” If the statement is true, then it’s false, and if it’s false, then it’s true. We get a logical infinite regression.

          We call it paradoxical. If we jump outside the system, we can understand what is happening. It is an oscillating condition which looks like this:

          The Theory of Logical Types was a basic concept in Bertrand Russell and Alfred North Whitehead’s Principia Mathematica, which was an attempt to codify Western scientific thought. When this book was finished the challenge went out to the world. Can we prove anything with these postulates which we know to be false, or conversely, is there anything which we know to be true that we can’t prove with these postulates?

          The mathematician Kurt Gödel used the axioms as stated in Principia Mathematica to prove that any system of logic which is consistent cannot be complete. The proof, called “Gödel’s A” or “Gödel’s Incompleteness Theorem,” was first published in 1931. Simply stated, Gödel proved that Western scientific thought could not provide a complete understanding of the world. Western science has yet to recover from this blow.

          Consistency in the scientific world view was based on the Theory of Logical Types which excluded the network or self-reflexive concept. Essentially, Gödel proved that a hierarchical system cannot be complete. But if we allow the network as well as the hierarchy, Gödel’s proof no longer applies.

          An example Russell used to illustrate the problem which led to the formulation of the Theory of Logical Types was the paradox of the librarian. Each librarian was to prepare a catalog containing a list of every book in the library. The catalog, once prepared, was itself a book, and the librarians were uncertain as to whether they should list the catalog itself in the catalog—a class as a member of itself. Some chose to list it, others chose not to. The master librarian was to prepare a list of all catalogs which did not list themselves—catalogs in which the class was not a member of itself. The master librarian could not decide whether to list the catalog itself in this catalog. If he did not list the catalog, then this master catalog did not contain itself which was the criterion for listing and hence the catalog was incomplete. If he did list the catalog, then the master catalog did contain itself and therefore did not meet the criterion for listing. A paradox.

          To resolve this problem, Russell said that no class can be a member of itself; none of the catalogs should be listed in themselves. The paradox didn’t go away, but the librarians at least had a consistent way to behave and could get on with their work.

          The introduction of the network implies an alternative solution. Every class is a member of itself; all of the catalogs should be listed in themselves. Once again the paradox doesn’t go away, but we have an alternative consistent way to deal with the world.

          “No class is a member of itself” is a hierarchy. “Every class is a member of itself” is a network. Both views are valid. By having both views as consistent behavior—the librarian can either list the catalog in itself or not list it—our world view is both consistent and complete.


  • Revise Cartesian world view of René Descrates and Rationalistic seperation of mind and matter.

  • The hierarchy is objective...the scientist is an observer.

  • The network is subjective...the scientist is a participant.

  • Both conditions are valid perceptions and must be allowed in the new.

  • Subjective and objective evaluations needed for both form and process.


Objective - Form

What are the results?

Subjective - Form


Objective - Process

Does it work?
Does it last?

Subjective - Process

How does it feel?



        The introduction of the network into the Western scientific paradigm requires the revision of a number of basic philosophical concepts.

          It seems to me that Russell’s Theory of Logical Types has its basis in the Cartesian world view of Rene Descartes and the rationalistic separation of mind and matter. This separation is hierarchical in nature and provides a philosophical basis for objective science.

          The network is subjective in nature—a self-referencing or inward perspective. The hierarchy is objective—an outward directed perspective. The introduction of the network implies that we must develop a subjective as well as objective view of science: the scientist as a participant, as well as the scientist as an observer.

          Eastern science in contrast to Western science is subjective in nature. The Eastern scientists use themselves as a basic measuring instrument. Western scientists always use some type of machine or device which is separate from themselves. As a result of these basic differences, the East and the West have had great difficulty communicating.

          A good example of this difference is Eastern and Western medical science. In the East, doctors train themselves to measure kundalini or ch’i. This phenomenon is one of the basic concepts of Eastern medicine and has been effectively used for thousands of years. We in the West have yet to find a way to objectively measure this phenomenon, so we say that it does not exist. The words are not even included in our dictionaries.

          The inability of objective science to solve social problems is another result of the exclusion of the network from Western science. Social issues by definition include the scientist; there is simply no way to deal with them from a totally objective point of view. “If we can go to the moon, why can’t we .........?” is a reflection of this problem.

          In general, Eastern and Western religion have the same division. Eastern religions are inward directed or self-reflective, whereas Western religions are more outward directed or messianic.

          The inclusion of the network and hierarchy in the relational paradigm implies that both conditions are valid and must be allowed in the new. A world view cannot be complete without both perceptions. It does not discard either the East or the West but integrates them into a greater whole.

          In a relational paradigm we must introduce subjective evaluations to understand and control the complex living systems of which we are a part. For a complete understanding of our reality, we need objective and subjective evaluations of both process and form. The basic questions that must be answered are:

What are the results? Objective evaluation of form.
Does it work? Subjective evaluation of form.
Does is last? Objective evaluation of process.
How does it feel? Subjective evaluation of process.
        The only evaluation which is regularly used in the Western economic paradigm is the objective evaluation of form. Even the objective evaluation of process is not commonly used.


Aristotle'sThree Laws of Logic

  • Law of Identity - A thing is what it is. A is A

  • Law of the Excluded Middle - Anything is either A or Non-A.

  • Law of Non-Contradiction - Something cannot be both A and Non-A.


An emergent 4th law

  • Law of Relativity - Whether a thing is A or Non-A is relative to the perception of the observer.



          Russell’s theory and the Cartesian world view are based upon the Greeks, specifically upon Aristotle’s Three Laws of Logic. It seems to me that all Western thought is ultimately based upon these three simple axioms.

          Briefly, the Three Laws of Logic are as follows:

- Law of identity. A thing is what it is. A is A.
- Law of excluded middle. Anything is either A or non-A.
- Law of non-contradiction. Something cannot be both A and non-A..

          If we add the network or self-referencing concept to the existing Western world view, then we must revise Aristotle’s three laws. The old laws are not wrong, they are just incomplete. We need a fourth law.

          My proposal for an emergent fourth law is:

- Law of relativity. Whether a thing is A or non-A is relative to the perception of the observer.

          The Law of Relativity provides a logical basis by which the paradoxical nature of our reality can be resolved. It does not throw out the paradox but rather gives us a way to deal with and integrate the paradox as a special case of the new.

          From one perspective it is hierarchical, linear and objective; no class is a member of itself. From another perspective it is a network, circular and subjective; every class is a member of itself. From a third perspective, it is both. All are valid views. None is more correct than the others; they are simply relative to the perception of the observer.

         The scientist must perceive himself for his world view to be complete.


  • Einstein's Theory of Relativity provides foundation for the 4th law. The speed of light is constant, relative to the position of the observer.

  • Norbert Wiener's cybernetics is mathematical study of the network (control).

  • G. Spenser-Brown's Laws of Form provides a base for handling the logic.

  • Whether a thing is A or Non-A is relative to the perception of the observer.

       I believe that the foundation for the introduction of the network into Western science has already been laid. The East has been working with the concept for centuries. Eastern science or Eastern mysticism (Western scientists usually refer to any other concept of science as mysticism) is well developed and many of the connections between the two world views have been documented. See specifically The Tao of Physics by Fritjof Capra and The Dancing Wu Li Masters by Gary Zukav.

          In Western science the network or self-referencing concept has already been introduced in many disciplines. Einstein’s Theory of Relativity provides a foundation for the fourth law. If one asks experts what Einstein’s theory is about, the usual response is that the speed of light is constant. They leave out the last half of the sentence, “...relative to the position of the observer.” It is clear which concept Einstein thought most important: he called it the theory of relativity, not the theory of constancy. The speed of light is relative. Even with the use of instruments, there is no objective measurement of the speed of light. All measurements of the speed of light are subjective.

          The self-referencing concept called "feedback" was the emergent factor in Norbert Wiener's mathematical work in cybernetics. Control is based on the concept of feedback. Control is a network, planning is a hierarchy.

          G. Spencer-Brown in Laws of Form provides a conceptual basis for the logic necessary to handle the relational paradigm. In addition to the existing logical states of true and false, the self-referencing logic results in an additional state which Spencer-Brown calls imaginary.

          Pictures illustrating the fourth law are shown in Exhibit 2.2.2 (4). The necker cube can switch positions relative to the perception of the observer. Faces/vase can be seen as a vase or as different faces. This picture is composed of exponentially decreasing and increasing sine waves, and contains eight separate faces.

          Another illustration of the fourth law is from language. In grammar, the subjective case is the subject of a sentence, as in, “I did it.” “I” is the subjective case. The objective case is the object, as in, “It happened to me.” “Me” is objective case. The objective case has the pattern of a network; it circles around and points back to me. It is subjective. The subjective case has a hierarchical pattern; it is the objective observer. Let’s see now, the subjective case is objective and the objective case is subjective. Like I said, it’s relative to the perception of the observer. No wonder we’ve been confused.

Form/Process • Hierarchy/Network • Sequence/JumpMeasure/CountKnown/RandomBalance/RelationshipSummary of Emergent Axioms

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