Self-Reference in a Three-Valued System

References:

  • Paradox
  • Brown 4 (Arthur Collins, 1 januari 2017)
  • Self-reference calculus (Varela, F.J., 1 januari 1975)
  • Form dynamics (Kauffman, L.H. and Varela, F.J., 1 januari 1980)
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Statement: Embrace the paradox, i.e., a difference in what was previously stated and therefore contradicting what was said before. Differences keep setting things in motion. Without differences we cease to exist. Therefore, change is inevitable, in fact, it is a necessity for living.

Aspect: Reflexive Domain, Principle: We got to move, Principle page: Principles and Ground Rules

Statement pageStatement
Exploring ChangeThe constant factor in life is movement.
Self-Reference in a Three-Valued SystemEmbrace the paradox, i.e., a difference in what was previously stated and therefore contradicting what was said before. Differences keep setting things in motion. Without differences we cease to exist. Therefore, change is inevitable, in fact, it is a necessity for living.
The Autopoietic TurnHumans and social systems operate autonomously and my react when irritated.

Principles, aspects and statements overview

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Perhaps the most important contribution of LoF is when Spencer-Brown discusses in chapter 11 and 12 self-referential form expressions. These are circular expressions that lead to recursion. A famous example for illustrating recursion is the “Droste effect”, named after a Dutch brand of cocoa. It is the effect of a picture recursively appearing within itself, in a place where a similar picture would realistically be expected to appear.

Droste.jpg

A self-referential expression may result in a paradox, as is the case in the well-known liar paradox.

This sentence is false.

The liar paradox, which is also known as “This sentence is a lie” or “I am lying”, switches between true and false continuously. If the sentence is true, then the sentence states that it is false. If false, then the statement “This sentence is false.” makes it true again. And so on, indefinitely. A paradox is defined as a seemingly absurd or contradictory statement or proposition which when investigated may prove to be well founded or true. (Source: https://www.lexico.com/en/definition/paradox) Mathematicians shy away from paradoxes because of their often inherent contradictions. But Spencer-Brown found a way to deal with seemingly contradictory statements. The paradox “This sentence is false.” can be rephrased as a recurrent form expression.

[math]\displaystyle{ \begin{array}{lcl}TSIF & = & \overline{TSIF|}\end{array} }[/math]

Note that the expression TSIF, short for “This sentence is false.”, occurs both on the Left Hand Side (LHS) and the Right Hand Side (RHS) of the equal (=) sign: LHS = RHS. This means that LHS (= RHS) can be substituted in the occurrence of LHS in the RHS leading to a recursion

[math]\displaystyle{ \begin{array}{lcl} TSIF & = & \overline{TSIF|} \\ & = & \overline{\overline{TSIF|}\Big|} \\ & = & \overline{\overline{\overline{TSIF|}\Big|}\bigg|} \\ & = & \overline{\overline{\overline{\cdots|}\Big|}\bigg|} \end{array} }[/math]

which is equivalent to an alternating time sequence

[math]\displaystyle{ TSIF \;\; = \;\; \cdots , \overline{\;\;|} , \overline{\overline{\;\;|}\Big|} , \overline{\;\;|} , \overline{\overline{\;\;|}\Big|} , \overline{\;\;|} , \overline{\overline{\;\;|}\Big|} , \cdots \;\; = \;\; \cdots , \mbox{marked, unmarked, marked, unmarked, marked, unmarked}, \cdots }[/math]

In a sense, the recurrent form can be regarded as a form expression that is re-entered in its own indicational space. And as a result, an oscillation my occur switching between the marked (true) and unmarked (false) state. Spencer-Brown devised a special symbol for re-entrance.

[math]\displaystyle{ f = \overline{f|} \;\;\; \stackrel{def}{=} \;\;\; \overline{\underline{_{\lfloor}\;\;}\bigg|} \;\;\; \mbox{with $f$ denoting an arbitrary expression, e.g., $TSIF$} }[/math]

So, Spencer-Brown discovered that the solution of paradoxical statement is an oscillation. That is to say, besides the markedness and unmarkedness sides of a distinction in space, a new dimension is introduced: time! Paradoxes are de-paradoxified in time. At one moment in time a statement may be true, while at another moment a statement may be false, which makes perfectly sense.

The impact of this discovery cannot be overstated. The idea of a paradox leading to an oscillation in time is key to understanding self-production (autopoiesis) in living organisms to sustain life as shown by Maturana and Varela (cite{?}). Luhmann applied this idea to social systems in his autopoietic turn to show how societies carry on (cite{?}).

Now consider this equation.

[math]\displaystyle{ x^2 = -1 }[/math]

It is well-known that this equation has no real solution because squaring a negative or a positive real number always yields a positive real number, so it can never be -1. The equation can be rewritten as a recursive expression, in which x is occurring in both the LHS and RHS of the equal (=) sign.

[math]\displaystyle{ x = \frac{-1}{x} }[/math]

This equation has the same paradoxical qualities of the liar paradox. Obviously, If a solution would exist, this can only be with x taken on the unity value of +1 or -1. But unfortunately, when +1 or -1 is substituted for x, the result is precisely the opposite, like an oscillation.

[math]\displaystyle{ +1 = \frac{-1}{+1} = -1 }[/math]
[math]\displaystyle{ -1 = \frac{-1}{-1} = +1 }[/math]

Perhaps unknown for readers not well-acquainted with mathematics, the equation

[math]\displaystyle{ x^2 = -1 }[/math]

can be solved by resorting to imaginary numbers. By definition

[math]\displaystyle{ i^2 = -1 }[/math]

with

[math]\displaystyle{ i = \sqrt{-1} }[/math]

How strange it may look, imaginary numbers are as real as real numbers. It took a while to get accustomed with imaginary numbers, but that was once also the case with zero and negative numbers. Nowadays, they are applied routinely in all kind of engineering domains.


Formulas

[math]\displaystyle{ \overline{\underline{_{\lfloor}\;\;}\bigg|} = \overline{\overline{\underline{_{\lfloor}\;\;}\bigg|}\Bigg|} }[/math]
[math]\displaystyle{ system \;\; \stackrel{def}{=} \;\; \overline{system|}environment }[/math]
[math]\displaystyle{ \overline{\underline{\overline{_{\lfloor}system|}environment}\Big|} \;\; = \;\; \overline{\overline{\overline{\overline{ \cdots system|}environment\Big|}system\bigg|}environment\Bigg|} }[/math]
[math]\displaystyle{ \mbox{indication} = \mbox{distinction} \stackrel{implies}{\longrightarrow} \mbox{indication} }[/math]
[math]\displaystyle{ \mbox{distinction} = \mbox{indication} \stackrel{implies}{\longrightarrow} \mbox{distinction} }[/math]























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