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.


Formulas

[math]\displaystyle{ \begin{array}{lcl}TSIF & = & \overline{TSIF|}\end{array} }[/math]
[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]
[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]
[math]\displaystyle{ f = \overline{f|} \;\;\; \stackrel{def}{=} \;\;\; \overline{\underline{_{\lfloor}\;\;}\bigg|} \;\;\; \mbox{with $f$ denoting an arbitrary expression, e.g., $TSIF$} }[/math]
[math]\displaystyle{ x^2 = -1 }[/math]
[math]\displaystyle{ x = \frac{-1}{x} }[/math]
[math]\displaystyle{ \begin{array}{lclcl} +1 & = & \frac{-1}{+1} & = & -1 \\ -1 & = & \frac{-1}{-1} & = & +1 \end{array} }[/math]
[math]\displaystyle{ +1 = \frac{-1}{+1} = -1 }[/math]
[math]\displaystyle{ -1 = \frac{-1}{-1} = +1 }[/math]
[math]\displaystyle{ x^2 = -1 }[/math]
[math]\displaystyle{ i^2 = -1 }[/math]
[math]\displaystyle{ i = \sqrt{-1} }[/math]
[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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