Double Negation • 1

The article and section linked below introduce the Double Negation Theorem (DNT) in the manner described as Consequence 1 (C_1) or Reflection by Spencer Brown.

Converting the planar figures used by Peirce and Spencer Brown to the graph-theoretic structures commonly used in mathematics and computer science, the double negation theorem takes the following form.

Double Negation Theorem

The formal system of logical graphs is defined by a foursome of formal equations, called initials when regarded purely formally, in abstraction from potential interpretations, and called axioms when interpreted as logical equivalences.  There are two arithmetic initials and two algebraic initials, as shown below.

Arithmetic Initials



Algebraic Initials



Using these equations as transformation rules permits the development of formal consequences which may be interpreted as logical theorems.  The double negation theorem is one such consequence.  The proof of the double negation theorem I give next time is adapted from the one Spencer Brown presents in his Laws of Form and credits to two of his students, John Dawes and D.A. Utting.


cc: CyberneticsLaws of FormFB | Logical GraphsOntolog ForumPeirce List
Structural ModelingSystems Science

This entry was posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Constraint Satisfaction Problems, Deduction, Diagrammatic Reasoning, Duality, Equational Inference, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Painted Cacti, Peirce, Proof Theory, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Theorem Proving, Visualization and tagged , , , , , , , , , , , , , , , , , , , , , , , , , . Bookmark the permalink.

1 Response to Double Negation • 1

  1. Pingback: Double Negation • 3 | Inquiry Into Inquiry

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