Animated Logical Graphs • 61

Re: Richard J. LiptonThe Art Of Math
Re: Animated Logical Graphs • (57)(58)(59)(60)

Anything called a duality is naturally associated with a transformation group of order 2, say a group G acting on a set X.  Transformation groupies normally refer to X as a set of “points” even when the elements have additional structure of their own, as they often do.  A group of order two has the form G = \{ 1, t \}, where 1 is the identity element and the remaining element t satisfies the equation t^2 = 1, being on that account self-inverse.

A first look at the dual interpretation of logical graphs from a group-theoretic point of view is provided by the following Table.  The sixteen boolean functions f : \mathbb{B} \times \mathbb{B} \to \mathbb{B} on two variables are listed in Column 1.  Column 2 lists the elements of the set X, specifically, the sixteen logical graphs \gamma giving canonical expression to the boolean functions in Column 1.  Column 2 shows the graphs in existential order but the order is arbitrary since only the transformations of the set X into itself are material in this setting.  Column 3 shows the result 1 \gamma of the group element 1 acting on each graph \gamma in X, which is of course the same graph \gamma back again.  Column 4 shows the result t \gamma of the group element t acting on each graph \gamma in X, which is the entitative graph dual to the existential graph in Column 2.

\text{Peirce Duality as Group Symmetry}

Peirce Duality as Group Symmetry

The last Row of the Table displays a statistic of considerable interest to transformation group theorists.  It is the total incidence of fixed points, in other words, the number of points in X left invariant or unchanged by the respective group actions.  I’ll explain the significance of the fixed point parameter next time.

Resources

cc: Cybernetics (1) (2)Laws of FormFB | Logical Graphs • Ontolog Forum (1) (2)
• Peirce List (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) • Structural Modeling (1) (2)
• Systems Science (1) (2)

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.

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