## Animated Logical Graphs • 49

Re: Richard J. LiptonThe Art Of Math
Re: Animated Logical Graphs • (30) (45) (46) (47) (48)

Dualities are symmetries of order two and symmetries bear on complexity by reducing its measure in proportion to their order.  The inverse relationship between symmetry and all those dissymmetries from dispersion and diversity to entropy and uncertainty is governed in cybernetics by the Law of Requisite Variety, the medium of which exchanges C.S. Peirce invested in his formula, Information = Comprehension × Extension.

The duality between entitative and existential interpretations of logical graphs is one example of a mathematical symmetry but it’s not unusual to find symmetries within symmetries and it’s always rewarding to find them where they exist.  To that end let’s take up our Table of Venn Diagrams and Logical Graphs on Two Variables and sort the rows to bring together diagrams and graphs having similar shapes.  What defines their similarity is the action of a mathematical group whose operations transform the elements of each class among one another but intermingle no dissimilar elements.  In the jargon of transformation groups these classes are called orbits.  We find the sixteen rows partition into seven orbits, as shown below.

Venn Diagrams and Logical Graphs on Two Variables • Orbit Order
$\text{Boolean Function}$ $\text{Entitative Graph}$ $\text{Existential Graph}$

$f_{0}$ $\text{false}$ $\text{false}$

$f_{1}$ $\lnot (x \lor y)$ $\lnot x \land \lnot y$

$f_{2}$ $\lnot x \land y$ $\lnot x \land y$

$f_{4}$ $x \land \lnot y$ $x \land \lnot y$

$f_{8}$ $x \land y$ $x \land y$

$f_{3}$ $\lnot x$ $\lnot x$

$f_{12}$ $x$ $x$

$f_{6}$ $x \ne y$ $x \ne y$

$f_{9}$ $x = y$ $x = y$

$f_{5}$ $\lnot y$ $\lnot y$

$f_{10}$ $y$ $y$

$f_{7}$ $\lnot x \lor \lnot y$ $\lnot (x \land y)$

$f_{11}$ $x \Rightarrow y$ $x \Rightarrow y$

$f_{13}$ $x \Leftarrow y$ $x \Leftarrow y$

$f_{14}$ $x \lor y$ $x \lor y$

$f_{15}$ $\text{true}$ $\text{true}$

### Resources

cc: Cybernetics Communications (1) (2)FB | Logical Graphs • Ontolog Forum (1) (2)
cc: Peirce (1) (2) (3) (4) (5) (6) (7) (8) (9) • Structural Modeling (1) (2) • Systems (1) (2)

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