Animated Logical Graphs • 30

Posted on August 25, 2019 by Jon Awbrey

The duality between Entitative and Existential interpretations of logical graphs is a good example of a mathematical symmetry, in this case a symmetry of order two.  Symmetries of this and higher orders give us conceptual handles on excess complexity in the manifold of sensuous impressions, making it well worth the effort to seek them out and grasp them where we find them.

In that vein, here’s a Rosetta Stone to give us a grounding in the relationship between boolean functions and our two readings of logical graphs.

\text{Boolean Functions on Two Variables}
\text{Boolean Function} \text{Entitative Graph} \text{Existential Graph}
f_{0} Cactus Root Cactus Stem
\text{false} \text{false} \text{false}
f_{1} Cactus (xy) Cactus (x)(y)
\text{neither}~ x ~\text{nor}~ y \lnot (x \lor y) \lnot x \land \lnot y
f_{2} Cactus (x(y)) Cactus (x)y
y ~\text{and not}~ x \lnot x \land y \lnot x \land y
f_{3} Cactus (x) Cactus (x)
\text{not}~ x \lnot x \lnot x
f_{4} Cactus ((x)y) Cactus x(y)
x ~\text{and not}~ y x \land \lnot y x \land \lnot y
f_{5} Cactus (y) Cactus (y)
\text{not}~ y \lnot y \lnot y
f_{6} Cactus ((x,y)) Cactus (x,y)
x ~\text{not equal to}~ y x \ne y x \ne y
f_{7} Cactus (x)(y) Cactus (xy)
\text{not both}~ x ~\text{and}~ y \lnot x \lor \lnot y \lnot (x \land y)
f_{8} Cactus ((x)(y)) Cactus xy
x ~\text{and}~ y x \land y x \land y
f_{9} Cactus (x,y) Cactus ((x,y))
x ~\text{equal to}~ y x = y x = y
f_{10} Cactus y Cactus y
y y y
f_{11} Cactus (x)y Cactus (x(y))
\text{if}~ x ~\text{then}~ y x \Rightarrow y x \Rightarrow y
f_{12} Cactus x Cactus x
x x x
f_{13} Cactus x(y) Cactus ((x)y)
\text{if}~ y ~\text{then}~ x x \Leftarrow y x \Leftarrow y
f_{14} Cactus xy Cactus ((x)(y))
x ~\text{or}~ y x \lor y x \lor y
f_{15} Cactus Stem Cactus Root
\text{true} \text{true} \text{true}

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