Animated Logical Graphs • 48

Posted on November 30, 2020 by Jon Awbrey

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

A more graphic picture of Peirce duality is given by the next Table, which shows how logical graphs map to venn diagrams under entitative and existential interpretations.  Column 1 shows the logical graphs for the sixteen boolean functions on two variables.  Column 2 shows the venn diagrams associated with the entitative interpretation and Column 3 shows the venn diagrams associated with the existential interpretation.

Logical Graphs • Entitative and Existential Venn Diagrams
\text{Logical Graph} \text{Entitative Interpretation} \text{Existential Interpretation}
Cactus Stem
 		 f₁₅(x,y) f₀(x,y)
\texttt{(} ~ \texttt{)}
 		 \text{true}
f_{15}		 \text{false}
f_{0}
Cactus (x)(y)
 		 f₇(x,y) f₁(x,y)
\texttt{(} x \texttt{)(} y \texttt{)}
 		 \lnot x \lor \lnot y
f_{7}		 \lnot x \land \lnot y
f_{1}
Cactus (x)y
 		 f₁₁(x,y) f₂(x,y)
\texttt{(} x \texttt{)} y
 		 x \Rightarrow y
f_{11}		 x \nLeftarrow y
f_{2}
Cactus (x)
 		 f₃(x,y) f₃(x,y)
\texttt{(} x \texttt{)}
 		 \lnot x
f_{3}		 \lnot x
f_{3}
Cactus x(y)
 		 f₁₃(x,y) f₄(x,y)
x \texttt{(} y \texttt{)}
 		 x \Leftarrow y
f_{13}		 x \nRightarrow y
f_{4}
Cactus (y)
 		 f₅(x,y) f₅(x,y)
\texttt{(} y \texttt{)}
 		 \lnot y
f_{5}		 \lnot y
f_{5}
Cactus (x,y)
 		 f₉(x,y) f₆(x,y)
\texttt{(} x \texttt{,} y \texttt{)}
 		 x = y
f_{9}		 x \ne y
f_{6}
Cactus (xy)
 		 f₁(x,y) f₇(x,y)
\texttt{(} x y \texttt{)}
 		 \lnot (x \lor y)
f_{1}		 \lnot (x \land y)
f_{7}
Cactus xy
 		 f₁₄(x,y) f₈(x,y)
x y
 		 x \lor y
f_{14}		 x \land y
f_{8}
Cactus ((x,y))
 		 f₆(x,y) f₉(x,y)
\texttt{((} x \texttt{,} y \texttt{))}
 		 x \ne y
f_{6}		 x = y
f_{9}
Cactus y
 		 f₁₀(x,y) f₁₀(x,y)
y
 		 y
f_{10}		 y
f_{10}
Cactus (x(y))
 		 f₂(x,y) f₁₁(x,y)
\texttt{(} x \texttt{(} y \texttt{))}
 		 x \nLeftarrow y
f_{2}		 x \Rightarrow y
f_{11}
Cactus x
 		 f₁₂(x,y) f₁₂(x,y)
x
 		 x
f_{12}		 x
f_{12}
Cactus ((x)y)
 		 f₄(x,y) f₁₃(x,y)
\texttt{((} x \texttt{)} y \texttt{)}
 		 x \nRightarrow y
f_{4}		 x \Leftarrow y
f_{13}
Cactus ((x)(y))
 		 f₈(x,y) f₁₄(x,y)
\texttt{((} x \texttt{)(} y \texttt{))}
 		 x \land y
f_{8}		 x \lor y
f_{14}
Cactus Root
 		 f₀(x,y) f₁₅(x,y)
 
 		 \text{false}
f_{0}		 \text{true}
f_{15}

Resources

cc: Cybernetics Communications (1) (2) • 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.

4 Responses to Animated Logical Graphs • 48

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