## Animated Logical Graphs • 48

Re: Richard J. LiptonThe 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
$\text{Logical Graph}$ $\text{Entitative Interpretation}$ $\text{Existential Interpretation}$

$\texttt{(} ~ \texttt{)}$

$\text{true}$
$f_{15}$
$\text{false}$
$f_{0}$

$\texttt{(} x \texttt{)(} y \texttt{)}$

$\lnot x \lor \lnot y$
$f_{7}$
$\lnot x \land \lnot y$
$f_{1}$

$\texttt{(} x \texttt{)} y$

$x \Rightarrow y$
$f_{11}$
$x \nLeftarrow y$
$f_{2}$

$\texttt{(} x \texttt{)}$

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

$x \texttt{(} y \texttt{)}$

$x \Leftarrow y$
$f_{13}$
$x \nRightarrow y$
$f_{4}$

$\texttt{(} y \texttt{)}$

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

$\texttt{(} x \texttt{,} y \texttt{)}$

$x = y$
$f_{9}$
$x \ne y$
$f_{6}$

$\texttt{(} x y \texttt{)}$

$\lnot (x \lor y)$
$f_{1}$
$\lnot (x \land y)$
$f_{7}$

$x y$

$x \lor y$
$f_{14}$
$x \land y$
$f_{8}$

$\texttt{((} x \texttt{,} y \texttt{))}$

$x \ne y$
$f_{6}$
$x = y$
$f_{9}$

$y$

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

$\texttt{(} x \texttt{(} y \texttt{))}$

$x \nLeftarrow y$
$f_{2}$
$x \Rightarrow y$
$f_{11}$

$x$

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

$\texttt{((} x \texttt{)} y \texttt{)}$

$x \nRightarrow y$
$f_{4}$
$x \Leftarrow y$
$f_{13}$

$\texttt{((} x \texttt{)(} y \texttt{))}$

$x \land y$
$f_{8}$
$x \lor y$
$f_{14}$

$\text{false}$
$f_{0}$
$\text{true}$
$f_{15}$

### 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)

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