Animated Logical Graphs • 48

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
$\text{Logical Graph}$	$\text{Entitative Interpretation}$	$\text{Existential Interpretation}$

$\texttt{(} ~ \texttt{)}$	$\text{true}$ $f_{15}$	$\text{false}$ $f_{0}$

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

$\texttt{(} x \texttt{)} y$	$x \Rightarrow y$ $f_{11}$	$x \nLeftarrow y$ $f_{2}$

$\texttt{(} x \texttt{)}$	$\lnot x$ $f_{3}$	$\lnot x$ $f_{3}$

$x \texttt{(} y \texttt{)}$	$x \Leftarrow y$ $f_{13}$	$x \nRightarrow y$ $f_{4}$

$\texttt{(} y \texttt{)}$	$\lnot y$ $f_{5}$	$\lnot y$ $f_{5}$

$\texttt{(} x \texttt{,} y \texttt{)}$	$x = y$ $f_{9}$	$x \ne y$ $f_{6}$

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

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

$\texttt{((} x \texttt{,} y \texttt{))}$	$x \ne y$ $f_{6}$	$x = y$ $f_{9}$

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

$\texttt{(} x \texttt{(} y \texttt{))}$	$x \nLeftarrow y$ $f_{2}$	$x \Rightarrow y$ $f_{11}$

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

$\texttt{((} x \texttt{)} y \texttt{)}$	$x \nRightarrow y$ $f_{4}$	$x \Leftarrow y$ $f_{13}$

$\texttt{((} x \texttt{)(} y \texttt{))}$	$x \land y$ $f_{8}$	$x \lor y$ $f_{14}$

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

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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