## Animated Logical Graphs • 30

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.

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

cc: Cybernetics (1) (2)Laws of FormFB | 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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