## Animated Logical Graphs • 66

Re: Richard J. LiptonThe Art Of Math
Re: Animated Logical Graphs • (57)(58)(59)(60)(61)(62)(63)(64)(65)

Once we bring the dual interpretations of logical graphs to the same Table and relate their parleys to the same objects, it is clear we are dealing with a triadic sign relation of the sort taken up in C.S. Peirce’s semiotics or theory of signs.

A sign relation $L \subseteq O \times S \times I,$ as a set $L$ embedded in a cartesian product $O \times S \times I,$ tells how the signs in $S$ and the interpretant signs in $I$ correlate with the objects or objective situations in $O.$

There are many ways of using sign relations to model various types of sign-theoretic situations and processes.  The following cases are often seen.

• Some sign relations model co‑referring signs or transitions between signs within a single language or symbol system.  In that event $L \subseteq O \times S \times I$ has $S = I.$
• Other sign relations model translations between different languages or different interpretations of the same language, in other words, different ways of referring the same set of signs to a shared object domain.

The next Table extracts the sign relation $L \subseteq O \times S \times I$ involved in switching between existential and entitative interpretations of logical graphs.

• Column 1 shows the object domain $O$ as the set of 16 boolean functions on 2 variables.
• Column 2 shows the sign domain $S$ as a representative set of logical graphs denoting the objects in $O$ according to the existential interpretation.
• Column 3 shows the interpretant domain $I$ as the same set of logical graphs denoting the objects in $O$ according to the entitative interpretation.

$\text{Peirce Duality as Sign Relation}$

### Resources

cc: Cybernetics (1) (2) • Peirce List (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)
cc: Ontolog Forum (1) (2) • Structural Modeling (1) (2) • Systems Science (1) (2)
cc: FB | Logical GraphsLaws of Form

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