Animated Logical Graphs • 70

Re: Richard J. LiptonThe Art Of Math
Re: Animated Logical Graphs • (57)(58)(59)(60)(61)(62)(63)(64)(65)(66)(69)

Our study of the duality between entitative and existential interpretations of logical graphs has brought to light its fully sign-relational character.  We can see this in the sign relation linking an object domain with two sign domains, whose signs denote the objects in two distinct ways.  We illustrated the general principle using an object domain consisting of the sixteen boolean functions on two variables and a pair of sign domains consisting of representative logical graphs for those functions, as shown in the following Table.

$\text{Peirce Duality as Sign Relation}$

• 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.

Additional aspects of the sign relation’s structure can be brought out by sorting the Table in accord with the orbits induced on the object domain by the action of the transformation group inherent in the dual interpretations.  Performing that sort produces the following Table.

$\text{Peirce Duality as Sign Relation} \stackrel{_\bullet}{} \text{Orbit Order}$

That’s enough bytes to chew on for one post — we’ll extract more information from the Tables next time.

Resources

cc: Peirce List (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15)
cc: Structural Modeling (1) (2) • Systems Science (1) (2)
cc: Cybernetics (1) (2) • Ontolog Forum (1) (2)
cc: FB | Logical GraphsLaws of Form

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