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}

Peirce Duality as Sign Relation


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

This entry was posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Constraint Satisfaction Problems, Deduction, Diagrammatic Reasoning, Duality, Equational Inference, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Painted Cacti, Peirce, Proof Theory, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Theorem Proving, Visualization and tagged , , , , , , , , , , , , , , , , , , , , , , , , , . Bookmark the permalink.

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