Animated Logical Graphs • 73

Re: Richard J. LiptonThe Art Of Math
Re: Animated Logical Graphs • (30)(45)(57)(58)(59)(60)(61)(62)(63)(64)(65)(66)(69)(70)(71)(72)

Last time we took up the four singleton orbits in the action of T on X and saw each consists of a single logical graph which T fixes, preserves, or transforms into itself.  On that account these four logical graphs are said to be self-dual or T-invariant.

In general terms, it is useful to think of the entitative and existential interpretations as two formal languages which happen to use the same set of signs, each in its own way, to denote the same set of formal objects.  Then T defines the translation between languages and the self-dual logical graphs are the points where the languages coincide, where the same signs denote the same objects in both.  Such constellations of “fixed stars” are indispensable to navigation between languages, as every argot-naut discovers in time.

Returning to the case at hand, where T acts on a selection of 16 logical graphs for the 16 boolean functions on two variables, the following Table shows the values of the denoted boolean function f : \mathbb{B} \times \mathbb{B} \to \mathbb{B} for each of the self-dual logical graphs.

Self-Dual Logical Graphs

The functions indexed here as f_{12} and f_{10} are known as the coordinate projections (x, y) \mapsto x and (x, y) \mapsto y on the first and second coordinates, respectively, and the functions indexed as f_{3} and f_{5} are the negations (x, y) \mapsto \tilde{x} and (x, y) \mapsto \tilde{y} of those projections, respectively.


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

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.

5 Responses to Animated Logical Graphs • 73

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