Animated Logical Graphs • 76

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)(73)(74)(75)

Taking from our wallets an old schedule of orbits, let’s review the classes of logical graphs we’ve covered so far.

Self-Dual Logical Graphs

Four orbits of self-dual logical graphs, x, y, \texttt{(} x \texttt{)}, \texttt{(} y \texttt{)}, were discussed in Episode 73.

Self-Dual Logical Graphs

The logical graphs x, y, \texttt{(} x \texttt{)}, \texttt{(} y \texttt{)} denote the boolean functions f_{12}, f_{10}, f_{3}, f_{5}, in that order, and the value of each function f at each point (x, y) in \mathbb{B} \times \mathbb{B} is shown in the Table above.

Constants and Amphecks

Two orbits of logical graphs called constants and amphecks were discussed in Episode 74.

Constants and Amphecks

The constant logical graphs denote the constant functions

f_{0} : \mathbb{B} \times \mathbb{B} \to 0 \quad \text{and} \quad f_{15} : \mathbb{B} \times \mathbb{B} \to 1.

  • Under \mathrm{Ex} the logical graph whose text form is “  ” denotes the function f_{15}
    and the logical graph whose text form is ``\texttt{(} ~ \texttt{)}" denotes the function f_{0}.
  • Under \mathrm{En} the logical graph whose text form is “  ” denotes the function f_{0}
    and the logical graph whose text form is ``\texttt{(} ~ \texttt{)}" denotes the function f_{15}.

The ampheck logical graphs denote the ampheck functions

f_{1}(x, y) = \textsc{nnor}(x, y) \quad \text{and} \quad f_{7}(x, y) = \textsc{nand}(x, y).

  • Under \mathrm{Ex} the logical graph \texttt{(} xy \texttt{)} denotes the function f_{7}(x, y) = \textsc{nand}(x, y)
    and the logical graph \texttt{(} x \texttt{)(} y \texttt{)} denotes the function f_{1}(x, y) = \textsc{nnor}(x, y).
  • Under \mathrm{En} the logical graph \texttt{(} xy \texttt{)} denotes the function f_{1}(x, y) = \textsc{nnor}(x, y)
    and the logical graph \texttt{(} x \texttt{)(} y \texttt{)} denotes the function f_{7}(x, y) = \textsc{nand}(x, y).

Subtractions and Implications

The logical graphs called subtractions and implications were discussed in Episode 75.

Subtractions and Implications

The subtraction logical graphs denote the subtraction functions

f_{2}(x, y) = y \lnot x \quad \text{and} \quad f_{4}(x, y) = x \lnot y.

The implication logical graphs denote the implication functions

f_{11}(x, y) = x \Rightarrow y \quad \text{and} \quad f_{13}(x, y) = y \Rightarrow x.

Under the action of the \mathrm{En} \leftrightarrow \mathrm{Ex} duality the logical graphs for the subtraction f_{2} and the implication f_{11} fall into one orbit while the logical graphs for the subtraction f_{4} and the implication f_{13} fall into another orbit, making these two partitions of the four functions orthogonal or transversal to each other.

Resources

cc: Peirce List (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17)
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.

1 Response to Animated Logical Graphs • 76

  1. Pingback: Survey of Animated Logical Graphs • 4 | Inquiry Into Inquiry

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