## 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. 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. 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. 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

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