Animated Logical Graphs • 76

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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

Logic Syllabus • Logical Implication • Truth Tables • Zeroth Order Logic
Survey of Animated Logical Graphs

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