User:Jon Awbrey/Figures and Tables 17

Peirce Duality as Group Symmetry

Index Order

PNG

$\text{Peirce Duality as Group Symmetry}$

WIKI + JPG

$\text{Peirce Duality as Group Symmetry}$

$f$	$\gamma$	$1\gamma$	$t\gamma$
$\begin{array}{c}{f}_0\\ \text{false}\end{array}$
$\begin{array}{c}{f}_1\\ \text{neither}x\text{nor}y\end{array}$
$\begin{array}{c}{f}_2\\ y\text{and not}x\end{array}$
$\begin{array}{c}{f}_3\\ \text{not}x\end{array}$
$\begin{array}{c}{f}_4\\ x\text{and not}y\end{array}$
$\begin{array}{c}{f}_5\\ \text{not}y\end{array}$
$\begin{array}{c}{f}_6\\ x\text{not equal to}y\end{array}$
$\begin{array}{c}{f}_7\\ \text{not both}x\text{and}y\end{array}$
$\begin{array}{c}{f}_8\\ x\text{and}y\end{array}$
$\begin{array}{c}{f}_9\\ x\text{equal to}y\end{array}$
$\begin{array}{c}{f}_{10}\\ y\end{array}$
$\begin{array}{c}{f}_{11}\\ \text{if}x\text{then}y\end{array}$
$\begin{array}{c}{f}_{12}\\ x\end{array}$
$\begin{array}{c}{f}_{13}\\ \text{if}y\text{then}x\end{array}$
$\begin{array}{c}{f}_{14}\\ x\text{or}y\end{array}$
$\begin{array}{c}{f}_{15}\\ \text{true}\end{array}$
$\text{Fixed Point Total}$		$16$	$4$

Orbit Order

PNG

$\text{Peirce Duality as Group Symmetry}\stackrel{{}_{\bullet }}{}\text{Orbit Order}$

WIKI + JPG

$\text{Peirce Duality as Group Symmetry}\stackrel{{}_{\bullet }}{}\text{Orbit Order}$

$f$	$\gamma$	$1\gamma$	$t\gamma$
$\begin{array}{c}{f}_{12}\\ x\end{array}$
$\begin{array}{c}{f}_{10}\\ y\end{array}$
$\begin{array}{c}{f}_3\\ \text{not}x\end{array}$
$\begin{array}{c}{f}_5\\ \text{not}y\end{array}$
$\begin{array}{c}{f}_{15}\\ \text{true}\end{array}$
$\begin{array}{c}{f}_0\\ \text{false}\end{array}$
$\begin{array}{c}{f}_7\\ \text{not both}x\text{and}y\end{array}$
$\begin{array}{c}{f}_1\\ \text{neither}x\text{nor}y\end{array}$
$\begin{array}{c}{f}_2\\ y\text{and not}x\end{array}$
$\begin{array}{c}{f}_{11}\\ \text{if}x\text{then}y\end{array}$
$\begin{array}{c}{f}_4\\ x\text{and not}y\end{array}$
$\begin{array}{c}{f}_{13}\\ \text{if}y\text{then}x\end{array}$
$\begin{array}{c}{f}_8\\ x\text{and}y\end{array}$
$\begin{array}{c}{f}_{14}\\ x\text{or}y\end{array}$
$\begin{array}{c}{f}_6\\ x\text{not equal to}y\end{array}$
$\begin{array}{c}{f}_9\\ x\text{equal to}y\end{array}$
$\text{Fixed Point Total}$		$16$	$4$

HTML + JPG

Ex → En

$\text{Boolean Functions, Existential Graphs, Entitative Graphs}$

$\text{Boolean Function}$	$\text{Existential Graph}$	$\text{Entitative Graph}$
$f_0$
$f_1$
$f_2$
$f_3$
$f_4$
$f_5$
$f_6$
$f_7$
$f_8$
$f_9$
$f_{10}$
$f_{11}$
$f_{12}$
$f_{13}$
$f_{14}$
$f_{15}$

En → Ex

$\text{Boolean Functions on Two Variables}$

$\text{Boolean Function}$	$\text{Entitative Graph}$	$\text{Existential Graph}$
$f_0$ $\text{false}$	$\text{false}$	$\text{false}$
$f_1$ $\text{neither}x\text{nor}y$	$\mathrm{\neg }(x\vee y)$	$\mathrm{\neg }x\wedge \mathrm{\neg }y$
$f_2$ $y\text{and not}x$	$\mathrm{\neg }x\wedge y$	$\mathrm{\neg }x\wedge y$
$f_3$ $\text{not}x$	$\mathrm{\neg }x$	$\mathrm{\neg }x$
$f_4$ $x\text{and not}y$	$x\wedge \mathrm{\neg }y$	$x\wedge \mathrm{\neg }y$
$f_5$ $\text{not}y$	$\mathrm{\neg }y$	$\mathrm{\neg }y$
$f_6$ $x\text{not equal to}y$	$x\ne y$	$x\ne y$
$f_7$ $\text{not both}x\text{and}y$	$\mathrm{\neg }x\vee \mathrm{\neg }y$	$\mathrm{\neg }(x\wedge y)$
$f_8$ $x\text{and}y$	$x\wedge y$	$x\wedge y$
$f_9$ $x\text{equal to}y$	$x=y$	$x=y$
$f_{10}$ $y$	$y$	$y$
$f_{11}$ $\text{if}x\text{then}y$	$x\Rightarrow y$	$x\Rightarrow y$
$f_{12}$ $x$	$x$	$x$
$f_{13}$ $\text{if}y\text{then}x$	$x\Leftarrow y$	$x\Leftarrow y$
$f_{14}$ $x\text{or}y$	$x\vee y$	$x\vee y$
$f_{15}$ $\text{true}$	$\text{true}$	$\text{true}$

Table A3. Ef Expanded Over Differential Features

$\text{Table A3.}\mathrm{E}f\text{Expanded Over Differential Features}\{\mathrm{d}x,\mathrm{d}y\}$

	$f$	$\begin{array}{c}{\mathrm{T}}_{11}f\\ \mathrm{E}f{|}_{\mathrm{d}x\mathrm{d}y}\end{array}$	$\begin{array}{c}{\mathrm{T}}_{10}f\\ \mathrm{E}f{|}_{\mathrm{d}x\text{<mo stretchy="false">(</mo>}\mathrm{d}y\text{<mo stretchy="false">)</mo>}}\end{array}$	$\begin{array}{c}{\mathrm{T}}_{01}f\\ \mathrm{E}f{|}_{\text{<mo stretchy="false">(</mo>}\mathrm{d}x\text{<mo stretchy="false">)</mo>}\mathrm{d}y}\end{array}$	$\begin{array}{c}{\mathrm{T}}_{00}f\\ \mathrm{E}f{|}_{\text{<mo stretchy="false">(</mo>}\mathrm{d}x\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">)</mo><mo stretchy="false">(</mo></mrow>}\mathrm{d}y\text{<mo stretchy="false">)</mo>}}\end{array}$
$f_0$	$0$	$0$	$0$	$0$	$0$
$\begin{array}{c}{f}_1\\ f_2\\ f_4\\ f_8\end{array}$	$\begin{array}{c}\text{<mo stretchy="false">(</mo>}x\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">)</mo><mo stretchy="false">(</mo></mrow>}y\text{<mo stretchy="false">)</mo>}\\ \text{<mo stretchy="false">(</mo>}x\text{<mo stretchy="false">)</mo>}y\\ x\text{<mo stretchy="false">(</mo>}y\text{<mo stretchy="false">)</mo>}\\ xy\end{array}$	$\begin{array}{c}xy\\ x\text{<mo stretchy="false">(</mo>}y\text{<mo stretchy="false">)</mo>}\\ \text{<mo stretchy="false">(</mo>}x\text{<mo stretchy="false">)</mo>}y\\ \text{<mo stretchy="false">(</mo>}x\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">)</mo><mo stretchy="false">(</mo></mrow>}y\text{<mo stretchy="false">)</mo>}\end{array}$	$\begin{array}{c}x\text{<mo stretchy="false">(</mo>}y\text{<mo stretchy="false">)</mo>}\\ xy\\ \text{<mo stretchy="false">(</mo>}x\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">)</mo><mo stretchy="false">(</mo></mrow>}y\text{<mo stretchy="false">)</mo>}\\ \text{<mo stretchy="false">(</mo>}x\text{<mo stretchy="false">)</mo>}y\end{array}$	$\begin{array}{c}\text{<mo stretchy="false">(</mo>}x\text{<mo stretchy="false">)</mo>}y\\ \text{<mo stretchy="false">(</mo>}x\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">)</mo><mo stretchy="false">(</mo></mrow>}y\text{<mo stretchy="false">)</mo>}\\ xy\\ x\text{<mo stretchy="false">(</mo>}y\text{<mo stretchy="false">)</mo>}\end{array}$	$\begin{array}{c}\text{<mo stretchy="false">(</mo>}x\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">)</mo><mo stretchy="false">(</mo></mrow>}y\text{<mo stretchy="false">)</mo>}\\ \text{<mo stretchy="false">(</mo>}x\text{<mo stretchy="false">)</mo>}y\\ x\text{<mo stretchy="false">(</mo>}y\text{<mo stretchy="false">)</mo>}\\ xy\end{array}$
$\begin{array}{c}{f}_3\\ f_{12}\end{array}$	$\begin{array}{c}\text{<mo stretchy="false">(</mo>}x\text{<mo stretchy="false">)</mo>}\\ x\end{array}$	$\begin{array}{c}x\\ \text{<mo stretchy="false">(</mo>}x\text{<mo stretchy="false">)</mo>}\end{array}$	$\begin{array}{c}x\\ \text{<mo stretchy="false">(</mo>}x\text{<mo stretchy="false">)</mo>}\end{array}$	$\begin{array}{c}\text{<mo stretchy="false">(</mo>}x\text{<mo stretchy="false">)</mo>}\\ x\end{array}$	$\begin{array}{c}\text{<mo stretchy="false">(</mo>}x\text{<mo stretchy="false">)</mo>}\\ x\end{array}$
$\begin{array}{c}{f}_6\\ f_9\end{array}$	$\begin{array}{c}\text{<mo stretchy="false">(</mo>}x\text{<mo>,</mo>}y\text{<mo stretchy="false">)</mo>}\\ \text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mo stretchy="false">(</mo></mrow>}x\text{<mo>,</mo>}y\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow>}\end{array}$	$\begin{array}{c}\text{<mo stretchy="false">(</mo>}x\text{<mo>,</mo>}y\text{<mo stretchy="false">)</mo>}\\ \text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mo stretchy="false">(</mo></mrow>}x\text{<mo>,</mo>}y\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow>}\end{array}$	$\begin{array}{c}\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mo stretchy="false">(</mo></mrow>}x\text{<mo>,</mo>}y\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow>}\\ \text{<mo stretchy="false">(</mo>}x\text{<mo>,</mo>}y\text{<mo stretchy="false">)</mo>}\end{array}$	$\begin{array}{c}\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mo stretchy="false">(</mo></mrow>}x\text{<mo>,</mo>}y\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow>}\\ \text{<mo stretchy="false">(</mo>}x\text{<mo>,</mo>}y\text{<mo stretchy="false">)</mo>}\end{array}$	$\begin{array}{c}\text{<mo stretchy="false">(</mo>}x\text{<mo>,</mo>}y\text{<mo stretchy="false">)</mo>}\\ \text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mo stretchy="false">(</mo></mrow>}x\text{<mo>,</mo>}y\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow>}\end{array}$
$\begin{array}{c}{f}_5\\ f_{10}\end{array}$	$\begin{array}{c}\text{<mo stretchy="false">(</mo>}y\text{<mo stretchy="false">)</mo>}\\ y\end{array}$	$\begin{array}{c}y\\ \text{<mo stretchy="false">(</mo>}y\text{<mo stretchy="false">)</mo>}\end{array}$	$\begin{array}{c}\text{<mo stretchy="false">(</mo>}y\text{<mo stretchy="false">)</mo>}\\ y\end{array}$	$\begin{array}{c}y\\ \text{<mo stretchy="false">(</mo>}y\text{<mo stretchy="false">)</mo>}\end{array}$	$\begin{array}{c}\text{<mo stretchy="false">(</mo>}y\text{<mo stretchy="false">)</mo>}\\ y\end{array}$
$\begin{array}{c}{f}_7\\ f_{11}\\ f_{13}\\ f_{14}\end{array}$	$\begin{array}{c}\text{<mo stretchy="false">(</mo>}xy\text{<mo stretchy="false">)</mo>}\\ \text{<mo stretchy="false">(</mo>}x\text{<mo stretchy="false">(</mo>}y\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow>}\\ \text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mo stretchy="false">(</mo></mrow>}x\text{<mo stretchy="false">)</mo>}y\text{<mo stretchy="false">)</mo>}\\ \text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mo stretchy="false">(</mo></mrow>}x\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">)</mo><mo stretchy="false">(</mo></mrow>}y\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow>}\end{array}$	$\begin{array}{c}\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mo stretchy="false">(</mo></mrow>}x\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">)</mo><mo stretchy="false">(</mo></mrow>}y\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow>}\\ \text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mo stretchy="false">(</mo></mrow>}x\text{<mo stretchy="false">)</mo>}y\text{<mo stretchy="false">)</mo>}\\ \text{<mo stretchy="false">(</mo>}x\text{<mo stretchy="false">(</mo>}y\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow>}\\ \text{<mo stretchy="false">(</mo>}xy\text{<mo stretchy="false">)</mo>}\end{array}$	$\begin{array}{c}\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mo stretchy="false">(</mo></mrow>}x\text{<mo stretchy="false">)</mo>}y\text{<mo stretchy="false">)</mo>}\\ \text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mo stretchy="false">(</mo></mrow>}x\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">)</mo><mo stretchy="false">(</mo></mrow>}y\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow>}\\ \text{<mo stretchy="false">(</mo>}xy\text{<mo stretchy="false">)</mo>}\\ \text{<mo stretchy="false">(</mo>}x\text{<mo stretchy="false">(</mo>}y\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow>}\end{array}$	$\begin{array}{c}\text{<mo stretchy="false">(</mo>}x\text{<mo stretchy="false">(</mo>}y\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow>}\\ \text{<mo stretchy="false">(</mo>}xy\text{<mo stretchy="false">)</mo>}\\ \text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mo stretchy="false">(</mo></mrow>}x\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">)</mo><mo stretchy="false">(</mo></mrow>}y\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow>}\\ \text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mo stretchy="false">(</mo></mrow>}x\text{<mo stretchy="false">)</mo>}y\text{<mo stretchy="false">)</mo>}\end{array}$	$\begin{array}{c}\text{<mo stretchy="false">(</mo>}xy\text{<mo stretchy="false">)</mo>}\\ \text{<mo stretchy="false">(</mo>}x\text{<mo stretchy="false">(</mo>}y\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow>}\\ \text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mo stretchy="false">(</mo></mrow>}x\text{<mo stretchy="false">)</mo>}y\text{<mo stretchy="false">)</mo>}\\ \text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mo stretchy="false">(</mo></mrow>}x\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">)</mo><mo stretchy="false">(</mo></mrow>}y\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow>}\end{array}$
$f_{15}$	$1$	$1$	$1$	$1$	$1$
$\text{Fixed Point Total}$		$4$	$4$	$4$	$16$