Draft

Version of 2018-12-05 or later

Draft Outline

The basic building block for Boolean geometry and logic
is the Boolean “object of truth values”: $\bftwo = \{\ladjbot,\radjtop\} = \{\text{false},\text{true}\}$.
We give this set the (linear) order $\ladjbot\lt\radjtop$, in accordance with the ex falso sequitur quodlibet principle of logic.

Boolean geometry then considers powers of $\bftwo$:

the Boolean point	$\bftwo^0 \cong 1$ ,
the Boolean line	$\bftwo^1 \cong \bftwo$ ,
the Boolean plane	$\bftwo^2 \cong \bftwo\times\bftwo$ ,
the Boolean cube	$\bftwo^3\cong \bftwo\times\bftwo\times\bftwo$ , and so on.

The next step is to consider Boolean (i.e., $\bftwo$-valued) functions (which we will call predicates) defined on these geometric objects.
Since (the Boolean object of truth values, $\bftwo$) is (the subobject classifier) for (the category of sets),
such predicates are in bijection with
the subsets of (the source (domain) of the predicate).

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A comparison of the traditional way of presenting truth tables to
the geometrical way (the graph of the function from $\bftwo\times\bftwo$ to $\bftwo$ determined by the truth table):

Truth table for $\objA\wedge\objB$

$\objA$ $\objB$ $\objA\wedge\objB$ $\radjtop$ $\radjtop$ $\radjtop$ $\ladjbot$ $\radjtop$ $\ladjbot$ $\radjtop$ $\ladjbot$ $\ladjbot$ $\ladjbot$ $\ladjbot$ $\ladjbot$

The graph of the function $\wedge : \bftwo\times\bftwo \rightarrow \bftwo$

$\objB$ $\radjtop$ $\ladjbot$ $\Rule{64px}{2px}{0px}$ $\radjtop$ $\neg\objA$ $\Rule{2px}{32px}{32px}$ $\objA\wedge\objB$ $\Rule{2px}{32px}{32px}$ $\objA$ $\ladjbot$ $\ladjbot$ $\Rule{64px}{2px}{0px}$ $\ladjbot$ $\ladjbot$ $\neg\objB$ $\radjtop$

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			$\objB\in\{\ladjbot,\radjtop\}$
			$\vary\in\{0,1\}$
			$\Big\uparrow$	$\objB$
$1$	$\top$		$\begin{array}{c}\langle \ladjbot, \radjtop \rangle\\\objB - \objA\\\end{array}$	$\Rule{64px}{2px}{0px}$	$\begin{array}{c}\langle \radjtop, \radjtop \rangle\\\objA\cap\objB\\\end{array}$
	$\neg\objA$		$\Rule{2px}{32px}{32px}$		$\Rule{2px}{32px}{32px}$	$\objA$
$0$	$\bot$		$\begin{array}{c}\langle \ladjbot, \ladjbot \rangle\\\end{array}$	$\Rule{64px}{2px}{0px}$	$\begin{array}{c}\langle \radjtop, \ladjbot \rangle\\\objA - \objB\\\end{array}$		$\longrightarrow \objA\in\{\ladjbot,\radjtop\}, \varx\in\{0,1\}$
			$\bot$	$\neg\objB$	$\top$
			$0$		$1$

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${2\times2 \choose 4} = 1$	all			$\objB$ $\radjtop$ $\radjtop$ $\Rule{64px}{2px}{0px}$ $\radjtop$ $\neg\objA$ $\Rule{2px}{32px}{32px}$ $\begin{array}{c}\radjtop\\1\end{array}$ $\Rule{2px}{32px}{32px}$ $\objA$ $\ladjbot$ $\radjtop$ $\Rule{64px}{2px}{0px}$ $\radjtop$ $\ladjbot$ $\neg\objB$ $\radjtop$
${2\times2 \choose 3} = 4$	ells angles		$\objB$ $\radjtop$ $\radjtop$ $\Rule{128px}{2px}{0px}$ $\radjtop$ $\neg\objA$ $\Rule{2px}{32px}{32px}$ $\begin{array}{c}\objA\vee\objB\\(\varx+1)(\vary+1)+1\\\varx\vary+\varx+\vary\end{array}$ $\Rule{2px}{32px}{32px}$ $\objA$ $\ladjbot$ $\ladjbot$ $\Rule{128px}{2px}{0px}$ $\radjtop$ $\ladjbot$ $\neg\objB$ $\radjtop$	$\objB$ $\radjtop$ $\radjtop$ $\Rule{96px}{2px}{0px}$ $\radjtop$ $\neg\objA$ $\Rule{2px}{32px}{32px}$ $\begin{array}{c}\neg\objA\vee\objB\\\objA\Rightarrow\objB\\\neg(\objA\wedge\neg\objB)\\\varx(\vary+1)+1\\\varx\vary+\varx+1\end{array}$ $\Rule{2px}{32px}{32px}$ $\objA$ $\ladjbot$ $\radjtop$ $\Rule{96px}{2px}{0px}$ $\ladjbot$ $\ladjbot$ $\neg\objB$ $\radjtop$	$\objB$ $\radjtop$ $\ladjbot$ $\Rule{96px}{2px}{0px}$ $\radjtop$ $\neg\objA$ $\Rule{2px}{32px}{32px}$ $\begin{array}{c}\objA\vee\neg\objB\\\objA\Leftarrow\objB\\\neg(\neg\objA\wedge\objB)\\(\varx+1)\vary+1\\\varx\vary+\vary+1\end{array}$ $\Rule{2px}{32px}{32px}$ $\objA$ $\ladjbot$ $\radjtop$ $\Rule{96px}{2px}{0px}$ $\radjtop$ $\ladjbot$ $\neg\objB$ $\radjtop$	$\objB$ $\radjtop$ $\radjtop$ $\Rule{128px}{2px}{0px}$ $\ladjbot$ $\neg\objA$ $\Rule{2px}{32px}{32px}$ $\begin{array}{c}\neg\objA\vee\neg\objB\\\neg(\objA\wedge\objB)\\\varx\vary+1\end{array}$ $\Rule{2px}{32px}{32px}$ $\objA$ $\ladjbot$ $\radjtop$ $\Rule{128px}{2px}{0px}$ $\radjtop$ $\ladjbot$ $\neg\objB$ $\radjtop$
${2\times2 \choose 2} = 6$	lines	$\objB$ $\radjtop$ $\ladjbot$ $\Rule{64px}{2px}{0px}$ $\radjtop$ $\neg\objA$ $\Rule{2px}{32px}{32px}$ $\begin{array}{c}\objA\\\varx\end{array}$ $\Rule{2px}{32px}{32px}$ $\objA$ $\ladjbot$ $\ladjbot$ $\Rule{64px}{2px}{0px}$ $\radjtop$ $\ladjbot$ $\neg\objB$ $\radjtop$	$\objB$ $\radjtop$ $\radjtop$ $\Rule{128px}{2px}{0px}$ $\radjtop$ $\neg\objA$ $\Rule{2px}{32px}{32px}$ $\begin{array}{c}\objB\\\vary\end{array}$ $\Rule{2px}{32px}{32px}$ $\objA$ $\ladjbot$ $\ladjbot$ $\Rule{128px}{2px}{0px}$ $\ladjbot$ $\ladjbot$ $\neg\objB$ $\radjtop$	$\objB$ $\radjtop$ $\ladjbot$ $\Rule{96px}{2px}{0px}$ $\radjtop$ $\neg\objA$ $\Rule{2px}{32px}{32px}$ $\begin{array}{c}\objA\Leftrightarrow\objB\\\varx+\vary+1\end{array}$ $\Rule{2px}{32px}{32px}$ $\objA$ $\ladjbot$ $\radjtop$ $\Rule{96px}{2px}{0px}$ $\ladjbot$ $\ladjbot$ $\neg\objB$ $\radjtop$	$\objB$ $\radjtop$ $\radjtop$ $\Rule{96px}{2px}{0px}$ $\ladjbot$ $\neg\objA$ $\Rule{2px}{32px}{32px}$ $\begin{array}{c}\objA \veebar\objB\\\varx+\vary\end{array}$ $\Rule{2px}{32px}{32px}$ $\objA$ $\ladjbot$ $\ladjbot$ $\Rule{96px}{2px}{0px}$ $\radjtop$ $\ladjbot$ $\neg\objB$ $\radjtop$	$\objB$ $\radjtop$ $\ladjbot$ $\Rule{128px}{2px}{0px}$ $\ladjbot$ $\neg\objA$ $\Rule{2px}{32px}{32px}$ $\begin{array}{c}\neg\objB\\\vary+1\end{array}$ $\Rule{2px}{32px}{32px}$ $\objA$ $\ladjbot$ $\radjtop$ $\Rule{128px}{2px}{0px}$ $\radjtop$ $\ladjbot$ $\neg\objB$ $\radjtop$	$\objB$ $\radjtop$ $\radjtop$ $\Rule{64px}{2px}{0px}$ $\ladjbot$ $\neg\objA$ $\Rule{2px}{32px}{32px}$ $\begin{array}{c}\neg\objA\\\varx+1\end{array}$ $\Rule{2px}{32px}{32px}$ $\objA$ $\ladjbot$ $\radjtop$ $\Rule{64px}{2px}{0px}$ $\ladjbot$ $\ladjbot$ $\neg\objB$ $\radjtop$
${2\times2 \choose 1} = 4$	points vertices		$\objB$ $\radjtop$ $\ladjbot$ $\Rule{128px}{2px}{0px}$ $\radjtop$ $\neg\objA$ $\Rule{2px}{32px}{32px}$ $\begin{array}{c}\objA\wedge\objB\\\varx\vary\end{array}$ $\Rule{2px}{32px}{32px}$ $\objA$ $\ladjbot$ $\ladjbot$ $\Rule{128px}{2px}{0px}$ $\ladjbot$ $\ladjbot$ $\neg\objB$ $\radjtop$	$\objB$ $\radjtop$ $\radjtop$ $\Rule{96px}{2px}{0px}$ $\ladjbot$ $\neg\objA$ $\Rule{2px}{32px}{32px}$ $\begin{array}{c}\neg\objA\wedge\objB\\(\varx+1)\vary\\\varx\vary+\vary\end{array}$ $\Rule{2px}{32px}{32px}$ $\objA$ $\ladjbot$ $\ladjbot$ $\Rule{96px}{2px}{0px}$ $\ladjbot$ $\ladjbot$ $\neg\objB$ $\radjtop$	$\objB$ $\radjtop$ $\ladjbot$ $\Rule{96px}{2px}{0px}$ $\ladjbot$ $\neg\objA$ $\Rule{2px}{32px}{32px}$ $\begin{array}{c}\objA\wedge\neg\objB\\\varx(\vary+1)\\\varx\vary+\varx\end{array}$ $\Rule{2px}{32px}{32px}$ $\objA$ $\ladjbot$ $\ladjbot$ $\Rule{96px}{2px}{0px}$ $\radjtop$ $\ladjbot$ $\neg\objB$ $\radjtop$	$\objB$ $\radjtop$ $\ladjbot$ $\Rule{128px}{2px}{0px}$ $\ladjbot$ $\neg\objA$ $\Rule{2px}{32px}{32px}$ $\begin{array}{c}\neg\objA\wedge\neg\objB\\(\varx+1)(\vary+1)\\\varx\vary+\varx+\vary+1\end{array}$ $\Rule{2px}{32px}{32px}$ $\objA$ $\ladjbot$ $\radjtop$ $\Rule{128px}{2px}{0px}$ $\ladjbot$ $\ladjbot$ $\neg\objB$ $\radjtop$
${2\times2 \choose 0} = 1$	none			$\objB$ $\radjtop$ $\ladjbot$ $\Rule{64px}{2px}{0px}$ $\ladjbot$ $\neg\objA$ $\Rule{2px}{32px}{32px}$ $\begin{array}{c}\ladjbot\\0\end{array}$ $\Rule{2px}{32px}{32px}$ $\objA$ $\ladjbot$ $\ladjbot$ $\Rule{64px}{2px}{0px}$ $\ladjbot$ $\ladjbot$ $\neg\objB$ $\radjtop$

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$$\begin{array}{} &&& dot & \unicode{0x2500} & dot \\ && \unicode{0x2571} && \unicode{0x2571} \\ & dot & \unicode{0x2500} & dot \\ \\ && dot & \unicode{0x2500} & dot \\ & \unicode{0x2571} && \unicode{0x2571} \\ dot & \unicode{0x2500} & dot \\ \end{array}$$