Changes

Propositional forms on one variable correspond to boolean functions <math>f : \mathbb{B}^1 \to \mathbb{B}.</math>  In Table&nbsp;6 these functions are listed in a variant form of [[truth table]], one in which the axes of the usual arrangement are rotated through a right angle.  Each function <math>f_i\!</math> is indexed by the string of values that it takes on the points of the universe <math>X^\circ = [x] \cong \mathbb{B}^1.</math>  The binary index generated in this way is converted to its decimal equivalent and these are used as conventional names for the <math>f_i,\!</math> as shown in the first column of the Table.  In their own right the <math>2^1\!</math> points of the universe <math>X^\circ</math> are coordinated as a space of type <math>\mathbb{B}^1,</math> this in light of the universe <math>X^\circ</math> being a functional domain where the coordinate projection <math>x\!</math> takes on its values in <math>\mathbb{B}.</math>

Propositional forms on one variable correspond to boolean functions <math>f : \mathbb{B}^1 \to \mathbb{B}.</math>  In Table&nbsp;6 these functions are listed in a variant form of [[truth table]], one in which the axes of the usual arrangement are rotated through a right angle.  Each function <math>f_i\!</math> is indexed by the string of values that it takes on the points of the universe <math>X^\circ = [x] \cong \mathbb{B}^1.</math>  The binary index generated in this way is converted to its decimal equivalent and these are used as conventional names for the <math>f_i,\!</math> as shown in the first column of the Table.  In their own right the <math>2^1\!</math> points of the universe <math>X^\circ</math> are coordinated as a space of type <math>\mathbb{B}^1,</math> this in light of the universe <math>X^\circ</math> being a functional domain where the coordinate projection <math>x\!</math> takes on its values in <math>\mathbb{B}.</math>

{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:96%"

|+ '''Table 6.  Propositional Forms on One Variable'''

|+ '''Table 6.  Propositional Forms on One Variable'''

|- style="background:ghostwhite"

|- style="background:ghostwhite"

! style="width:16%" | L₁<br>Decimal

| style="width:16%" |

! style="width:16%" | L₂<br>Binary

<math>\begin{matrix}\mathcal{L}_1 \\ \mbox{Decimal}\end{matrix}</math>

! style="width:16%" | L₃<br>Vector

| style="width:16%" |

! style="width:16%" | L₄<br>Cactus

<math>\begin{matrix}\mathcal{L}_2 \\ \mbox{Binary}\end{matrix}</math>

! style="width:16%" | L₅<br>English

| style="width:16%" |  

! style="width:16%" | L₆<br>Ordinary

<math>\begin{matrix}\mathcal{L}_3 \\ \mbox{Vector}\end{matrix}</math>

| style="width:16%" |

<math>\begin{matrix}\mathcal{L}_4 \\ \mbox{Cactus}\end{matrix}</math>

| style="width:16%" |

<math>\begin{matrix}\mathcal{L}_5 \\ \mbox{English}\end{matrix}</math>

| style="width:16%" |

<math>\begin{matrix}\mathcal{L}_6 \\ \mbox{Ordinary}\end{matrix}</math>

|- style="background:ghostwhite"

|- style="background:ghostwhite"

| &nbsp;

| <math>~</math>

| align="right" | x :

| align="right" | <math>x :\!</math>

| 1 0  

| <math>1~0</math>

| &nbsp;

| <math>~</math>

| &nbsp;

| <math>~</math>

| &nbsp;

| <math>~</math>

|-

|-

| f₀

| <math>f_0\!</math>

| f₀₀

| <math>f_{00}\!</math>

| 0 0

| <math>0~0</math>

| ( )

| <math>(~)\!</math>

| false

| <math>\mbox{false}\!</math>

| 0

| <math>0\!</math>

|-

|-

| f₁

| <math>f_1\!</math>

| f₀₁

| <math>f_{01}\!</math>

| 0 1

| <math>0~1</math>

| (x)

| <math>(x)\!</math>

| not x

| <math>\mbox{not}\ x</math>

| ~x

| <math>\lnot x</math>

|-

|-

| f₂

| <math>f_2\!</math>

| f₁₀

| <math>f_{10}\!</math>

| 1 0

| <math>1~0</math>

| x

| <math>x\!</math>

| x

| <math>x\!</math>

| x

| <math>x\!</math>

|-

|-

| f₃

| <math>f_3\!</math>

| f₁₁

| <math>f_{11}\!</math>

| 1 1

| <math>1~1</math>

| (( ))

| <math>((~))\!</math>

| true

| <math>\mbox{true}\!</math>

| 1

| <math>1\!</math>

|}<br>

|}<br>