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==Differential Logic==

===Ascii Tables===

<pre>
Table 1. Propositional Forms On Two Variables
o---------o---------o---------o----------o------------------o----------o
| L_1 | L_2 | L_3 | L_4 | L_5 | L_6 |
| | | | | | |
| Decimal | Binary | Vector | Cactus | English | Ordinary |
o---------o---------o---------o----------o------------------o----------o
| | x : 1 1 0 0 | | | |
| | y : 1 0 1 0 | | | |
o---------o---------o---------o----------o------------------o----------o
| | | | | | |
| f_0 | f_0000 | 0 0 0 0 | () | false | 0 |
| | | | | | |
| f_1 | f_0001 | 0 0 0 1 | (x)(y) | neither x nor y | ~x & ~y |
| | | | | | |
| f_2 | f_0010 | 0 0 1 0 | (x) y | y and not x | ~x & y |
| | | | | | |
| f_3 | f_0011 | 0 0 1 1 | (x) | not x | ~x |
| | | | | | |
| f_4 | f_0100 | 0 1 0 0 | x (y) | x and not y | x & ~y |
| | | | | | |
| f_5 | f_0101 | 0 1 0 1 | (y) | not y | ~y |
| | | | | | |
| f_6 | f_0110 | 0 1 1 0 | (x, y) | x not equal to y | x + y |
| | | | | | |
| f_7 | f_0111 | 0 1 1 1 | (x y) | not both x and y | ~x v ~y |
| | | | | | |
| f_8 | f_1000 | 1 0 0 0 | x y | x and y | x & y |
| | | | | | |
| f_9 | f_1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y |
| | | | | | |
| f_10 | f_1010 | 1 0 1 0 | y | y | y |
| | | | | | |
| f_11 | f_1011 | 1 0 1 1 | (x (y)) | not x without y | x => y |
| | | | | | |
| f_12 | f_1100 | 1 1 0 0 | x | x | x |
| | | | | | |
| f_13 | f_1101 | 1 1 0 1 | ((x) y) | not y without x | x <= y |
| | | | | | |
| f_14 | f_1110 | 1 1 1 0 | ((x)(y)) | x or y | x v y |
| | | | | | |
| f_15 | f_1111 | 1 1 1 1 | (()) | true | 1 |
| | | | | | |
o---------o---------o---------o----------o------------------o----------o
</pre>
<pre>
Table 2. Ef Expanded Over Ordinary Features {x, y}
o------o------------o------------o------------o------------o------------o
| | | | | | |
| | f | Ef | xy | Ef | x(y) | Ef | (x)y | Ef | (x)(y)|
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_0 | () | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_1 | (x)(y) | dx dy | dx (dy) | (dx) dy | (dx)(dy) |
| | | | | | |
| f_2 | (x) y | dx (dy) | dx dy | (dx)(dy) | (dx) dy |
| | | | | | |
| f_4 | x (y) | (dx) dy | (dx)(dy) | dx dy | dx (dy) |
| | | | | | |
| f_8 | x y | (dx)(dy) | (dx) dy | dx (dy) | dx dy |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_3 | (x) | dx | dx | (dx) | (dx) |
| | | | | | |
| f_12 | x | (dx) | (dx) | dx | dx |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_6 | (x, y) | (dx, dy) | ((dx, dy)) | ((dx, dy)) | (dx, dy) |
| | | | | | |
| f_9 | ((x, y)) | ((dx, dy)) | (dx, dy) | (dx, dy) | ((dx, dy)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_5 | (y) | dy | (dy) | dy | (dy) |
| | | | | | |
| f_10 | y | (dy) | dy | (dy) | dy |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_7 | (x y) | ((dx)(dy)) | ((dx) dy) | (dx (dy)) | (dx dy) |
| | | | | | |
| f_11 | (x (y)) | ((dx) dy) | ((dx)(dy)) | (dx dy) | (dx (dy)) |
| | | | | | |
| f_13 | ((x) y) | (dx (dy)) | (dx dy) | ((dx)(dy)) | ((dx) dy) |
| | | | | | |
| f_14 | ((x)(y)) | (dx dy) | (dx (dy)) | ((dx) dy) | ((dx)(dy)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_15 | (()) | (()) | (()) | (()) | (()) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
</pre>
<pre>
Table 3. Df Expanded Over Ordinary Features {x, y}
o------o------------o------------o------------o------------o------------o
| | | | | | |
| | f | Df | xy | Df | x(y) | Df | (x)y | Df | (x)(y)|
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_0 | () | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_1 | (x)(y) | dx dy | dx (dy) | (dx) dy | ((dx)(dy)) |
| | | | | | |
| f_2 | (x) y | dx (dy) | dx dy | ((dx)(dy)) | (dx) dy |
| | | | | | |
| f_4 | x (y) | (dx) dy | ((dx)(dy)) | dx dy | dx (dy) |
| | | | | | |
| f_8 | x y | ((dx)(dy)) | (dx) dy | dx (dy) | dx dy |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_3 | (x) | dx | dx | dx | dx |
| | | | | | |
| f_12 | x | dx | dx | dx | dx |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_6 | (x, y) | (dx, dy) | (dx, dy) | (dx, dy) | (dx, dy) |
| | | | | | |
| f_9 | ((x, y)) | (dx, dy) | (dx, dy) | (dx, dy) | (dx, dy) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_5 | (y) | dy | dy | dy | dy |
| | | | | | |
| f_10 | y | dy | dy | dy | dy |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_7 | (x y) | ((dx)(dy)) | (dx) dy | dx (dy) | dx dy |
| | | | | | |
| f_11 | (x (y)) | (dx) dy | ((dx)(dy)) | dx dy | dx (dy) |
| | | | | | |
| f_13 | ((x) y) | dx (dy) | dx dy | ((dx)(dy)) | (dx) dy |
| | | | | | |
| f_14 | ((x)(y)) | dx dy | dx (dy) | (dx) dy | ((dx)(dy)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_15 | (()) | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
</pre>
<pre>
Table 4. Ef Expanded Over Differential Features {dx, dy}
o------o------------o------------o------------o------------o------------o
| | | | | | |
| | f | T_11 f | T_10 f | T_01 f | T_00 f |
| | | | | | |
| | | Ef| dx dy | Ef| dx(dy) | Ef| (dx)dy | Ef|(dx)(dy)|
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_0 | () | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_1 | (x)(y) | x y | x (y) | (x) y | (x)(y) |
| | | | | | |
| f_2 | (x) y | x (y) | x y | (x)(y) | (x) y |
| | | | | | |
| f_4 | x (y) | (x) y | (x)(y) | x y | x (y) |
| | | | | | |
| f_8 | x y | (x)(y) | (x) y | x (y) | x y |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_3 | (x) | x | x | (x) | (x) |
| | | | | | |
| f_12 | x | (x) | (x) | x | x |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_6 | (x, y) | (x, y) | ((x, y)) | ((x, y)) | (x, y) |
| | | | | | |
| f_9 | ((x, y)) | ((x, y)) | (x, y) | (x, y) | ((x, y)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_5 | (y) | y | (y) | y | (y) |
| | | | | | |
| f_10 | y | (y) | y | (y) | y |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_7 | (x y) | ((x)(y)) | ((x) y) | (x (y)) | (x y) |
| | | | | | |
| f_11 | (x (y)) | ((x) y) | ((x)(y)) | (x y) | (x (y)) |
| | | | | | |
| f_13 | ((x) y) | (x (y)) | (x y) | ((x)(y)) | ((x) y) |
| | | | | | |
| f_14 | ((x)(y)) | (x y) | (x (y)) | ((x) y) | ((x)(y)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_15 | (()) | (()) | (()) | (()) | (()) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | |
| Fixed Point Total | 4 | 4 | 4 | 16 |
| | | | | |
o-------------------o------------o------------o------------o------------o
</pre>
<pre>
Table 5. Df Expanded Over Differential Features {dx, dy}
o------o------------o------------o------------o------------o------------o
| | | | | | |
| | f | Df| dx dy | Df| dx(dy) | Df| (dx)dy | Df|(dx)(dy)|
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_0 | () | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_1 | (x)(y) | ((x, y)) | (y) | (x) | () |
| | | | | | |
| f_2 | (x) y | (x, y) | y | (x) | () |
| | | | | | |
| f_4 | x (y) | (x, y) | (y) | x | () |
| | | | | | |
| f_8 | x y | ((x, y)) | y | x | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_3 | (x) | (()) | (()) | () | () |
| | | | | | |
| f_12 | x | (()) | (()) | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_6 | (x, y) | () | (()) | (()) | () |
| | | | | | |
| f_9 | ((x, y)) | () | (()) | (()) | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_5 | (y) | (()) | () | (()) | () |
| | | | | | |
| f_10 | y | (()) | () | (()) | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_7 | (x y) | ((x, y)) | y | x | () |
| | | | | | |
| f_11 | (x (y)) | (x, y) | (y) | x | () |
| | | | | | |
| f_13 | ((x) y) | (x, y) | y | (x) | () |
| | | | | | |
| f_14 | ((x)(y)) | ((x, y)) | (y) | (x) | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_15 | (()) | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
</pre>

===Wiki Tables===

{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ '''Table 1. Propositional Forms on Two Variables'''
|- style="background:paleturquoise"
! style="width:15%" | L<sub>1</sub>
! style="width:15%" | L<sub>2</sub>
! style="width:15%" | L<sub>3</sub>
! style="width:15%" | L<sub>4</sub>
! style="width:15%" | L<sub>5</sub>
! style="width:15%" | L<sub>6</sub>
|- style="background:paleturquoise"
| &nbsp;
| align="right" | x :
| 1 1 0 0
| &nbsp;
| &nbsp;
| &nbsp;
|- style="background:paleturquoise"
| &nbsp;
| align="right" | y :
| 1 0 1 0
| &nbsp;
| &nbsp;
| &nbsp;
|-
| f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || (&nbsp;) || false || 0
|-
| f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || &not;x &and; &not;y
|-
| f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || &not;x &and; y
|-
| f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || &not;x
|-
| f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x &and; &not;y
|-
| f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || &not;y
|-
| f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x &ne; y
|-
| f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x&nbsp;y) || not both x and y || &not;x &or; &not;y
|-
| f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x&nbsp;y || x and y || x &and; y
|-
| f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y
|-
| f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y
|-
| f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x &rarr; y
|-
| f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x
|-
| f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x &larr; y
|-
| f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y || x &or; y
|-
| f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || ((&nbsp;)) || true || 1
|}
<br>

==Differential Logic and Dynamic Systems==

===Table 1. Syntax & Semantics of a Calculus for Propositional Logic===

<pre>
Table 1. Syntax & Semantics of a Calculus for Propositional Logic
o-------------------o-------------------o-------------------o
| Expression | Interpretation | Other Notations |
o-------------------o-------------------o-------------------o
| " " | True. | 1 |
o-------------------o-------------------o-------------------o
| () | False. | 0 |
o-------------------o-------------------o-------------------o
| A | A. | A |
o-------------------o-------------------o-------------------o
| (A) | Not A. | A' |
| | | ~A |
o-------------------o-------------------o-------------------o
| A B C | A and B and C. | A & B & C |
o-------------------o-------------------o-------------------o
| ((A)(B)(C)) | A or B or C. | A v B v C |
o-------------------o-------------------o-------------------o
| (A (B)) | A implies B. | A => B |
| | If A then B. | |
o-------------------o-------------------o-------------------o
| (A, B) | A not equal to B. | A =/= B |
| | A exclusive-or B. | A + B |
o-------------------o-------------------o-------------------o
| ((A, B)) | A is equal to B. | A = B |
| | A if & only if B. | A <=> B |
o-------------------o-------------------o-------------------o
| (A, B, C) | Just one of | A'B C v |
| | A, B, C | A B'C v |
| | is false. | A B C' |
o-------------------o-------------------o-------------------o
| ((A),(B),(C)) | Just one of | A B'C' v |
| | A, B, C | A'B C' v |
| | is true. | A'B'C |
| | | |
| | Partition all | |
| | into A, B, C. | |
o-------------------o-------------------o-------------------o
| ((A, B), C) | Oddly many of | A + B + C |
| (A, (B, C)) | A, B, C | |
| | are true. | A B C v |
| | | A B'C' v |
| | | A'B C' v |
| | | A'B'C |
o-------------------o-------------------o-------------------o
| (Q, (A),(B),(C)) | Partition Q | Q'A'B'C' v |
| | into A, B, C. | Q A B'C' v |
| | | Q A'B C' v |
| | Genus Q comprises | Q A'B'C |
| | species A, B, C. | |
o-------------------o-------------------o-------------------o
</pre>

<font face="courier new">
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
|+ '''Table 1. Syntax and Semantics of a Calculus for Propositional Logic'''
|- style="background:paleturquoise"
! Expression
! Interpretation
! Other Notations
|-
| "&nbsp;"
| True.
| 1
|-
| (&nbsp;)
| False.
| 0
|-
| A
| A.
| A
|-
| (A)
| Not A.
| &nbsp;A’ <br> ~A <br> &not;A
|-
| A B C
| A and B and C.
| A &and; B &and; C
|-
| ((A)(B)(C))
| A or B or C.
| A &or; B &or; C
|-
| (A (B))
| A implies B. <br> If A then B.
| A &rArr; B
|-
| (A, B)
| A not equal to B. <br> A exclusive-or B.
| A &ne; B <br> A + B
|-
| ((A, B))
| A is equal to B. <br> A if & only if B.
| A = B <br> A &hArr; B
|-
| (A, B, C)
| Just one of <br> A, B, C <br> is false.
|
A’B C &or;<br>
A B’C &or;<br>
A B C’
|-
| ((A),(B),(C))
| Just one of <br> A, B, C <br> is true. <br><br>
Partition all <br> into A, B, C.
|
A B’C’ &or;<br>
A’B C’ &or;<br>
A’B’C
|-
| ((A, B), C) <br> &nbsp; <br> (A, (B, C))
| Oddly many of <br> A, B, C <br> are true.
|
A + B + C<br>&nbsp;<br>
A B C &nbsp;&or;<br>
A B’C’ &or;<br>
A’B C’ &or;<br>
A’B’C
|-
| (Q, (A),(B),(C))
| Partition Q <br> into A, B, C.<br>
Genus Q comprises <br> species A, B, C.
|
Q’A’B’C’ &or;<br>
Q A B’C’ &or;<br>
Q A’B C’ &or;<br>
Q A’B’C
|}
</font><br>

===Table 2. Fundamental Notations for Propositional Calculus===

<pre>
Table 2. Fundamental Notations for Propositional Calculus
o---------o-------------------o-------------------o-------------------o
| Symbol | Notation | Description | Type |
o---------o-------------------o-------------------o-------------------o
| !A! | {a_1, ..., a_n} | Alphabet | [n] = #n# |
o---------o-------------------o-------------------o-------------------o
| A_i | {(a_i), a_i} | Dimension i | B |
o---------o-------------------o-------------------o-------------------o
| A | <|!A!|> | Set of cells, | B^n |
| | <|a_i, ..., a_n|> | coordinate tuples,| |
| | {<a_i, ..., a_n>} | interpretations, | |
| | A_1 x ... x A_n | points, or vectors| |
| | Prod_i A_i | in the universe | |
o---------o-------------------o-------------------o-------------------o
| A* | (hom : A -> B) | Linear functions | (B^n)* = B^n |
o---------o-------------------o-------------------o-------------------o
| A^ | (A -> B) | Boolean functions | B^n -> B |
o---------o-------------------o-------------------o-------------------o
| A% | [!A!] | Universe of Disc. | (B^n, (B^n -> B)) |
| | (A, A^) | based on features | (B^n +-> B) |
| | (A +-> B) | {a_1, ..., a_n} | [B^n] |
| | (A, (A -> B)) | | |
| | [a_1, ..., a_n] | | |
o---------o-------------------o-------------------o-------------------o
</pre>

<font face="courier new">
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
|+ '''Table 2. Fundamental Notations for Propositional Calculus'''
|- style="background:paleturquoise"
! Symbol
! Notation
! Description
! Type
|-
| <font face="lucida calligraphy">A<font>
| {''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>}
| Alphabet
| [''n''] = '''n'''
|-
| ''A''<sub>''i''</sub>
| {(''a''<sub>''i''</sub>), ''a''<sub>''i''</sub>}
| Dimension ''i''
| '''B'''
|-
| ''A''
|
〈<font face="lucida calligraphy">A</font>〉<br>
〈''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>〉<br>
{‹''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>›}<br>
''A''<sub>1</sub> &times; &hellip; &times; ''A''<sub>''n''</sub><br>
&prod;<sub>''i''</sub> ''A''<sub>''i''</sub>
|
Set of cells,<br>
coordinate tuples,<br>
points, or vectors<br>
in the universe<br>
of discourse
| '''B'''<sup>''n''</sup>
|-
| ''A''*
| (hom : ''A'' &rarr; '''B''')
| Linear functions
| ('''B'''<sup>''n''</sup>)* = '''B'''<sup>''n''</sup>
|-
| ''A''^
| (''A'' &rarr; '''B''')
| Boolean functions
| '''B'''<sup>''n''</sup> &rarr; '''B'''
|-
| ''A''<sup>&bull;</sup>
|
[<font face="lucida calligraphy">A</font>]<br>
(''A'', ''A''^)<br>
(''A'' +&rarr; '''B''')<br>
(''A'', (''A'' &rarr; '''B'''))<br>
[''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>]
|
Universe of discourse<br>
based on the features<br>
{''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>}
|
('''B'''<sup>''n''</sup>, ('''B'''<sup>''n''</sup> &rarr; '''B'''))<br>
('''B'''<sup>''n''</sup> +&rarr; '''B''')<br>
['''B'''<sup>''n''</sup>]
|}</font><br>

===Table 3. Analogy of Real and Boolean Types===

<pre>
Table 3. Analogy of Real and Boolean Types
o-------------------------o-------------------------o-------------------------o
| Real Domain R | <-> | Boolean Domain B |
o-------------------------o-------------------------o-------------------------o
| R^n | Basic Space | B^n |
o-------------------------o-------------------------o-------------------------o
| R^n -> R | Function Space | B^n -> B |
o-------------------------o-------------------------o-------------------------o
| (R^n -> R) -> R | Tangent Vector | (B^n -> B) -> B |
o-------------------------o-------------------------o-------------------------o
| R^n -> ((R^n -> R) -> R)| Vector Field | B^n -> ((B^n -> B) -> B)|
o-------------------------o-------------------------o-------------------------o
| (R^n x (R^n -> R)) -> R | ditto | (B^n x (B^n -> B)) -> B |
o-------------------------o-------------------------o-------------------------o
| ((R^n -> R) x R^n) -> R | ditto | ((B^n -> B) x B^n) -> B |
o-------------------------o-------------------------o-------------------------o
| (R^n -> R) -> (R^n -> R)| Derivation | (B^n -> B) -> (B^n -> B)|
o-------------------------o-------------------------o-------------------------o
| R^n -> R^m | Basic Transformation | B^n -> B^m |
o-------------------------o-------------------------o-------------------------o
| (R^n -> R) -> (R^m -> R)| Function Transformation | (B^n -> B) -> (B^m -> B)|
o-------------------------o-------------------------o-------------------------o
| ... | ... | ... |
o-------------------------o-------------------------o-------------------------o
</pre>

<font face="courier new">
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|+ '''Table 3. Analogy of Real and Boolean Types'''
|- style="background:paleturquoise"
! Real Domain '''R'''
! &larr;&rarr;
! Boolean Domain '''B'''
|-
| '''R'''<sup>''n''</sup>
| Basic Space
| '''B'''<sup>''n''</sup>
|-
| '''R'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''R'''
| Function Space
| '''B'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''B'''
|-
| ('''R'''<sup>''n''</sup>&rarr;'''R''')&nbsp;&rarr;&nbsp;'''R'''
| Tangent Vector
| ('''B'''<sup>''n''</sup>&rarr;'''B''')&nbsp;&rarr;&nbsp;'''B'''
|-
| '''R'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;(('''R'''<sup>''n''</sup>&rarr;'''R''')&rarr;'''R''')
| Vector Field
| '''B'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;(('''B'''<sup>''n''</sup>&rarr;'''B''')&rarr;'''B''')
|-
| ('''R'''<sup>''n''</sup>&nbsp;&times;&nbsp;('''R'''<sup>''n''</sup>&rarr; '''R'''))&nbsp;&rarr;&nbsp;'''R'''
| ditto
| ('''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;('''B'''<sup>''n''</sup>&rarr; '''B'''))&nbsp;&rarr;&nbsp;'''B'''
|-
| (('''R'''<sup>''n''</sup>&rarr;'''R''')&nbsp;&times;&nbsp;'''R'''<sup>''n''</sup>)&nbsp;&rarr;&nbsp;'''R'''
| ditto
| (('''B'''<sup>''n''</sup>&rarr;'''B''')&nbsp;&times;&nbsp;'''B'''<sup>''n''</sup>)&nbsp;&rarr;&nbsp;'''B'''
|-
| ('''R'''<sup>''n''</sup>&rarr;'''R''')&nbsp;&rarr;&nbsp;('''R'''<sup>''n''</sup>&rarr;'''R''')
| Derivation
| ('''B'''<sup>''n''</sup>&rarr;'''B''')&nbsp;&rarr;&nbsp;('''B'''<sup>''n''</sup>&rarr;'''B''')
|-
| '''R'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''R'''<sup>''m''</sup>
| Basic Transformation
| '''B'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''B'''<sup>''m''</sup>
|-
| ('''R'''<sup>''n''</sup>&rarr;'''R''')&nbsp;&rarr;&nbsp;('''R'''<sup>''m''</sup>&rarr;'''R''')
| Function Transformation
| ('''B'''<sup>''n''</sup>&rarr;'''B''')&nbsp;&rarr;&nbsp;('''B'''<sup>''m''</sup>&rarr;'''B''')
|-
| ...
| ...
| ...
|}
</font><br>

===Table 4. An Equivalence Based on the Propositions as Types Analogy===

<pre>
Table 4. An Equivalence Based on the Propositions as Types Analogy
o-------------------------o------------------------o--------------------------o
| Pattern | Construction | Instance |
o-------------------------o------------------------o--------------------------o
| X -> (Y -> Z) | Vector Field | K^n -> ((K^n -> K) -> K) |
o-------------------------o------------------------o--------------------------o
| (X x Y) -> Z | | (K^n x (K^n -> K)) -> K |
o-------------------------o------------------------o--------------------------o
| (Y x X) -> Z | | ((K^n -> K) x K^n) -> K |
o-------------------------o------------------------o--------------------------o
| Y -> (X -> Z) | Derivation | (K^n -> K) -> (K^n -> K) |
o-------------------------o------------------------o--------------------------o
</pre>

<font face="courier new">
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
|+ '''Table 4. An Equivalence Based on the Propositions as Types Analogy
'''
|- style="background:paleturquoise"
! Pattern
! Construction
! Instance
|-
| ''X''&nbsp;&rarr;&nbsp;(''Y''&nbsp;&rarr;&nbsp;''Z'')
| Vector Field
| '''K'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;(('''K'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''K''')&nbsp;&rarr;&nbsp;'''K''')
|-
|(''X''&nbsp;&times;&nbsp;''Y'')&nbsp;&rarr;&nbsp;''Z''
| &nbsp;
| ('''K'''<sup>''n''</sup>&nbsp;&times;&nbsp;('''K'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''K'''))&nbsp;&rarr;&nbsp;'''K'''
|-
| (''Y''&nbsp;&times;&nbsp;''X'')&nbsp;&rarr;&nbsp;''Z''
| &nbsp;
| (('''K'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''K''')&nbsp;&times;&nbsp;'''K'''<sup>''n''</sup>)&nbsp;&rarr;&nbsp;'''K'''
|-
| ''Y''&nbsp;&rarr;&nbsp;(''X''&nbsp;&rarr;&nbsp;''Z'')
| Derivation
| ('''K'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''K''')&nbsp;&rarr;&nbsp;('''K'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''K''')
|}
</font><br>

===Table 5. A Bridge Over Troubled Waters===

<pre>
Table 5. A Bridge Over Troubled Waters
o-------------------------o-------------------------o-------------------------o
| Linear Space | Liminal Space | Logical Space |
o-------------------------o-------------------------o-------------------------o
| | | |
| !X! | !`X`! | !A! |
| | | |
| {x_1, ..., x_n} | {`x`_1, ..., `x`_n} | {a_1, ..., a_n} |
| | | |
| cardinality n | cardinality n | cardinality n |
o-------------------------o-------------------------o-------------------------o
| | | |
| X_i | `X`_i | A_i |
| | | |
| <|x_i|> | {(`x`_i), `x`_i} | {(a_i), a_i} |
| | | |
| isomorphic to K | isomorphic to B | isomorphic to B |
o-------------------------o-------------------------o-------------------------o
| | | |
| X | `X` | A |
| | | |
| <|!X!|> | <|!`X`!|> | <|!A!|> |
| | | |
| <|x_1, ..., x_n|> | <|`x`_1, ..., `x`_n|> | <|a_1, ..., a_n|> |
| | | |
| {<x_1, ..., x_n>} | {<`x`_1, ..., `x`_n>} | {<a_1, ..., a_n>} |
| | | |
| X_1 x ... x X_n | `X`_1 x ... x `X`_n | A_1 x ... x A_n |
| | | |
| Prod_i X_i | Prod_i `X`_i | Prod_i A_i |
| | | |
| isomorphic to K^n | isomorphic to B^n | isomorphic to B^n |
o-------------------------o-------------------------o-------------------------o
| | | |
| X* | `X`* | A* |
| | | |
| (hom : X -> K) | (hom : `X` -> B) | (hom : A -> B) |
| | | |
| isomorphic to K^n | isomorphic to B^n | isomorphic to B^n |
o-------------------------o-------------------------o-------------------------o
| | | |
| X^ | `X`^ | A^ |
| | | |
| (X -> K) | (`X` -> B) | (A -> B) |
| | | |
| isomorphic to (K^n -> K)| isomorphic to (B^n -> B)| isomorphic to (B^n -> B)|
o-------------------------o-------------------------o-------------------------o
| | | |
| X% | `X`% | A% |
| | | |
| [!X!] | [!`X`!] | [!A!] |
| | | |
| [x_1, ..., x_n] | [`x`_1, ..., `x`_n] | [a_1, ..., a_n] |
| | | |
| (X, X^) | (`X`, `X`^) | (A, A^) |
| | | |
| (X +-> K) | (`X` +-> B) | (A +-> B) |
| | | |
| (X, (X -> K)) | (`X`, (`X` -> B)) | (A, (A -> B)) |
| | | |
| isomorphic to: | isomorphic to: | isomorphic to: |
| | | |
| (K^n, (K^n -> K)) | (B^n, (B^n -> B)) | (B^n, (B^n -> K)) |
| | | |
| (K^n +-> K) | (B^n +-> B) | (B^n +-> B) |
| | | |
| [K^n] | [B^n] | [B^n] |
o-------------------------o-------------------------o-------------------------o
</pre>

<font face="courier new">
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
|+ '''Table 5. A Bridge Over Troubled Waters'''
|- style="background:paleturquoise"
! Linear Space
! Liminal Space
! Logical Space
|-
|
<font face="lucida calligraphy">X</font><br>
{''x''<sub>1</sub>, &hellip;, ''x''<sub>''n''</sub>}<br>
cardinality ''n''
|
<font face="lucida calligraphy"><u>X</u></font><br>
{<u>''x''</u><sub>1</sub>, &hellip;, <u>''x''</u><sub>''n''</sub>}<br>
cardinality ''n''
|
<font face="lucida calligraphy">A</font><br>
{''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>}<br>
cardinality ''n''
|-
|
''X''<sub>''i''</sub><br>
〈''x''<sub>''i''</sub>〉<br>
isomorphic to '''K'''
|
<u>''X''</u><sub>''i''</sub><br>
{(<u>''x''</u><sub>''i''</sub>), <u>''x''</u><sub>''i''</sub>}<br>
isomorphic to '''B'''
|
''A''<sub>''i''</sub><br>
{(''a''<sub>''i''</sub>), ''a''<sub>''i''</sub>}<br>
isomorphic to '''B'''
|-
|
''X''<br>
〈<font face="lucida calligraphy">X</font>〉<br>
〈''x''<sub>1</sub>, &hellip;, ''x''<sub>''n''</sub>〉<br>
{‹''x''<sub>1</sub>, &hellip;, ''x''<sub>''n''</sub>›}<br>
''X''<sub>1</sub> &times; &hellip; &times; ''X''<sub>''n''</sub><br>
&prod;<sub>''i''</sub> ''X''<sub>''i''</sub><br>
isomorphic to '''K'''<sup>''n''</sup>
|
<u>''X''</u><br>
〈<font face="lucida calligraphy"><u>X</u></font>〉<br>
〈<u>''x''</u><sub>1</sub>, &hellip;, <u>''x''</u><sub>''n''</sub>〉<br>
{‹<u>''x''</u><sub>1</sub>, &hellip;, <u>''x''</u><sub>''n''</sub>›}<br>
<u>''X''</u><sub>1</sub> &times; &hellip; &times; <u>''X''</u><sub>''n''</sub><br>
&prod;<sub>''i''</sub> <u>''X''</u><sub>''i''</sub><br>
isomorphic to '''B'''<sup>''n''</sup>
|
''A''<br>
〈<font face="lucida calligraphy">A</font>〉<br>
〈''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>〉<br>
{‹''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>›}<br>
''A''<sub>1</sub> &times; &hellip; &times; ''A''<sub>''n''</sub><br>
&prod;<sub>''i''</sub> ''A''<sub>''i''</sub><br>
isomorphic to '''B'''<sup>''n''</sup>
|-
|
''X''*<br>
(hom : ''X'' &rarr; '''K''')<br>
isomorphic to '''K'''<sup>''n''</sup>
|
<u>''X''</u>*<br>
(hom : <u>''X''</u> &rarr; '''B''')<br>
isomorphic to '''B'''<sup>''n''</sup>
|
''A''*<br>
(hom : ''A'' &rarr; '''B''')<br>
isomorphic to '''B'''<sup>''n''</sup>
|-
|
''X''^<br>
(''X'' &rarr; '''K''')<br>
isomorphic to:<br>
('''K'''<sup>''n''</sup> &rarr; '''K''')
|
<u>''X''</u>^<br>
(<u>''X''</u> &rarr; '''B''')<br>
isomorphic to:<br>
('''B'''<sup>''n''</sup> &rarr; '''B''')
|
''A''^<br>
(''A'' &rarr; '''B''')<br>
isomorphic to:<br>
('''B'''<sup>''n''</sup> &rarr; '''B''')
|-
|
''X''<sup>&bull;</sup><br>
[<font face="lucida calligraphy">X</font>]<br>
[''x''<sub>1</sub>, &hellip;, ''x''<sub>''n''</sub>]<br>
(''X'', ''X''^)<br>
(''X'' +&rarr; '''K''')<br>
(''X'', (''X'' &rarr; '''K'''))<br>
isomorphic to:<br>
('''K'''<sup>''n''</sup>, ('''K'''<sup>''n''</sup> &rarr; '''K'''))<br>
('''K'''<sup>''n''</sup> +&rarr; '''K''')<br>
['''K'''<sup>''n''</sup>]
|
<u>''X''</u><sup>&bull;</sup><br>
[<font face="lucida calligraphy"><u>X</u></font>]<br>
[<u>''x''</u><sub>1</sub>, &hellip;, <u>''x''</u><sub>''n''</sub>]<br>
(<u>''X''</u>, <u>''X''</u>^)<br>
(<u>''X''</u> +&rarr; '''B''')<br>
(<u>''X''</u>, (<u>''X''</u> &rarr; '''B'''))<br>
isomorphic to:<br>
('''B'''<sup>''n''</sup>, ('''B'''<sup>''n''</sup> &rarr; '''B'''))<br>
('''B'''<sup>''n''</sup> +&rarr; '''B''')<br>
['''B'''<sup>''n''</sup>]
|
''A''<sup>&bull;</sup><br>
[<font face="lucida calligraphy">A</font>]<br>
[''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>]<br>
(''A'', ''A''^)<br>
(''A'' +&rarr; '''B''')<br>
(''A'', (''A'' &rarr; '''B'''))<br>
isomorphic to:<br>
('''B'''<sup>''n''</sup>, ('''B'''<sup>''n''</sup> &rarr; '''B'''))<br>
('''B'''<sup>''n''</sup> +&rarr; '''B''')<br>
['''B'''<sup>''n''</sup>]
|}
</font><br>

===Table 6. Propositional Forms on One Variable===

<pre>
Table 6. Propositional Forms on One Variable
o---------o---------o---------o----------o------------------o----------o
| L_1 | L_2 | L_3 | L_4 | L_5 | L_6 |
| | | | | | |
| Decimal | Binary | Vector | Cactus | English | Ordinary |
o---------o---------o---------o----------o------------------o----------o
| | x : 1 0 | | | |
o---------o---------o---------o----------o------------------o----------o
| | | | | | |
| f_0 | f_00 | 0 0 | ( ) | false | 0 |
| | | | | | |
| f_1 | f_01 | 0 1 | (x) | not x | ~x |
| | | | | | |
| f_2 | f_10 | 1 0 | x | x | x |
| | | | | | |
| f_3 | f_11 | 1 1 | (( )) | true | 1 |
| | | | | | |
o---------o---------o---------o----------o------------------o----------o
</pre>

{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|+ '''Table 6. Propositional Forms on One Variable'''
|- style="background:paleturquoise"
! style="width:16%" | L<sub>1</sub><br>Decimal
! style="width:16%" | L<sub>2</sub><br>Binary
! style="width:16%" | L<sub>3</sub><br>Vector
! style="width:16%" | L<sub>4</sub><br>Cactus
! style="width:16%" | L<sub>5</sub><br>English
! style="width:16%" | L<sub>6</sub><br>Ordinary
|- style="background:paleturquoise"
| &nbsp;
| align="right" | x :
| 1 0
| &nbsp;
| &nbsp;
| &nbsp;
|-
| f<sub>0</sub>
| f<sub>00</sub>
| 0 0
| ( )
| false
| 0
|-
| f<sub>1</sub>
| f<sub>01</sub>
| 0 1
| (x)
| not x
| ~x
|-
| f<sub>2</sub>
| f<sub>10</sub>
| 1 0
| x
| x
| x
|-
| f<sub>3</sub>
| f<sub>11</sub>
| 1 1
| (( ))
| true
| 1
|}
<br>

===Table 7. Propositional Forms on Two Variables===

<pre>
Table 7. Propositional Forms on Two Variables
o---------o---------o---------o----------o------------------o----------o
| L_1 | L_2 | L_3 | L_4 | L_5 | L_6 |
| | | | | | |
| Decimal | Binary | Vector | Cactus | English | Ordinary |
o---------o---------o---------o----------o------------------o----------o
| | x : 1 1 0 0 | | | |
| | y : 1 0 1 0 | | | |
o---------o---------o---------o----------o------------------o----------o
| | | | | | |
| f_0 | f_0000 | 0 0 0 0 | () | false | 0 |
| | | | | | |
| f_1 | f_0001 | 0 0 0 1 | (x)(y) | neither x nor y | ~x & ~y |
| | | | | | |
| f_2 | f_0010 | 0 0 1 0 | (x) y | y and not x | ~x & y |
| | | | | | |
| f_3 | f_0011 | 0 0 1 1 | (x) | not x | ~x |
| | | | | | |
| f_4 | f_0100 | 0 1 0 0 | x (y) | x and not y | x & ~y |
| | | | | | |
| f_5 | f_0101 | 0 1 0 1 | (y) | not y | ~y |
| | | | | | |
| f_6 | f_0110 | 0 1 1 0 | (x, y) | x not equal to y | x + y |
| | | | | | |
| f_7 | f_0111 | 0 1 1 1 | (x y) | not both x and y | ~x v ~y |
| | | | | | |
| f_8 | f_1000 | 1 0 0 0 | x y | x and y | x & y |
| | | | | | |
| f_9 | f_1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y |
| | | | | | |
| f_10 | f_1010 | 1 0 1 0 | y | y | y |
| | | | | | |
| f_11 | f_1011 | 1 0 1 1 | (x (y)) | not x without y | x => y |
| | | | | | |
| f_12 | f_1100 | 1 1 0 0 | x | x | x |
| | | | | | |
| f_13 | f_1101 | 1 1 0 1 | ((x) y) | not y without x | x <= y |
| | | | | | |
| f_14 | f_1110 | 1 1 1 0 | ((x)(y)) | x or y | x v y |
| | | | | | |
| f_15 | f_1111 | 1 1 1 1 | (()) | true | 1 |
| | | | | | |
o---------o---------o---------o----------o------------------o----------o
</pre>

{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|+ '''Table 7. Propositional Forms on Two Variables'''
|- style="background:paleturquoise"
! style="width:16%" | L<sub>1</sub><br>Decimal
! style="width:16%" | L<sub>2</sub><br>Binary
! style="width:16%" | L<sub>3</sub><br>Vector
! style="width:16%" | L<sub>4</sub><br>Cactus
! style="width:16%" | L<sub>5</sub><br>English
! style="width:16%" | L<sub>6</sub><br>Ordinary
|- style="background:paleturquoise"
| &nbsp;
| align="right" | x :
| 1 1 0 0
| &nbsp;
| &nbsp;
| &nbsp;
|- style="background:paleturquoise"
| &nbsp;
| align="right" | y :
| 1 0 1 0
| &nbsp;
| &nbsp;
| &nbsp;
|-
| f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || (&nbsp;) || false || 0
|-
| f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || &not;x &and; &not;y
|-
| f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || &not;x &and; y
|-
| f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || &not;x
|-
| f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x &and; &not;y
|-
| f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || &not;y
|-
| f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x &ne; y
|-
| f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x&nbsp;y) || not both x and y || &not;x &or; &not;y
|-
| f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x&nbsp;y || x and y || x &and; y
|-
| f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y
|-
| f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y
|-
| f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x &rarr; y
|-
| f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x
|-
| f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x &larr; y
|-
| f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y || x &or; y
|-
| f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || ((&nbsp;)) || true || 1
|}
<br>

===Table 8. Notation for the Differential Extension of Propositional Calculus===

<pre>
Table 8. Notation for the Differential Extension of Propositional Calculus
o---------o-------------------o-------------------o-------------------o
| Symbol | Notation | Description | Type |
o---------o-------------------o-------------------o-------------------o
| d!A! | {da_1, ..., da_n} | Alphabet of | [n] = #n# |
| | | differential | |
| | | features | |
o---------o-------------------o-------------------o-------------------o
| dA_i | {(da_i), da_i} | Differential | D |
| | | dimension i | |
o---------o-------------------o-------------------o-------------------o
| dA | <|d!A!|> | Tangent space | D^n |
| | <|da_i,...,da_n|> | at a point: | |
| | {<da_i,...,da_n>} | Set of changes, | |
| | dA_1 x ... x dA_n | motions, steps, | |
| | Prod_i dA_i | tangent vectors | |
| | | at a point | |
o---------o-------------------o-------------------o-------------------o
| dA* | (hom : dA -> B) | Linear functions | (D^n)* ~=~ D^n |
| | | on dA | |
o---------o-------------------o-------------------o-------------------o
| dA^ | (dA -> B) | Boolean functions | D^n -> B |
| | | on dA | |
o---------o-------------------o-------------------o-------------------o
| dA% | [d!A!] | Tangent universe | (D^n, (D^n -> B)) |
| | (dA, dA^) | at a point of A%, | (D^n +-> B) |
| | (dA +-> B) | based on the | [D^n] |
| | (dA, (dA -> B)) | tangent features | |
| | [da_1, ..., da_n] | {da_1, ..., da_n} | |
o---------o-------------------o-------------------o-------------------o
</pre>

<font face="courier new">
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
|+ '''Table 8. Notation for the Differential Extension of Propositional Calculus'''
|- style="background:paleturquoise"
! Symbol
! Notation
! Description
! Type
|-
| d<font face="lucida calligraphy">A<font>
| {d''a''<sub>1</sub>, &hellip;, d''a''<sub>''n''</sub>}
|
Alphabet of<br>
differential<br>
features
| [''n''] = '''n'''
|-
| d''A''<sub>''i''</sub>
| {(d''a''<sub>''i''</sub>), d''a''<sub>''i''</sub>}
|
Differential<br>
dimension ''i''
| '''D'''
|-
| d''A''
|
〈d<font face="lucida calligraphy">A</font>〉<br>
〈d''a''<sub>1</sub>, &hellip;, d''a''<sub>''n''</sub>〉<br>
{‹d''a''<sub>1</sub>, &hellip;, d''a''<sub>''n''</sub>›}<br>
d''A''<sub>1</sub> &times; &hellip; &times; d''A''<sub>''n''</sub><br>
&prod;<sub>''i''</sub> d''A''<sub>''i''</sub>
|
Tangent space<br>
at a point:<br>
Set of changes,<br>
motions, steps,<br>
tangent vectors<br>
at a point
| '''D'''<sup>''n''</sup>
|-
| d''A''*
| (hom : d''A'' &rarr; '''B''')
|
Linear functions<br>
on d''A''
| ('''D'''<sup>''n''</sup>)* = '''D'''<sup>''n''</sup>
|-
| d''A''^
| (d''A'' &rarr; '''B''')
|
Boolean functions<br>
on d''A''
| '''D'''<sup>''n''</sup> &rarr; '''B'''
|-
| d''A''<sup>&bull;</sup>
|
[d<font face="lucida calligraphy">A</font>]<br>
(d''A'', d''A''^)<br>
(d''A'' +&rarr; '''B''')<br>
(d''A'', (d''A'' &rarr; '''B'''))<br>
[d''a''<sub>1</sub>, &hellip;, d''a''<sub>''n''</sub>]
|
Tangent universe<br>
at a point of ''A''<sup>&bull;</sup>,<br>
based on the<br>
tangent features<br>
{d''a''<sub>1</sub>, &hellip;, d''a''<sub>''n''</sub>}
|
('''D'''<sup>''n''</sup>, ('''D'''<sup>''n''</sup> &rarr; '''B'''))<br>
('''D'''<sup>''n''</sup> +&rarr; '''B''')<br>
['''D'''<sup>''n''</sup>]
|}
</font><br>

===Table 9. Higher Order Differential Features===

<pre>
Table 9. Higher Order Differential Features
o----------------------------------------o----------------------------------------o
| | |
| !A! = d^0.!A! = {a_1, ..., a_n} | E^0.!A! = d^0.!A! |
| | |
| d!A! = d^1.!A! = {da_1, ..., da_n} | E^1.!A! = d^0.!A! |_| d^1.!A! |
| | |
| d^k.!A! = {d^k.a_1,...,d^k.a_n}| E^k.!A! = d^0.!A! |_| ... |_| d^k.!A! |
| | |
| d*!A! = {d^0.!A!, ..., d^k.!A!, ...} | E^oo.!A! = |_| d*!A! |
| | |
o----------------------------------------o----------------------------------------o
</pre>

<font face="courier new">
{| align="center" border="1" cellpadding="10" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
|+ '''Table 9. Higher Order Differential Features'''
| width=50% |
<font face="lucida calligraphy">A</font> = d<sup>0</sup><font face="lucida calligraphy">A</font> = {''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>}<br><br>
d<font face="lucida calligraphy">A</font> = d<sup>1</sup><font face="lucida calligraphy">A</font> = {d''a''<sub>1</sub>, &hellip;, d''a''<sub>''n''</sub>}<br><br>
d<sup>''k''</sup><font face="lucida calligraphy">A</font> = {d<sup>''k''</sup>''a''<sub>''1''</sub>, &hellip;, d<sup>''k''</sup>''a''<sub>''n''</sub>}<br><br>
d<sup>*</sup><font face="lucida calligraphy">A</font> = {d<sup>0</sup><font face="lucida calligraphy">A</font>, &hellip;, d<sup>''k''</sup><font face="lucida calligraphy">A</font>, &hellip;}
| width=50% |
E<sup>0</sup><font face="lucida calligraphy">A</font> = d<sup>0</sup><font face="lucida calligraphy">A</font><br><br>
E<sup>1</sup><font face="lucida calligraphy">A</font> = d<sup>0</sup><font face="lucida calligraphy">A</font> &cup; d<sup>1</sup><font face="lucida calligraphy">A</font><br><br>
E<sup>''k''</sup><font face="lucida calligraphy">A</font> = d<sup>0</sup><font face="lucida calligraphy">A</font> &cup; &hellip; &cup; d<sup>''k''</sup><font face="lucida calligraphy">A</font><br><br>
E<sup>&infin;</sup><font face="lucida calligraphy">A</font> = &cup; d<sup>*</sup><font face="lucida calligraphy">A</font>
|}
</font><br>

<font face="courier new">
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
|+ '''Table 9. Higher Order Differential Features'''
| width=50% |
{| cellpadding="4" style="background:lightcyan"
| <font face="lucida calligraphy">A</font>
| =
| d<sup>0</sup><font face="lucida calligraphy">A</font>
| =
| {''a''<sub>1</sub>,
| &hellip;,
| ''a''<sub>''n''</sub>}
|-
| d<font face="lucida calligraphy">A</font>
| =
| d<sup>1</sup><font face="lucida calligraphy">A</font>
| =
| {d''a''<sub>1</sub>,
| &hellip;,
| d''a''<sub>''n''</sub>}
|-
| &nbsp;
| &nbsp;
| d<sup>''k''</sup><font face="lucida calligraphy">A</font>
| =
| {d<sup>''k''</sup>''a''<sub>''1''</sub>,
| &hellip;,
| d<sup>''k''</sup>''a''<sub>''n''</sub>}
|-
| d<sup>*</sup><font face="lucida calligraphy">A</font>
| =
| {d<sup>0</sup><font face="lucida calligraphy">A</font>,
| &hellip;,
| d<sup>''k''</sup><font face="lucida calligraphy">A</font>,
| &hellip;}
|}
| width=50% |
{| cellpadding="4" style="background:lightcyan"
| E<sup>0</sup><font face="lucida calligraphy">A</font>
| =
| d<sup>0</sup><font face="lucida calligraphy">A</font>
|-
| E<sup>1</sup><font face="lucida calligraphy">A</font>
| =
| d<sup>0</sup><font face="lucida calligraphy">A</font> &cup; d<sup>1</sup><font face="lucida calligraphy">A</font>
|-
| E<sup>''k''</sup><font face="lucida calligraphy">A</font>
| =
| d<sup>0</sup><font face="lucida calligraphy">A</font> &cup; &hellip; &cup; d<sup>''k''</sup><font face="lucida calligraphy">A</font>
|-
| E<sup>&infin;</sup><font face="lucida calligraphy">A</font>
| =
| &cup; d<sup>*</sup><font face="lucida calligraphy">A</font>
|}
|}
</font><br>

===Table 10. A Realm of Intentional Features===

<pre>
Table 10. A Realm of Intentional Features
o---------------------------------------o----------------------------------------o
| | |
| p^0.!A! = !A! = {a_1, ..., a_n} | Q^0.!A! = !A! |
| | |
| p^1.!A! = !A!' = {a_1', ..., a_n'} | Q^1.!A! = !A! |_| !A!' |
| | |
| p^2.!A! = !A!" = {a_1", ..., a_n"} | Q^2.!A! = !A! |_| !A!' |_| !A!" |
| | |
| ... ... ... | ... ... |
| | |
| p^k.!A! = {p^k.a_1, ..., p^k.a_n} | Q^k.!A! = !A! |_| ... |_| p^k.!A! |
| | |
o---------------------------------------o----------------------------------------o
</pre>

<font face="courier new">
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
|+ '''Table 10. A Realm of Intentional Features'''
| width=50% |
{| cellpadding="4" style="background:lightcyan"
| p<sup>0</sup><font face="lucida calligraphy">A</font>
| =
| <font face="lucida calligraphy">A</font>
| =
| {''a''<sub>1</sub>&nbsp;,
| &hellip;,
| ''a''<sub>''n''</sub>&nbsp;}
|-
| p<sup>1</sup><font face="lucida calligraphy">A</font>
| =
| <font face="lucida calligraphy">A</font>&prime;
| =
| {''a''<sub>1</sub>&prime;,
| &hellip;,
| ''a''<sub>''n''</sub>&prime;}
|-
| p<sup>2</sup><font face="lucida calligraphy">A</font>
| =
| <font face="lucida calligraphy">A</font>&Prime;
| =
| {''a''<sub>1</sub>&Prime;,
| &hellip;,
| ''a''<sub>''n''</sub>&Prime;}
|-
| ...
| &nbsp;
| &nbsp;
| &nbsp;
| ...
|-
| p<sup>''k''</sup><font face="lucida calligraphy">A</font>
| =
| &nbsp;
| &nbsp;
| {p<sup>''k''</sup>''a''<sub>1</sub>,
| &hellip;,
| p<sup>''k''</sup>''a''<sub>''n''</sub>}
|}
| width=50% |
{| cellpadding="4" style="background:lightcyan"
| Q<sup>0</sup><font face="lucida calligraphy">A</font>
| =
| <font face="lucida calligraphy">A</font>
|-
| Q<sup>1</sup><font face="lucida calligraphy">A</font>
| =
| <font face="lucida calligraphy">A</font> &cup; <font face="lucida calligraphy">A</font>&prime;
|-
| Q<sup>2</sup><font face="lucida calligraphy">A</font>
| =
| <font face="lucida calligraphy">A</font> &cup; <font face="lucida calligraphy">A</font>&prime; &cup; <font face="lucida calligraphy">A</font>&Prime;
|-
| ...
| &nbsp;
| ...
|-
| Q<sup>''k''</sup><font face="lucida calligraphy">A</font>
| =
| <font face="lucida calligraphy">A</font> &cup; <font face="lucida calligraphy">A</font>&prime; &cup; &hellip; &cup; p<sup>''k''</sup><font face="lucida calligraphy">A</font>
|}
|}
</font><br>

===Formula Display 1===

<pre>
o-------------------------------------------------o
| |
| From (A) & (dA) infer (A) next. |
| |
| From (A) & dA infer A next. |
| |
| From A & (dA) infer A next. |
| |
| From A & dA infer (A) next. |
| |
o-------------------------------------------------o
</pre>

<br><font face="courier new">
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
|
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
| &nbsp; || From || (''A'') || and || (d''A'') || infer || (''A'') || next. || &nbsp;
|-
| &nbsp; || From || (''A'') || and || d''A'' || infer || ''A'' || next. || &nbsp;
|-
| &nbsp; || From || ''A'' || and || (d''A'') || infer || ''A'' || next. || &nbsp;
|-
| &nbsp; || From || ''A'' || and || d''A'' || infer || (''A'') || next. || &nbsp;
|}
|}
</font><br>

===Table 11. A Pair of Commodious Trajectories===

<pre>
Table 11. A Pair of Commodious Trajectories
o---------o-------------------o-------------------o
| Time | Trajectory 1 | Trajectory 2 |
o---------o-------------------o-------------------o
| | | |
| 0 | A dA (d^2.A) | (A) (dA) d^2.A |
| | | |
| 1 | (A) dA d^2.A | (A) dA d^2.A |
| | | |
| 2 | A (dA) (d^2.A) | A (dA) (d^2.A) |
| | | |
| 3 | A (dA) (d^2.A) | A (dA) (d^2.A) |
| | | |
| 4 | " " " | " " " |
| | | |
o---------o-------------------o-------------------o
</pre>

<font face="courier new">
{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
|+ '''Table 11. A Pair of Commodious Trajectories'''
|- style="background:paleturquoise"
! Time
! Trajectory 1
! Trajectory 2
|-
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; text-align:center"
| 0
|-
| 1
|-
| 2
|-
| 3
|-
| 4
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; text-align:center"
| ''A'' || d''A'' || (d<sup>2</sup>''A'')
|-
| (''A'') || d''A'' || d<sup>2</sup>''A''
|-
| ''A'' || (d''A'') || (d<sup>2</sup>''A'')
|-
| ''A'' || (d''A'') || (d<sup>2</sup>''A'')
|-
| " || " || "
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; text-align:center"
| (''A'') || (d''A'') || d<sup>2</sup>''A''
|-
| (''A'') || d''A'' || d<sup>2</sup>''A''
|-
| ''A'' || (d''A'') || (d<sup>2</sup>''A'')
|-
| ''A'' || (d''A'') || (d<sup>2</sup>''A'')
|-
| " || " || "
|}
|}
</font><br>

===Figure 12. The Anchor===

<pre>
o-------------------------------------------------o
| E^2.X |
| |
| o-------------o |
| / \ |
| / A \ |
| / \ |
| / ->- \ |
| o / \ o |
| | \ / | |
| | -o- | |
| | ^ | |
| o---o---------o | o---------o---o |
| / \ \|/ / \ |
| / \ o | / \ |
| / \ | /|\ / \ |
| / \ | / | \ / \ |
| o o-|-o--|--o---o o |
| | | | | | | |
| | ---->o<----o | |
| | | | | |
| o dA o o d^2.A o |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| o-------------o o-------------o |
| |
| |
o-------------------------------------------------o
Figure 12. The Anchor
</pre>

===Figure 13. The Tiller===

<pre>
o-------------------------------------------------o
| |
| ->- |
| / \ |
| \ / |
| o-------------o -o- |
| / \ ^ |
| / dA \/ A |
| / /\ |
| / / \ |
| o o / o |
| | \ / | |
| | \ / | |
o------------|-------\-------/-------|------------o
| | \ / | |
| | \ / | |
| o v / o |
| \ o / |
| \ ^ / |
| \ | / d^2.A |
| \ | / |
| o------|------o |
| | |
| | |
| o |
| |
o-------------------------------------------------o
Figure 13. The Tiller
</pre>

===Table 14. Differential Propositions===

<pre>
Table 14. Differential Propositions
o-------o--------o---------o-----------o-------------------o----------o
| | A : 1 1 0 0 | | | |
| | dA : 1 0 1 0 | | | |
o-------o--------o---------o-----------o-------------------o----------o
| | | | | | |
| f_0 | g_0 | 0 0 0 0 | () | False | 0 |
| | | | | | |
o-------o--------o---------o-----------o-------------------o----------o
| | | | | | |
| | g_1 | 0 0 0 1 | (A)(dA) | Neither A nor dA | ~A & ~dA |
| | | | | | |
| | g_2 | 0 0 1 0 | (A) dA | Not A but dA | ~A & dA |
| | | | | | |
| | g_4 | 0 1 0 0 | A (dA) | A but not dA | A & ~dA |
| | | | | | |
| | g_8 | 1 0 0 0 | A dA | A and dA | A & dA |
| | | | | | |
o-------o--------o---------o-----------o-------------------o----------o
| | | | | | |
| f_1 | g_3 | 0 0 1 1 | (A) | Not A | ~A |
| | | | | | |
| f_2 | g_12 | 1 1 0 0 | A | A | A |
| | | | | | |
o-------o--------o---------o-----------o-------------------o----------o
| | | | | | |
| | g_6 | 0 1 1 0 | (A, dA) | A not equal to dA | A + dA |
| | | | | | |
| | g_9 | 1 0 0 1 | ((A, dA)) | A equal to dA | A = dA |
| | | | | | |
o-------o--------o---------o-----------o-------------------o----------o
| | | | | | |
| | g_5 | 0 1 0 1 | (dA) | Not dA | ~dA |
| | | | | | |
| | g_10 | 1 0 1 0 | dA | dA | dA |
| | | | | | |
o-------o--------o---------o-----------o-------------------o----------o
| | | | | | |
| | g_7 | 0 1 1 1 | (A dA) | Not both A and dA | ~A v ~dA |
| | | | | | |
| | g_11 | 1 0 1 1 | (A (dA)) | Not A without dA | A => dA |
| | | | | | |
| | g_13 | 1 1 0 1 | ((A) dA) | Not dA without A | A <= dA |
| | | | | | |
| | g_14 | 1 1 1 0 | ((A)(dA)) | A or dA | A v dA |
| | | | | | |
o-------o--------o---------o-----------o-------------------o----------o
| | | | | | |
| f_3 | g_15 | 1 1 1 1 | (()) | True | 1 |
| | | | | | |
o-------o--------o---------o-----------o-------------------o----------o
</pre>

{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|+ '''Table 14. Differential Propositions'''
|- style="background:paleturquoise"
| &nbsp;
| align="right" | A :
| 1 1 0 0
| &nbsp;
| &nbsp;
| &nbsp;
|- style="background:paleturquoise"
| &nbsp;
| align="right" | dA :
| 1 0 1 0
| &nbsp;
| &nbsp;
| &nbsp;
|-
| f<sub>0</sub>
| g<sub>0</sub>
| 0 0 0 0
| (&nbsp;)
| False
| 0
|-
| &nbsp;
| g<sub>1</sub>
| 0 0 0 1
| (A)(dA)
| Neither A nor dA
| &not;A &and; &not;dA
|-
| &nbsp;
| g<sub>2</sub>
| 0 0 1 0
| (A) dA
| Not A but dA
| &not;A &and; dA
|-
| &nbsp;
| g<sub>4</sub>
| 0 1 0 0
| A (dA)
| A but not dA
| A &and; &not;dA
|-
| &nbsp;
| g<sub>8</sub>
| 1 0 0 0
| A dA
| A and dA
| A &and; dA
|-
| f<sub>1</sub>
| g<sub>3</sub>
| 0 0 1 1
| (A)
| Not A
| &not;A
|-
| f<sub>2</sub>
| g<sub>12</sub>
| 1 1 0 0
| A
| A
| A
|-
| &nbsp;
| g<sub>6</sub>
| 0 1 1 0
| (A, dA)
| A not equal to dA
| A &ne; dA
|-
| &nbsp;
| g<sub>9</sub>
| 1 0 0 1
| ((A, dA))
| A equal to dA
| A = dA
|-
| &nbsp;
| g<sub>5</sub>
| 0 1 0 1
| (dA)
| Not dA
| &not;dA
|-
| &nbsp;
| g<sub>10</sub>
| 1 0 1 0
| dA
| dA
| dA
|-
| &nbsp;
| g<sub>7</sub>
| 0 1 1 1
| (A dA)
| Not both A and dA
| &not;A &or; &not;dA
|-
| &nbsp;
| g<sub>11</sub>
| 1 0 1 1
| (A (dA))
| Not A without dA
| A &rarr; dA
|-
| &nbsp;
| g<sub>13</sub>
| 1 1 0 1
| ((A) dA)
| Not dA without A
| A &larr; dA
|-
| &nbsp;
| g<sub>14</sub>
| 1 1 1 0
| ((A)(dA))
| A or dA
| A &or; dA
|-
| f<sub>3</sub>
| g<sub>15</sub>
| 1 1 1 1
| ((&nbsp;))
| True
| 1
|}
<br>

{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|+ '''Table 14. Differential Propositions'''
|- style="background:paleturquoise"
| &nbsp;
| align="right" | A :
| 1 1 0 0
| &nbsp;
| &nbsp;
| &nbsp;
|- style="background:paleturquoise"
| &nbsp;
| align="right" | dA :
| 1 0 1 0
| &nbsp;
| &nbsp;
| &nbsp;
|-
| f<sub>0</sub>
| g<sub>0</sub>
| 0 0 0 0
| (&nbsp;)
| False
| 0
|-
|
{| style="background:lightcyan"
|
&nbsp;<br>
&nbsp;<br>
&nbsp;<br>
&nbsp;
|}
|
{| style="background:lightcyan"
|
g<sub>1</sub><br>
g<sub>2</sub><br>
g<sub>4</sub><br>
g<sub>8</sub>
|}
|
{| style="background:lightcyan"
|
0 0 0 1<br>
0 0 1 0<br>
0 1 0 0<br>
1 0 0 0
|}
|
{| style="background:lightcyan"
|
(A)(dA)<br>
(A) dA <br>
A (dA)<br>
A dA
|}
|
{| style="background:lightcyan"
|
Neither A nor dA<br>
Not A but dA<br>
A but not dA<br>
A and dA
|}
|
{| style="background:lightcyan"
|
&not;A &and; &not;dA<br>
&not;A &and; dA<br>
A &and; &not;dA<br>
A &and; dA
|}
|-
|
{| style="background:lightcyan"
|
f<sub>1</sub><br>
f<sub>2</sub>
|}
|
{| style="background:lightcyan"
|
g<sub>3</sub><br>
g<sub>12</sub>
|}
|
{| style="background:lightcyan"
|
0 0 1 1<br>
1 1 0 0
|}
|
{| style="background:lightcyan"
|
(A)<br>
A
|}
|
{| style="background:lightcyan"
|
Not A<br>
A
|}
|
{| style="background:lightcyan"
|
&not;A<br>
A
|}
|-
|
{| style="background:lightcyan"
|
&nbsp;<br>
&nbsp;
|}
|
{| style="background:lightcyan"
|
g<sub>6</sub><br>
g<sub>9</sub>
|}
|
{| style="background:lightcyan"
|
0 1 1 0<br>
1 0 0 1
|}
|
{| style="background:lightcyan"
|
(A, dA)<br>
((A, dA))
|}
|
{| style="background:lightcyan"
|
A not equal to dA<br>
A equal to dA
|}
|
{| style="background:lightcyan"
|
A &ne; dA<br>
A = dA
|}
|-
|
{| style="background:lightcyan"
|
&nbsp;<br>
&nbsp;
|}
|
{| style="background:lightcyan"
|
g<sub>5</sub><br>
g<sub>10</sub>
|}
|
{| style="background:lightcyan"
|
0 1 0 1<br>
1 0 1 0
|}
|
{| style="background:lightcyan"
|
(dA)<br>
dA
|}
|
{| style="background:lightcyan"
|
Not dA<br>
dA
|}
|
{| style="background:lightcyan"
|
&not;dA<br>
dA
|}
|-
|
{| style="background:lightcyan"
|
&nbsp;<br>
&nbsp;<br>
&nbsp;<br>
&nbsp;
|}
|
{| style="background:lightcyan"
|
g<sub>7</sub><br>
g<sub>11</sub><br>
g<sub>13</sub><br>
g<sub>14</sub>
|}
|
{| style="background:lightcyan"
|
0 1 1 1<br>
1 0 1 1<br>
1 1 0 1<br>
1 1 1 0
|}
|
{| style="background:lightcyan"
|
(A dA)<br>
(A (dA))<br>
((A) dA)<br>
((A)(dA))
|}
|
{| style="background:lightcyan"
|
Not both A and dA<br>
Not A without dA<br>
Not dA without A<br>
A or dA
|}
|
{| style="background:lightcyan"
|
&not;A &or; &not;dA<br>
A &rarr; dA<br>
A &larr; dA<br>
A &or; dA
|}
|-
| f<sub>3</sub>
| g<sub>15</sub>
| 1 1 1 1
| ((&nbsp;))
| True
| 1
|}
<br>

===Table 15. Tacit Extension of [''A''] to [''A'', d''A'']===

<pre>
Table 15. Tacit Extension of [A] to [A, dA]
o---------------------------------------------------------------------o
| |
| 0 = 0 . ((dA), dA) = 0 |
| |
| (A) = (A) . ((dA), dA) = (A)(dA) + (A) dA |
| |
| A = A . ((dA), dA) = A (dA) + A dA |
| |
| 1 = 1 . ((dA), dA) = 1 |
| |
o---------------------------------------------------------------------o
</pre>

<font face="courier new">
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
|+ '''Table 15. Tacit Extension of [''A''] to [''A'', d''A'']'''
|
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
| &nbsp;
| 0
| =
| 0
| &middot;
| ((d''A''),&nbsp;d''A'')
| =
| 0
| &nbsp;
|-
| &nbsp;
| (''A'')
| =
| (''A'')
| &middot;
| ((d''A''),&nbsp;d''A'')
| =
| (''A'')(d''A'') + (''A'') d''A''&nbsp;
| &nbsp;
|-
| &nbsp;
| ''A''
| =
| ''A''
| &middot;
| ((d''A''),&nbsp;d''A'')
| =
| &nbsp;''A'' (d''A'') + &nbsp;''A''&nbsp;&nbsp;d''A''&nbsp;
| &nbsp;
|-
| &nbsp;
| 1
| =
| 1
| &middot;
| ((d''A''),&nbsp;d''A'')
| =
| 1
|}
|}
</font><br>

===Figure 16-a. A Couple of Fourth Gear Orbits: 1===

<pre>
o-------------------------------------------------o
| |
| o |
| / \ |
| / \ |
| / \ |
| / \ |
| o o |
| / \ / \ |
| / \ / \ |
| / \ / \ |
| / \ / \ |
| o o o |
| / \ / \ / \ |
| / \ / \ / \ |
| / \ / \ / \ |
| / \ / \ / \ |
| o 5 o 7 o o |
| / \ ^| / \ ^| / \ / \ |
| / \/ | / \/ | / \ / \ |
| / /\ | / /\ | / \ / \ |
| / / \|/ / \|/ \ / \ |
| o 4<---|----/----|----3 o o |
| |\ /|\ / /|\ ^ / \ /| |
| | \ / | \/ / | \/ / \ / | |
| | \ / | /\ / | /\ / \ / | |
| | \ / v/ \ / |/ \ / \ / | |
| | o 6 o | o o | |
| | |\ / \ /| / \ /| | |
| | | \ / \/ | / \ / | | |
| | | \ / /\ | / \ / | | |
| | d^0.A \ / / \|/ \ / d^1.A | |
| o----+----o 2<---|----1 o----+----o |
| | \ /|\ ^ / | |
| | \ / | \/ / | |
| | \ / | /\ / | |
| | d^2.A \ / v/ \ / d^3.A | |
| o---------o 0 o---------o |
| \ / |
| \ / |
| \ / |
| \ / |
| o |
| |
o-------------------------------------------------o
Figure 16-a. A Couple of Fourth Gear Orbits: 1
</pre>

===Figure 16-b. A Couple of Fourth Gear Orbits: 2===

<pre>
o-------------------------------------------------o
| |
| o |
| / \ |
| / \ |
| / \ |
| / \ |
| o 0 o |
| / \ / \ |
| / \ / \ |
| / \ / \ |
| / \ / \ |
| o 5 o 2 o |
| / \ / \ / \ |
| / \ / \ / \ |
| / \ / \ / \ |
| / \ / \ / \ |
| o o o 6 o |
| / \ / \ / \ / \ |
| / \ / \ / \ / \ |
| / \ / \ / \ / \ |
| / \ / \ / \ / \ |
| o o 7 o o 4 o |
| |\ / \ / \ / \ /| |
| | \ / \ / \ / \ / | |
| | \ / \ / \ / \ / | |
| | \ / \ / \ / \ / | |
| | o o 3 o 1 o | |
| | |\ / \ / \ /| | |
| | | \ / \ / \ / | | |
| | | \ / \ / \ / | | |
| | d^0.A \ / \ / \ / d^1.A | |
| o----+----o o o----+----o |
| | \ / \ / | |
| | \ / \ / | |
| | \ / \ / | |
| | d^2.A \ / \ / d^3.A | |
| o---------o o---------o |
| \ / |
| \ / |
| \ / |
| \ / |
| o |
| |
o-------------------------------------------------o
Figure 16-b. A Couple of Fourth Gear Orbits: 2
</pre>

===Formula Display 2===

<pre>
o-------------------------------------------------------------------------------o
| |
| r(q) = Sum_k d_k . 2^(-k) = Sum_k d^k.A(q) . 2^(-k) |
| |
| = |
| |
| s(q)/t = (Sum_k d_k . 2^(m-k)) / 2^m = (Sum_k d^k.A(q) . 2^(m-k)) / 2^m |
| |
o-------------------------------------------------------------------------------o
</pre>

<br><font face="courier new">
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
|
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
| ''r''(''q'')
| =
| &sum;<sub>''k''</sub> ''d''<sub>''k''</sub> &middot; 2<sup>-''k''</sup>
| =
| &sum;<sub>''k''</sub> d<sup>''k''</sup>''A''(''q'') &middot; 2<sup>-''k''</sup>
|-
| =
|-
| ''s''(''q'')/''t''
| =
| (&sum;<sub>''k''</sub> ''d''<sub>''k''</sub> &middot; 2<sup>(''m''-''k'')</sup>) / 2<sup>''m''</sup>
| =
| (&sum;<sub>''k''</sub> d<sup>''k''</sup>''A''(''q'') &middot; 2<sup>(''m''-''k'')</sup>) / 2<sup>''m''</sup>
|}
|}
</font><br>

<br><font face="courier new">
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
|
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
| <math>r(q)\!</math>
| <math>=</math>
| <math>\sum_k d_k \cdot 2^{-k}</math>
| <math>=</math>
| <math>\sum_k \mbox{d}^k A(q) \cdot 2^{-k}</math>
|-
| <math>=</math>
|-
| <math>\frac{s(q)}{t}</math>
| <math>=</math>
| <math>\frac{\sum_k d_k \cdot 2^{(m-k)}}{2^m}</math>
| <math>=</math>
| <math>\frac{\sum_k \mbox{d}^k A(q) \cdot 2^{(m-k)}}{2^m}</math>
|}
|}
</font><br>

===Table 17-a. A Couple of Orbits in Fourth Gear: Orbit 1===

<pre>
Table 17-a. A Couple of Orbits in Fourth Gear: Orbit 1
o---------o---------o---------o---------o---------o---------o---------o
| Time | State | A | dA | | | |
| p_i | q_j | d^0.A | d^1.A | d^2.A | d^3.A | d^4.A |
o---------o---------o---------o---------o---------o---------o---------o
| | | |
| p_0 | q_01 | 0. 0 0 0 1 |
| | | |
| p_1 | q_03 | 0. 0 0 1 1 |
| | | |
| p_2 | q_05 | 0. 0 1 0 1 |
| | | |
| p_3 | q_15 | 0. 1 1 1 1 |
| | | |
| p_4 | q_17 | 1. 0 0 0 1 |
| | | |
| p_5 | q_19 | 1. 0 0 1 1 |
| | | |
| p_6 | q_21 | 1. 0 1 0 1 |
| | | |
| p_7 | q_31 | 1. 1 1 1 1 |
| | | |
o---------o---------o---------o---------o---------o---------o---------o
</pre>

{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|+ '''Table 17-a. A Couple of Orbits in Fourth Gear: Orbit 1'''
|- style="background:paleturquoise"
| Time
| State
| ''A''
| d''A''
| &nbsp;
| &nbsp;
| &nbsp;
|- style="background:paleturquoise"
| ''p''<sub>''i''</sub>
| ''q''<sub>''j''</sub>
| d<sup>0</sup>''A''
| d<sup>1</sup>''A''
| d<sup>2</sup>''A''
| d<sup>3</sup>''A''
| d<sup>4</sup>''A''
|-
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center"
| ''p''<sub>0</sub>
|-
| ''p''<sub>1</sub>
|-
| ''p''<sub>2</sub>
|-
| ''p''<sub>3</sub>
|-
| ''p''<sub>4</sub>
|-
| ''p''<sub>5</sub>
|-
| ''p''<sub>6</sub>
|-
| ''p''<sub>7</sub>
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center"
| ''q''<sub>01</sub>
|-
| ''q''<sub>03</sub>
|-
| ''q''<sub>05</sub>
|-
| ''q''<sub>15</sub>
|-
| ''q''<sub>17</sub>
|-
| ''q''<sub>19</sub>
|-
| ''q''<sub>21</sub>
|-
| ''q''<sub>31</sub>
|}
| colspan="5" |
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0. || 0 || 0 || 0 || 1
|-
| 0. || 0 || 0 || 1 || 1
|-
| 0. || 0 || 1 || 0 || 1
|-
| 0. || 1 || 1 || 1 || 1
|-
| 1. || 0 || 0 || 0 || 1
|-
| 1. || 0 || 0 || 1 || 1
|-
| 1. || 0 || 1 || 0 || 1
|-
| 1. || 1 || 1 || 1 || 1
|}
|}
<br>

===Table 17-b. A Couple of Orbits in Fourth Gear: Orbit 2===

<pre>
Table 17-b. A Couple of Orbits in Fourth Gear: Orbit 2
o---------o---------o---------o---------o---------o---------o---------o
| Time | State | A | dA | | | |
| p_i | q_j | d^0.A | d^1.A | d^2.A | d^3.A | d^4.A |
o---------o---------o---------o---------o---------o---------o---------o
| | | |
| p_0 | q_25 | 1. 1 0 0 1 |
| | | |
| p_1 | q_11 | 0. 1 0 1 1 |
| | | |
| p_2 | q_29 | 1. 1 1 0 1 |
| | | |
| p_3 | q_07 | 0. 0 1 1 1 |
| | | |
| p_4 | q_09 | 0. 1 0 0 1 |
| | | |
| p_5 | q_27 | 1. 1 0 1 1 |
| | | |
| p_6 | q_13 | 0. 1 1 0 1 |
| | | |
| p_7 | q_23 | 1. 0 1 1 1 |
| | | |
o---------o---------o---------o---------o---------o---------o---------o
</pre>

{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|+ '''Table 17-b. A Couple of Orbits in Fourth Gear: Orbit 2'''
|- style="background:paleturquoise"
| Time
| State
| ''A''
| d''A''
| &nbsp;
| &nbsp;
| &nbsp;
|- style="background:paleturquoise"
| ''p''<sub>''i''</sub>
| ''q''<sub>''j''</sub>
| d<sup>0</sup>''A''
| d<sup>1</sup>''A''
| d<sup>2</sup>''A''
| d<sup>3</sup>''A''
| d<sup>4</sup>''A''
|-
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center"
| ''p''<sub>0</sub>
|-
| ''p''<sub>1</sub>
|-
| ''p''<sub>2</sub>
|-
| ''p''<sub>3</sub>
|-
| ''p''<sub>4</sub>
|-
| ''p''<sub>5</sub>
|-
| ''p''<sub>6</sub>
|-
| ''p''<sub>7</sub>
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center"
| ''q''<sub>25</sub>
|-
| ''q''<sub>11</sub>
|-
| ''q''<sub>29</sub>
|-
| ''q''<sub>07</sub>
|-
| ''q''<sub>09</sub>
|-
| ''q''<sub>27</sub>
|-
| ''q''<sub>13</sub>
|-
| ''q''<sub>23</sub>
|}
| colspan="5" |
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1. || 1 || 0 || 0 || 1
|-
| 0. || 1 || 0 || 1 || 1
|-
| 1. || 1 || 1 || 0 || 1
|-
| 0. || 0 || 1 || 1 || 1
|-
| 0. || 1 || 0 || 0 || 1
|-
| 1. || 1 || 0 || 1 || 1
|-
| 0. || 1 || 1 || 0 || 1
|-
| 1. || 0 || 1 || 1 || 1
|}
|}
<br>

===Figure 18-a. Extension from 1 to 2 Dimensions: Areal===

<pre>
o-----------------------------------------------------------o
| |
| o o |
| / \ / \ |
| / \ / \ |
| / \ / \ |
| / \ / \ |
| / o o 1 1 o |
| / / \ / \ / \ |
| / / \ / \ / \ |
| / 1 / \ / \ / \ |
| / / \ !e! / \ / \ |
| o / o ----> o 1 0 o 0 1 o |
| |\ / / |\ / \ /| |
| | \ / 0 / | \ / \ / | |
| | \ / / | \ / \ / | |
| |x_1\ / / |x_1\ / \ /x_2| |
| o----o / o----o 0 0 o----o |
| \ / \ / |
| \ / \ / |
| \ / \ / |
| \ / \ / |
| o o |
| |
o-----------------------------------------------------------o
Figure 18-a. Extension from 1 to 2 Dimensions: Areal
</pre>

===Figure 18-b. Extension from 1 to 2 Dimensions: Bundle===

<pre>
o-----------------------------o o-------------------o
| | | |
| | | o-------o |
| o---------o | | / \ |
| / \ | | o o |
| / o------------------------| | dx | |
| / \ | | o o |
| / \ | | \ / |
| o o | | o-------o |
| | | | | |
| | | | o-------------------o
| | x | |
| | | | o-------------------o
| | | | | |
| o o | | o-------o |
| \ / | | / \ |
| \ / | | o o |
| \ / o------------| | dx | |
| \ / | | o o |
| o---------o | | \ / |
| | | o-------o |
| | | |
o-----------------------------o o-------------------o
Figure 18-b. Extension from 1 to 2 Dimensions: Bundle
</pre>

===Figure 18-c. Extension from 1 to 2 Dimensions: Compact===

<pre>
o-----------------------------------------------------------o
| |
| |
| o-----------------o |
| / o \ |
| / (dx) / \ \ dx |
| / v o--------------------->o |
| / \ / \ |
| / o \ |
| o o |
| | | |
| | | |
| | x | (x) |
| | | |
| | | |
| o o |
| \ / o |
| \ / / \ |
| \ o<---------------------o v |
| \ / dx \ / (dx) |
| \ / o |
| o-----------------o |
| |
| |
o-----------------------------------------------------------o
Figure 18-c. Extension from 1 to 2 Dimensions: Compact
</pre>

===Figure 18-d. Extension from 1 to 2 Dimensions: Digraph===

<pre>
o-----------------------------------------------------------o
| |
| |
| dx |
| .--. .---------->----------. .--. |
| | \ / \ / | |
| (dx) ^ @ x (x) @ v (dx) |
| | / \ / \ | |
| *--* *----------<----------* *--* |
| dx |
| |
| |
o-----------------------------------------------------------o
Figure 18-d. Extension from 1 to 2 Dimensions: Digraph
</pre>

===Figure 19-a. Extension from 2 to 4 Dimensions: Areal===

<pre>
o-------------------------------------------------------------------------------o
| |
| o o |
| / \ / \ |
| / \ / \ |
| / \ / \ |
| / \ o 1100 o |
| / \ / \ / \ |
| / \ / \ / \ |
| / \ !e! / \ / \ |
| o 1 1 o ----> o 1101 o 1110 o |
| / \ / \ / \ / \ / \ |
| / \ / \ / \ / \ / \ |
| / \ / \ / \ / \ / \ |
| / \ / \ o 1001 o 1111 o 0110 o |
| / \ / \ / \ / \ / \ / \ |
| / \ / \ / \ / \ / \ / \ |
| / \ / \ / \ / \ / \ / \ |
| o 1 0 o 0 1 o o 1000 o 1011 o 0111 o 0100 o |
| |\ / \ /| |\ / \ / \ / \ /| |
| | \ / \ / | | \ / \ / \ / \ / | |
| | \ / \ / | | \ / \ / \ / \ / | |
| | \ / \ / | | o 1010 o 0011 o 0101 o | |
| | \ / \ / | | |\ / \ / \ /| | |
| | \ / \ / | | | \ / \ / \ / | | |
| | x_1 \ / \ / x_2 | |x_1| \ / \ / \ / |x_2| |
| o-------o 0 0 o-------o o---+---o 0010 o 0001 o---+---o |
| \ / | \ / \ / | |
| \ / | \ / \ / | |
| \ / | x_3 \ / \ / x_4 | |
| \ / o-------o 0000 o-------o |
| \ / \ / |
| \ / \ / |
| \ / \ / |
| o o |
| |
o-------------------------------------------------------------------------------o
Figure 19-a. Extension from 2 to 4 Dimensions: Areal
</pre>

===Figure 19-b. Extension from 2 to 4 Dimensions: Bundle===

<pre>
o-----------------------------o
| o-----o o-----o |
| / \ / \ |
| / o \ |
| / / \ \ |
| o o o o |
@ | du | | dv | |
/| o o o o |
/ | \ \ / / |
/ | \ o / |
/ | \ / \ / |
/ | o-----o o-----o |
/ o-----------------------------o
/
o-----------------------------------------/---o o-----------------------------o
| / | | o-----o o-----o |
| @ | | / \ / \ |
| o---------o o---------o | | / o \ |
| / \ / \ | | / / \ \ |
| / o \ | | o o o o |
| / / \ @-------\-----------@ | du | | dv | |
| / / @ \ \ | | o o o o |
| / / \ \ \ | | \ \ / / |
| / / \ \ \ | | \ o / |
| o o \ o o | | \ / \ / |
| | | \| | | | o-----o o-----o |
| | | | | | o-----------------------------o
| | u | |\ v | |
| | | | \ | | o-----------------------------o
| | | | \ | | | o-----o o-----o |
| o o o \ o | | / \ / \ |
| \ \ / \ / | | / o \ |
| \ \ / \ / | | / / \ \ |
| \ \ / \ / | | o o o o |
| \ @-----\-/-----------\-------------@ | du | | dv | |
| \ o / | | o o o o |
| \ / \ / \ | | \ \ / / |
| o---------o o---------o \ | | \ o / |
| \ | | \ / \ / |
| \ | | o-----o o-----o |
o-----------------------------------------\---o o-----------------------------o
\
\ o-----------------------------o
\ | o-----o o-----o |
\ | / \ / \ |
\ | / o \ |
\ | / / \ \ |
\| o o o o |
@ | du | | dv | |
| o o o o |
| \ \ / / |
| \ o / |
| \ / \ / |
| o-----o o-----o |
o-----------------------------o
Figure 19-b. Extension from 2 to 4 Dimensions: Bundle
</pre>

===Figure 19-c. Extension from 2 to 4 Dimensions: Compact===

<pre>
o---------------------------------------------------------------------o
| |
| |
| o-------------------o o-------------------o |
| / \ / \ |
| / o \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| o o (du).(dv) o o |
| | | -->-- | | |
| | | \ / | | |
| | dv .(du) | \ / | du .(dv) | |
| | u <---------------@---------------> v | |
| | | | | | |
| | | | | | |
| | | | | | |
| o o | o o |
| \ \ | / / |
| \ \ | / / |
| \ \ | / / |
| \ \ | / / |
| \ \|/ / |
| \ | / |
| \ /|\ / |
| o-------------------o | o-------------------o |
| | |
| du . dv |
| | |
| V |
| |
o---------------------------------------------------------------------o
Figure 19-c. Extension from 2 to 4 Dimensions: Compact
</pre>

===Figure 19-d. Extension from 2 to 4 Dimensions: Digraph===

<pre>
o-----------------------------------------------------------o
| |
| .->-. |
| | | |
| * * |
| \ / |
| .-->--@--<--. |
| / / \ \ |
| / / \ \ |
| / . . \ |
| / | | \ |
| / | | \ |
| / | | \ |
| . | | . |
| | | | | |
| v | | v |
| .--. | .---------->----------. | .--. |
| | \|/ | | \|/ | |
| ^ @ ^ v @ v |
| | /|\ | | /|\ | |
| *--* | *----------<----------* | *--* |
| ^ | | ^ |
| | | | | |
| * | | * |
| \ | | / |
| \ | | / |
| \ | | / |
| \ . . / |
| \ \ / / |
| \ \ / / |
| *-->--@--<--* |
| / \ |
| . . |
| | | |
| *-<-* |
| |
o-----------------------------------------------------------o
Figure 19-d. Extension from 2 to 4 Dimensions: Digraph
</pre>

===Figure 20-i. Thematization of Conjunction (Stage 1)===

<pre>
o-------------------------------o o-------------------------------o
| | | |
| o-----o o-----o | | o-----o o-----o |
| / \ / \ | | / \ / \ |
| / o \ | | / o \ |
| / /`\ \ | | / /`\ \ |
| o o```o o | | o o```o o |
| | u |```| v | | | | u |```| v | |
| o o```o o | | o o```o o |
| \ \`/ / | | \ \`/ / |
| \ o / | | \ o / |
| \ / \ / | | \ / \ / |
| o-----o o-----o | | o-----o o-----o |
| | | |
o-------------------------------o o-------------------------------o
\ /
\ /
\ /
u v \ J /
\ /
\ /
\ /
\ /
o
Figure 20-i. Thematization of Conjunction (Stage 1)
</pre>

===Figure 20-ii. Thematization of Conjunction (Stage 2)===

<pre>
o-------------------------------o o-------------------------------o
| | | |
| o-----o o-----o | | o-----o o-----o |
| / \ / \ | | / \ / \ |
| / o \ | | / o \ |
| / /`\ \ | | / /`\ \ |
| o o```o o | | o o```o o |
| | u |```| v | | | | u |```| v | |
| o o```o o | | o o```o o |
| \ \`/ / | | \ \`/ / |
| \ o / | | \ o / |
| \ / \ / | | \ / \ / |
| o-----o o-----o | | o-----o o-----o |
| | | |
o-------------------------------o o-------------------------------o
\ / \ /
\ / \ /
\ / \ J /
\ / \ /
\ / \ /
o----------\---------/----------o o----------\---------/----------o
| \ / | | \ / |
| \ / | | \ / |
| o-----@-----o | | o-----@-----o |
| /`````````````\ | | /`````````````\ |
| /```````````````\ | | /```````````````\ |
| /`````````````````\ | | /`````````````````\ |
| o```````````````````o | | o```````````````````o |
| |```````````````````| | | |```````````````````| |
| |```````` J ````````| | | |```````` x ````````| |
| |```````````````````| | | |```````````````````| |
| o```````````````````o | | o```````````````````o |
| \`````````````````/ | | \`````````````````/ |
| \```````````````/ | | \```````````````/ |
| \`````````````/ | | \`````````````/ |
| o-----------o | | o-----------o |
| | | |
| | | |
o-------------------------------o o-------------------------------o
J = u v x = J<u, v>

Figure 20-ii. Thematization of Conjunction (Stage 2)
</pre>

===Figure 20-iii. Thematization of Conjunction (Stage 3)===

<pre>
o-------------------------------o o-------------------------------o
| | |```````````````````````````````|
| | |````````````o-----o````````````|
| | |```````````/ \```````````|
| | |``````````/ \``````````|
| | |`````````/ \`````````|
| | |````````/ \````````|
| J | |```````o x o```````|
| | |```````| |```````|
| | |```````| |```````|
| | |```````| |```````|
| o-----o o-----o | |```````o-----o o-----o```````|
| / \ / \ | |``````/`\ \ / /`\``````|
| / o \ | |`````/```\ o /```\`````|
| / /`\ \ | |````/`````\ /`\ /`````\````|
| / /```\ \ | |```/```````\ /```\ /```````\```|
| o o`````o o | |``o`````````o-----o`````````o``|
| | u |`````| v | | |``|`````````| |`````````|``|
o--o---------o-----o---------o--o |``|``` u ```| |``` v ```|``|
|``|`````````| |`````````|``| |``|`````````| |`````````|``|
|``o`````````o o`````````o``| |``o`````````o o`````````o``|
|```\`````````\ /`````````/```| |```\`````````\ /`````````/```|
|````\`````````\ /`````````/````| |````\`````````\ /`````````/````|
|`````\`````````o`````````/`````| |`````\`````````o`````````/`````|
|``````\```````/`\```````/``````| |``````\```````/`\```````/``````|
|```````o-----o```o-----o```````| |```````o-----o```o-----o```````|
|```````````````````````````````| |```````````````````````````````|
o-------------------------------o o-------------------------------o
\ /
\ /
J = u v \ /
\ !j! /
\ /
!j! = (( x , u v )) \ /
\ /
\ /
@
Figure 20-iii. Thematization of Conjunction (Stage 3)
</pre>

===Figure 21. Thematization of Disjunction and Equality===

<pre>
f g
o-------------------------------o o-------------------------------o
| | |```````````````````````````````|
| o-----o o-----o | |```````o-----o```o-----o```````|
| /```````\ /```````\ | |``````/ \`/ \``````|
| /`````````o`````````\ | |`````/ o \`````|
| /`````````/`\`````````\ | |````/ /`\ \````|
| /`````````/```\`````````\ | |```/ /```\ \```|
| o`````````o`````o```````` o | |``o o`````o o``|
| |`````````|`````|`````````| | |``| |`````| |``|
| |``` u ```|`````|``` v ```| | |``| u |`````| v |``|
| |`````````|`````|`````````| | |``| |`````| |``|
| o`````````o`````o`````````o | |``o o`````o o``|
| \`````````\```/`````````/ | |```\ \```/ /```|
| \`````````\`/`````````/ | |````\ \`/ /````|
| \`````````o`````````/ | |`````\ o /`````|
| \```````/ \```````/ | |``````\ /`\ /``````|
| o-----o o-----o | |```````o-----o```o-----o```````|
| | |```````````````````````````````|
o-------------------------------o o-------------------------------o
((u)(v)) ((u , v))

| |
| |
theta theta
| |
| |
v v

!f! !g!
o-------------------------------o o-------------------------------o
|```````````````````````````````| | |
|````````````o-----o````````````| | o-----o |
|```````````/ \```````````| | /```````\ |
|``````````/ \``````````| | /`````````\ |
|`````````/ \`````````| | /```````````\ |
|````````/ \````````| | /`````````````\ |
|```````o f o```````| | o`````` g ``````o |
|```````| |```````| | |```````````````| |
|```````| |```````| | |```````````````| |
|```````| |```````| | |```````````````| |
|```````o-----o o-----o```````| | o-----o```o-----o |
|``````/ \`````\ /`````/ \``````| | /`\ \`/ /`\ |
|`````/ \`````o`````/ \`````| | /```\ o /```\ |
|````/ \```/`\```/ \````| | /`````\ /`\ /`````\ |
|```/ \`/```\`/ \```| | /```````\ /```\ /```````\ |
|``o o-----o o``| | o`````````o-----o`````````o |
|``| | | |``| | |`````````| |`````````| |
|``| u | | v |``| | |``` u ```| |``` v ```| |
|``| | | |``| | |`````````| |`````````| |
|``o o o o``| | o`````````o o`````````o |
|```\ \ / /```| | \`````````\ /`````````/ |
|````\ \ / /````| | \`````````\ /`````````/ |
|`````\ o /`````| | \`````````o`````````/ |
|``````\ /`\ /``````| | \```````/ \```````/ |
|```````o-----o```o-----o```````| | o-----o o-----o |
|```````````````````````````````| | |
o-------------------------------o o-------------------------------o
((f , ((u)(v)) )) ((g , ((u , v)) ))

Figure 21. Thematization of Disjunction and Equality
</pre>

===Table 22. Disjunction ''f'' and Equality ''g''===

<pre>
Table 22. Disjunction f and Equality g
o-------------------o-------------------o
| u v | f g |
o-------------------o-------------------o
| | |
| 0 0 | 0 1 |
| | |
| 0 1 | 1 0 |
| | |
| 1 0 | 1 0 |
| | |
| 1 1 | 1 1 |
| | |
o-------------------o-------------------o
</pre>

<font face="courier new">
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|+ '''Table 22. Disjunction ''f'' and Equality ''g'' '''
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| ''u'' || ''v''
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| ''f'' || ''g''
|}
|-
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 0 || 1
|-
| 1 || 0
|-
| 1 || 1
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 1
|-
| 1 || 0
|-
| 1 || 0
|-
| 1 || 1
|}
|}
</font><br>

===Tables 23-i and 23-ii. Thematics of Disjunction and Equality (1)===

<pre>
Tables 23-i and 23-ii. Thematics of Disjunction and Equality (1)
o-----------------o-----------o o-----------------o-----------o
| u v f | x !f! | | u v g | y !g! |
o-----------------o-----------o o-----------------o-----------o
| | | | | |
| 0 0 --> | 0 1 | | 0 0 --> | 1 1 |
| | | | | |
| 0 1 --> | 1 1 | | 0 1 --> | 0 1 |
| | | | | |
| 1 0 --> | 1 1 | | 1 0 --> | 0 1 |
| | | | | |
| 1 1 --> | 1 1 | | 1 1 --> | 1 1 |
| | | | | |
o-----------------o-----------o o-----------------o-----------o
| | | | | |
| 0 0 | 1 0 | | 0 0 | 0 0 |
| | | | | |
| 0 1 | 0 0 | | 0 1 | 1 0 |
| | | | | |
| 1 0 | 0 0 | | 1 0 | 1 0 |
| | | | | |
| 1 1 | 0 0 | | 1 1 | 0 0 |
| | | | | |
o-----------------o-----------o o-----------------o-----------o
</pre>

{| align="center" style="width:96%"
|+ '''Tables 23-i and 23-ii. Thematics of Disjunction and Equality (1)'''
|
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ '''Table 23-i. Disjunction ''f'' '''
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| ''u'' || ''v'' || ''f''
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| ''x'' || &phi;
|}
|-
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0 || &rarr;
|-
| 0 || 1 || &rarr;
|-
| 1 || 0 || &rarr;
|-
| 1 || 1 || &rarr;
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 1
|-
| 1 || 1
|-
| 1 || 1
|-
| 1 || 1
|}
|-
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0 || &nbsp;&nbsp;
|-
| 0 || 1 || &nbsp;&nbsp;
|-
| 1 || 0 || &nbsp;&nbsp;
|-
| 1 || 1 || &nbsp;&nbsp;
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1 || 0
|-
| 0 || 0
|-
| 0 || 0
|-
| 0 || 0
|}
|}
|
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ '''Table 23-ii. Equality ''g'' '''
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| ''u'' || ''v'' || ''g''
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| ''y'' || &gamma;
|}
|-
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0 || &rarr;
|-
| 0 || 1 || &rarr;
|-
| 1 || 0 || &rarr;
|-
| 1 || 1 || &rarr;
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1 || 1
|-
| 0 || 1
|-
| 0 || 1
|-
| 1 || 1
|}
|-
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0 || &nbsp;&nbsp;
|-
| 0 || 1 || &nbsp;&nbsp;
|-
| 1 || 0 || &nbsp;&nbsp;
|-
| 1 || 1 || &nbsp;&nbsp;
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 1 || 0
|-
| 1 || 0
|-
| 0 || 0
|}
|}
|}
<br>

===Tables 24-i and 24-ii. Thematics of Disjunction and Equality (2)===

<pre>
Tables 24-i and 24-ii. Thematics of Disjunction and Equality (2)
o-----------------------o-----o o-----------------------o-----o
| u v f x | !f! | | u v g y | !g! |
o-----------------------o-----o o-----------------------o-----o
| | | | | |
| 0 0 --> 0 | 1 | | 0 0 0 | 0 |
| | | | | |
| 0 0 1 | 0 | | 0 0 --> 1 | 1 |
| | | | | |
| 0 1 0 | 0 | | 0 1 --> 0 | 1 |
| | | | | |
| 0 1 --> 1 | 1 | | 0 1 1 | 0 |
| | | | | |
o-----------------------o-----o o-----------------------o-----o
| | | | | |
| 1 0 0 | 0 | | 1 0 --> 0 | 1 |
| | | | | |
| 1 0 --> 1 | 1 | | 1 0 1 | 0 |
| | | | | |
| 1 1 0 | 0 | | 1 1 0 | 0 |
| | | | | |
| 1 1 --> 1 | 1 | | 1 1 --> 1 | 1 |
| | | | | |
o-----------------------o-----o o-----------------------o-----o
</pre>

{| align="center" style="width:96%"
|+ '''Tables 24-i and 24-ii. Thematics of Disjunction and Equality (2)'''
|
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ '''Table 24-i. Disjunction ''f'' '''
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| ''u'' || ''v'' || ''f'' || ''x''
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| &phi;
|}
|-
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0 || &rarr; || 0
|-
| 0 || 0 || &nbsp;&nbsp; || 1
|-
| 0 || 1 || &nbsp;&nbsp; || 0
|-
| 0 || 1 || &rarr; || 1
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1
|-
| 0
|-
| 0
|-
| 1
|}
|-
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1 || 0 || &nbsp;&nbsp; || 0
|-
| 1 || 0 || &rarr; || 1
|-
| 1 || 1 || &nbsp;&nbsp; || 0
|-
| 1 || 1 || &rarr; || 1
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0
|-
| 1
|-
| 0
|-
| 1
|}
|}
|
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ '''Table 24-ii. Equality ''g'' '''
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| ''u'' || ''v'' || ''g'' || ''y''
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| &gamma;
|}
|-
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0 || &nbsp;&nbsp; || 0
|-
| 0 || 0 || &rarr; || 1
|-
| 0 || 1 || &rarr; || 0
|-
| 0 || 1 || &nbsp;&nbsp; || 1
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0
|-
| 1
|-
| 1
|-
| 0
|}
|-
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1 || 0 || &rarr; || 0
|-
| 1 || 0 || &nbsp;&nbsp; || 1
|-
| 1 || 1 || &nbsp;&nbsp; || 0
|-
| 1 || 1 || &rarr; || 1
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1
|-
| 0
|-
| 0
|-
| 1
|}
|}
|}
<br>

===Tables 25-i and 25-ii. Thematics of Disjunction and Equality (3)===

<pre>
Tables 25-i and 25-ii. Thematics of Disjunction and Equality (3)
o-----------------------o-----o o-----------------------o-----o
| u v f x | !f! | | u v g y | !g! |
o-----------------------o-----o o-----------------------o-----o
| | | | | |
| 0 0 --> 0 | 1 | | 0 0 0 | 0 |
| | | | | |
| 0 1 0 | 0 | | 0 1 --> 0 | 1 |
| | | | | |
| 1 0 0 | 0 | | 1 0 --> 0 | 1 |
| | | | | |
| 1 1 0 | 0 | | 1 1 0 | 0 |
| | | | | |
o-----------------------o-----o o-----------------------o-----o
| | | | | |
| 0 0 1 | 0 | | 0 0 --> 1 | 1 |
| | | | | |
| 0 1 --> 1 | 1 | | 0 1 1 | 0 |
| | | | | |
| 1 0 --> 1 | 1 | | 1 0 1 | 0 |
| | | | | |
| 1 1 --> 1 | 1 | | 1 1 --> 1 | 1 |
| | | | | |
o-----------------------o-----o o-----------------------o-----o
</pre>

{| align="center" style="width:96%"
|+ '''Tables 25-i and 25-ii. Thematics of Disjunction and Equality (3)'''
|
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ '''Table 25-i. Disjunction ''f'' '''
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| ''u'' || ''v'' || ''f'' || ''x''
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| &phi;
|}
|-
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0 || &rarr; || 0
|-
| 0 || 1 || &nbsp;&nbsp; || 0
|-
| 1 || 0 || &nbsp;&nbsp; || 0
|-
| 1 || 1 || &nbsp;&nbsp; || 0
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1
|-
| 0
|-
| 0
|-
| 0
|}
|-
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0 || &nbsp;&nbsp; || 1
|-
| 0 || 1 || &rarr; || 1
|-
| 1 || 0 || &rarr; || 1
|-
| 1 || 1 || &rarr; || 1
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0
|-
| 1
|-
| 1
|-
| 1
|}
|}
|
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ '''Table 25-ii. Equality ''g'' '''
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| ''u'' || ''v'' || ''g'' || ''y''
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| &gamma;
|}
|-
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0 || &nbsp;&nbsp; || 0
|-
| 0 || 1 || &rarr; || 0
|-
| 1 || 0 || &rarr; || 0
|-
| 1 || 1 || &nbsp;&nbsp; || 0
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0
|-
| 1
|-
| 1
|-
| 0
|}
|-
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0 || &rarr; || 1
|-
| 0 || 1 || &nbsp;&nbsp; || 1
|-
| 1 || 0 || &nbsp;&nbsp; || 1
|-
| 1 || 1 || &rarr; || 1
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1
|-
| 0
|-
| 0
|-
| 1
|}
|}
|}
<br>

===Tables 26-i and 26-ii. Tacit Extension and Thematization===

<pre>
Tables 26-i and 26-ii. Tacit Extension and Thematization
o-----------------o-----------o o-----------------o-----------o
| u v x | !e!f !f! | | u v y | !e!g !g! |
o-----------------o-----------o o-----------------o-----------o
| | | | | |
| 0 0 0 | 0 1 | | 0 0 0 | 1 0 |
| | | | | |
| 0 0 1 | 0 0 | | 0 0 1 | 1 1 |
| | | | | |
| 0 1 0 | 1 0 | | 0 1 0 | 0 1 |
| | | | | |
| 0 1 1 | 1 1 | | 0 1 1 | 0 0 |
| | | | | |
o-----------------o-----------o o-----------------o-----------o
| | | | | |
| 1 0 0 | 1 0 | | 1 0 0 | 0 1 |
| | | | | |
| 1 0 1 | 1 1 | | 1 0 1 | 0 0 |
| | | | | |
| 1 1 0 | 1 0 | | 1 1 0 | 1 0 |
| | | | | |
| 1 1 1 | 1 1 | | 1 1 1 | 1 1 |
| | | | | |
o-----------------o-----------o o-----------------o-----------o
</pre>

{| align="center" style="width:96%"
|+ '''Tables 26-i and 26-ii. Tacit Extension and Thematization'''
|
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ '''Table 26-i. Disjunction ''f'' '''
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| ''u'' || ''v'' || ''x''
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| &epsilon;''f'' || &theta;''f''
|}
|-
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0 || 0
|-
| 0 || 0 || 1
|-
| 0 || 1 || 0
|-
| 0 || 1 || 1
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 1
|-
| 0 || 0
|-
| 1 || 0
|-
| 1 || 1
|}
|-
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1 || 0 || 0
|-
| 1 || 0 || 1
|-
| 1 || 1 || 0
|-
| 1 || 1 || 1
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1 || 0
|-
| 1 || 1
|-
| 1 || 0
|-
| 1 || 1
|}
|}
|
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ '''Table 26-ii. Equality ''g'' '''
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| ''u'' || ''v'' || ''y''
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| &epsilon;''g'' || &theta;''g''
|}
|-
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0 || 0
|-
| 0 || 0 || 1
|-
| 0 || 1 || 0
|-
| 0 || 1 || 1
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1 || 0
|-
| 1 || 1
|-
| 0 || 1
|-
| 0 || 0
|}
|-
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1 || 0 || 0
|-
| 1 || 0 || 1
|-
| 1 || 1 || 0
|-
| 1 || 1 || 1
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 1
|-
| 0 || 0
|-
| 1 || 0
|-
| 1 || 1
|}
|}
|}
<br>

===Table 27. Thematization of Bivariate Propositions===

<pre>
Table 27. Thematization of Bivariate Propositions
o---------o---------o----------o--------------------o--------------------o
| u : 1 1 0 0 | f | theta (f) | theta (f) |
| v : 1 0 1 0 | | | |
o---------o---------o----------o--------------------o--------------------o
| | | | | |
| f_0 | 0 0 0 0 | () | (( f , () )) | f + 1 |
| | | | | |
| f_1 | 0 0 0 1 | (u)(v) | (( f , (u)(v) )) | f + u + v + uv |
| | | | | |
| f_2 | 0 0 1 0 | (u) v | (( f , (u) v )) | f + v + uv + 1 |
| | | | | |
| f_3 | 0 0 1 1 | (u) | (( f , (u) )) | f + u |
| | | | | |
| f_4 | 0 1 0 0 | u (v) | (( f , u (v) )) | f + u + uv + 1 |
| | | | | |
| f_5 | 0 1 0 1 | (v) | (( f , (v) )) | f + v |
| | | | | |
| f_6 | 0 1 1 0 | (u, v) | (( f , (u, v) )) | f + u + v + 1 |
| | | | | |
| f_7 | 0 1 1 1 | (u v) | (( f , (u v) )) | f + uv |
| | | | | |
o---------o---------o----------o--------------------o--------------------o
| | | | | |
| f_8 | 1 0 0 0 | u v | (( f , u v )) | f + uv + 1 |
| | | | | |
| f_9 | 1 0 0 1 | ((u, v)) | (( f , ((u, v)) )) | f + u + v |
| | | | | |
| f_10 | 1 0 1 0 | v | (( f , v )) | f + v + 1 |
| | | | | |
| f_11 | 1 0 1 1 | (u (v)) | (( f , (u (v)) )) | f + u + uv |
| | | | | |
| f_12 | 1 1 0 0 | u | (( f , u )) | f + u + 1 |
| | | | | |
| f_13 | 1 1 0 1 | ((u) v) | (( f , ((u) v) )) | f + v + uv |
| | | | | |
| f_14 | 1 1 1 0 | ((u)(v)) | (( f , ((u)(v)) )) | f + u + v + uv + 1 |
| | | | | |
| f_15 | 1 1 1 1 | (()) | (( f , (()) )) | f |
| | | | | |
o---------o---------o----------o--------------------o--------------------o
</pre>

===Table 28. Propositions on Two Variables===

<pre>
Table 28. Propositions on Two Variables
o-------o-----o----------------------------------------------------------------o
| u v | | f f f f f f f f f f f f f f f f |
| | | 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 |
o-------o-----o----------------------------------------------------------------o
| | | |
| 0 0 | | 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 |
| | | |
| 0 1 | | 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 |
| | | |
| 1 0 | | 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 |
| | | |
| 1 1 | | 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 |
| | | |
o-------o-----o----------------------------------------------------------------o
</pre>

===Table 29. Thematic Extensions of Bivariate Propositions===

<pre>
Table 29. Thematic Extensions of Bivariate Propositions
o-------o-----o----------------------------------------------------------------o
| u v | f^¢ |!f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! |
| | | 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 |
o-------o-----o----------------------------------------------------------------o
| | | |
| 0 0 | 0 | 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 |
| | | |
| 0 0 | 1 | 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 |
| | | |
| 0 1 | 0 | 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 |
| | | |
| 0 1 | 1 | 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 |
| | | |
| 1 0 | 0 | 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 |
| | | |
| 1 0 | 1 | 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 |
| | | |
| 1 1 | 0 | 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 |
| | | |
| 1 1 | 1 | 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 |
| | | |
o-------o-----o----------------------------------------------------------------o
</pre>

===Figure 30. Generic Frame of a Logical Transformation===

<pre>
o-------------------------------------------------------o
| U |
| |
| o-----------o o-----------o |
| / \ / \ |
| / o \ |
| / / \ \ |
| / / \ \ |
| o o o o |
| | | | | |
| | u | | v | |
| | | | | |
| o o o o |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| o-----------o o-----------o |
| |
| |
o---------------------------o---------------------------o
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
o-------------------------o o-------------------------o o-------------------------o
| U | | U | | U |
| o---o o---o | | o---o o---o | | o---o o---o |
| / \ / \ | | / \ / \ | | / \ / \ |
| / o \ | | / o \ | | / o \ |
| / / \ \ | | / / \ \ | | / / \ \ |
| o o o o | | o o o o | | o o o o |
| | u | | v | | | | u | | v | | | | u | | v | |
| o o o o | | o o o o | | o o o o |
| \ \ / / | | \ \ / / | | \ \ / / |
| \ o / | | \ o / | | \ o / |
| \ / \ / | | \ / \ / | | \ / \ / |
| o---o o---o | | o---o o---o | | o---o o---o |
| | | | | |
o-------------------------o o-------------------------o o-------------------------o
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ g | \ f / | h /
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ o----------|-----------\-----/-----------|----------o /
\ | X | \ / | | /
\ | | \ / | | /
\ | | o-----o-----o | | /
\| | / \ | |/
\ | / \ | /
|\ | / \ | /|
| \ | / \ | / |
| \ | / \ | / |
| \ | o x o | / |
| \ | | | | / |
| \ | | | | / |
| \ | | | | / |
| \ | | | | / |
| \ | | | | / |
| \| | | |/ |
| o--o--------o o--------o--o |
| / \ \ / / \ |
| / \ \ / / \ |
| / \ o / \ |
| / \ / \ / \ |
| / \ / \ / \ |
| o o--o-----o--o o |
| | | | | |
| | | | | |
| | | | | |
| | y | | z | |
| | | | | |
| | | | | |
| o o o o |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| \ / \ / |
| o-----------o o-----------o |
| |
| |
o---------------------------------------------------o
\ /
\ /
\ /
\ /
\ /
\ p , q /
\ /
\ /
\ /
\ /
\ /
\ /
\ /
o

Figure 30. Generic Frame of a Logical Transformation
</pre>

===Formula Display 3===

<pre>
o-------------------------------------------------o
| |
| x = f<u, v> |
| |
| y = g<u, v> |
| |
| z = h<u, v> |
| |
o-------------------------------------------------o
</pre>

<br><font face="courier new">
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
|
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
| width="20%" | &nbsp;
| width="20%" | ''x''
| width="20%" | =
| width="20%" | ''f''‹''u'', ''v''›
| width="20%" | &nbsp;
|-
| &nbsp; || ''y'' || = || ''g''‹''u'', ''v''› || &nbsp;
|-
| &nbsp; || ''z'' || = || ''h''‹''u'', ''v''› || &nbsp;
|}
|}
</font><br>

===Figure 31. Operator Diagram (1)===

<pre>
o---------------------------------------o
| |
| |
| U% F X% |
| o------------------>o |
| | | |
| | | |
| | | |
| | | |
| !W! | | !W! |
| | | |
| | | |
| | | |
| v v |
| o------------------>o |
| !W!U% !W!F !W!X% |
| |
| |
o---------------------------------------o
Figure 31. Operator Diagram (1)
</pre>

===Figure 32. Operator Diagram (2)===

<pre>
o---------------------------------------o
| |
| |
| U% !W! !W!U% |
| o------------------>o |
| | | |
| | | |
| | | |
| | | |
| F | | !W!F |
| | | |
| | | |
| | | |
| v v |
| o------------------>o |
| X% !W! !W!X% |
| |
| |
o---------------------------------------o
Figure 32. Operator Diagram (2)
</pre>

===Figure 33-i. Analytic Diagram (1)===

<pre>
U% $E$ $E$U% $E$U% $E$U%
o------------------>o============o============o
| | | |
| | | |
| | | |
| | | |
F | | $E$F = | $d$^0.F + | $r$^0.F
| | | |
| | | |
| | | |
v v v v
o------------------>o============o============o
X% $E$ $E$X% $E$X% $E$X%

Figure 33-i. Analytic Diagram (1)
</pre>

===Figure 33-ii. Analytic Diagram (2)===

<pre>
U% $E$ $E$U% $E$U% $E$U% $E$U%
o------------------>o============o============o============o
| | | | |
| | | | |
| | | | |
| | | | |
F | | $E$F = | $d$^0.F + | $d$^1.F + | $r$^1.F
| | | | |
| | | | |
| | | | |
v v v v v
o------------------>o============o============o============o
X% $E$ $E$X% $E$X% $E$X% $E$X%

Figure 33-ii. Analytic Diagram (2)
</pre>

===Formula Display 4===

<pre>
o--------------------------------------------------------------------------------------o
| |
| x_1 = !e!F_1 <u_1, ..., u_n, du_1, ..., du_n> = F_1 <u_1, ..., u_n> |
| |
| ... |
| |
| x_k = !e!F_k <u_1, ..., u_n, du_1, ..., du_n> = F_k <u_1, ..., u_n> |
| |
| |
| dx_1 = EF_1 <u_1, ..., u_n, du_1, ..., du_n> = F_1 <u_1 + du_1, ..., u_n + du_n> |
| |
| ... |
| |
| dx_k = EF_k <u_1, ..., u_n, du_1, ..., du_n> = F_k <u_1 + du_1, ..., u_n + du_n> |
| |
o--------------------------------------------------------------------------------------o
</pre>

<br><font face="courier new">
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
|
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
| width="8%" | ''x''<sub>1</sub>
| width="4%" | =
| width="44%" | <math>\epsilon</math>''F''<sub>1</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, &hellip;, d''u''<sub>''n''</sub>›
| width="4%" | =
| width="40%" | ''F''<sub>1</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>›
|-
| ...
|-
| width="8%" | ''x''<sub>''k''</sub>
| width="4%" | =
| width="44%" | <math>\epsilon</math>''F''<sub>''k''</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, &hellip;, d''u''<sub>''n''</sub>›
| width="4%" | =
| width="40%" | ''F''<sub>''k''</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>›
|}
|-
|
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
| width="8%" | d''x''<sub>1</sub>
| width="4%" | =
| width="44%" | E''F''<sub>1</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, &hellip;, d''u''<sub>''n''</sub>›
| width="4%" | =
| width="40%" | ''F''<sub>1</sub>‹''u''<sub>1</sub> + d''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub> + d''u''<sub>''n''</sub>›
|-
| ...
|-
| width="8%" | d''x''<sub>''k''</sub>
| width="4%" | =
| width="44%" | E''F''<sub>''k''</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, &hellip;, d''u''<sub>''n''</sub>›
| width="4%" | =
| width="40%" | ''F''<sub>''k''</sub>‹''u''<sub>1</sub> + d''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub> + d''u''<sub>''n''</sub>›
|}
|}
</font><br>

===Formula Display 5===

<pre>
o--------------------------------------------------------------------------------o
| |
| x_1 = !e!F_1 <u_1, ..., u_n, du_1, ..., du_n> = F_1 <u_1, ..., u_n> |
| |
| ... |
| |
| x_k = !e!F_k <u_1, ..., u_n, du_1, ..., du_n> = F_k <u_1, ..., u_n> |
| |
| |
| dx_1 = !e!F_1 <u_1, ..., u_n, du_1, ..., du_n> = F_1 <u_1, ..., u_n> |
| |
| ... |
| |
| dx_k = !e!F_k <u_1, ..., u_n, du_1, ..., du_n> = F_k <u_1, ..., u_n> |
| |
o--------------------------------------------------------------------------------o
</pre>

<br><font face="courier new">
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
|
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
| width="8%" | ''x''<sub>1</sub>
| width="4%" | =
| width="44%" | <math>\epsilon</math>''F''<sub>1</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, &hellip;, d''u''<sub>''n''</sub>›
| width="4%" | =
| width="40%" | ''F''<sub>1</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>›
|-
| ...
|-
| width="8%" | ''x''<sub>''k''</sub>
| width="4%" | =
| width="44%" | <math>\epsilon</math>''F''<sub>''k''</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, &hellip;, d''u''<sub>''n''</sub>›
| width="4%" | =
| width="40%" | ''F''<sub>''k''</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>›
|}
|-
|
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
| width="8%" | d''x''<sub>1</sub>
| width="4%" | =
| width="44%" | <math>\epsilon</math>''F''<sub>1</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, &hellip;, d''u''<sub>''n''</sub>›
| width="4%" | =
| width="40%" | ''F''<sub>1</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>›
|-
| ...
|-
| width="8%" | d''x''<sub>''k''</sub>
| width="4%" | =
| width="44%" | <math>\epsilon</math>''F''<sub>''k''</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, &hellip;, d''u''<sub>''n''</sub>›
| width="4%" | =
| width="40%" | ''F''<sub>''k''</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>›
|}
|}
</font><br>

===Formula Display 6===

<pre>
o--------------------------------------------------------------------------------o
| |
| dx_1 = !e!F_1 <u_1, ..., u_n, du_1, ..., du_n> = F_1 <u_1, ..., u_n> |
| |
| ... |
| |
| dx_k = !e!F_k <u_1, ..., u_n, du_1, ..., du_n> = F_k <u_1, ..., u_n> |
| |
o--------------------------------------------------------------------------------o
</pre>

<br><font face="courier new">
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
|
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
| width="8%" | d''x''<sub>1</sub>
| width="4%" | =
| width="44%" | <math>\epsilon</math>''F''<sub>1</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, &hellip;, d''u''<sub>''n''</sub>›
| width="4%" | =
| width="40%" | ''F''<sub>1</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>›
|-
| ...
|-
| width="8%" | d''x''<sub>''k''</sub>
| width="4%" | =
| width="44%" | <math>\epsilon</math>''F''<sub>''k''</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, &hellip;, d''u''<sub>''n''</sub>›
| width="4%" | =
| width="40%" | ''F''<sub>''k''</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>›
|}
|}
</font><br>

===Formula Display 7===

<pre>
o-------------------------------------------------o
| |
| $D$ = $E$ - $e$ |
| |
| = $r$^0 |
| |
| = $d$^1 + $r$^1 |
| |
| = $d$^1 + ... + $d$^m + $r$^m |
| |
| = Sum_(i = 1 ... m) $d$^i + $r$^m |
| |
o-------------------------------------------------o
</pre>

<br><font face="courier new">
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
|
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
| <font face=georgia>'''D'''</font>
| =
| <font face=georgia>'''E'''</font> &ndash; <font face=georgia>'''e'''</font>
|-
| &nbsp;
| =
| <font face=georgia>'''r'''</font><sup>0</sup>
|-
| &nbsp;
| =
| <font face=georgia>'''d'''</font><sup>1</sup> + <font face=georgia>'''r'''</font><sup>1</sup>
|-
| &nbsp;
| =
| <font face=georgia>'''d'''</font><sup>1</sup> + &hellip; + <font face=georgia>'''d'''</font><sup>''m''</sup> + <font face=georgia>'''r'''</font><sup>''m''</sup>
|-
| &nbsp;
| =
| <font size="+2">&sum;</font><sub>(''i'' = 1 &hellip; ''m'')</sub> <font face=georgia>'''d'''</font><sup>''i''</sup> + <font face=georgia>'''r'''</font><sup>''m''</sup>
|}
|}
</font><br>

===Figure 34. Tangent Functor Diagram===

<pre>
U% $T$ $T$U% $T$U%
o------------------>o============o
| | |
| | |
| | |
| | |
F | | $T$F = | <!e!, d> F
| | |
| | |
| | |
v v v
o------------------>o============o
X% $T$ $T$X% $T$X%

Figure 34. Tangent Functor Diagram
</pre>

===Figure 35. Conjunction as Transformation===

<pre>
o---------------------------------------o
| |
| |
| o---------o o---------o |
| / \ / \ |
| / o \ |
| / /`\ \ |
| / /```\ \ |
| o o`````o o |
| | |`````| | |
| | u |`````| v | |
| | |`````| | |
| o o`````o o |
| \ \```/ / |
| \ \`/ / |
| \ o / |
| \ / \ / |
| o---------o o---------o |
| |
| |
o---------------------------------------o
\ /
\ /
\ /
\ J /
\ /
\ /
\ /
o--------------\---------/--------------o
| \ / |
| \ / |
| o------@------o |
| /```````````````\ |
| /`````````````````\ |
| /```````````````````\ |
| /`````````````````````\ |
| o```````````````````````o |
| |```````````````````````| |
| |`````````` x ``````````| |
| |```````````````````````| |
| o```````````````````````o |
| \`````````````````````/ |
| \```````````````````/ |
| \`````````````````/ |
| \```````````````/ |
| o-------------o |
| |
| |
o---------------------------------------o
Figure 35. Conjunction as Transformation
</pre>

===Table 36. Computation of !e!J===

<pre>
Table 36. Computation of !e!J
o---------------------------------------------------------------------o
| |
| !e!J = J<u, v> |
| |
| = u v |
| |
| = u v (du)(dv) + u v (du) dv + u v du (dv) + u v du dv |
| |
o---------------------------------------------------------------------o
| |
| !e!J = u v (du)(dv) + |
| u v (du) dv + |
| u v du (dv) + |
| u v du dv |
| |
o---------------------------------------------------------------------o
</pre>

===Figure 37-a. Tacit Extension of J (Areal)===

<pre>
o---------------------------------------o
| |
| o |
| /%\ |
| /%%%\ |
| /%%%%%\ |
| o%%%%%%%o |
| /%\%%%%%/%\ |
| /%%%\%%%/%%%\ |
| /%%%%%\%/%%%%%\ |
| o%%%%%%%o%%%%%%%o |
| / \%%%%%/%\%%%%%/ \ |
| / \%%%/%%%\%%%/ \ |
| / \%/%%%%%\%/ \ |
| o o%%%%%%%o o |
| / \ / \%%%%%/ \ / \ |
| / \ / \%%%/ \ / \ |
| / \ / \%/ \ / \ |
| o o o o o |
| |\ / \ / \ / \ /| |
| | \ / \ / \ / \ / | |
| | \ / \ / \ / \ / | |
| | o o o o | |
| | |\ / \ / \ /| | |
| | | \ / \ / \ / | | |
| | u | \ / \ / \ / | v | |
| o---+---o o o---+---o |
| | \ / \ / | |
| | \ / \ / | |
| | du \ / \ / dv | |
| o-------o o-------o |
| \ / |
| \ / |
| \ / |
| o |
| |
o---------------------------------------o
Figure 37-a. Tacit Extension of J (Areal)
</pre>

===Figure 37-b. Tacit Extension of J (Bundle)===

<pre>
o-----------------------------o
| |
| o-----o o-----o |
| / \ / \ |
| / o \ |
| / / \ \ |
| o o o o |
@ | du | | dv | |
/| o o o o |
/ | \ \ / / |
/ | \ o / |
/ | \ / \ / |
/ | o-----o o-----o |
/ | |
/ o-----------------------------o
/
o----------------------------------------/----o o-----------------------------o
| / | | |
| @ | | o-----o o-----o |
| | | / \ / \ |
| o---------o o---------o | | / o \ |
| / \ / \ | | / / \ \ |
| / o \ | | o o o o |
| / /`\ @------\-----------@ | du | | dv | |
| / /```\ \ | | o o o o |
| / /`````\ \ | | \ \ / / |
| / /```````\ \ | | \ o / |
| o o`````````o o | | \ / \ / |
| | |````@````| | | | o-----o o-----o |
| | |`````\```| | | | |
| | |``````\``| | | o-----------------------------o
| | u |```````\`| v | |
| | |````````\| | | o-----------------------------o
| | |`````````| | | | |
| | |`````````|\ | | | o-----o o-----o |
| o o`````````o \ o | | / \ / \ |
| \ \```````/ \ / | | / o \ |
| \ \`````/ \ / | | / / \ \ |
| \ \```/ \ / | | o o o o |
| \ @------\-/---------\---------------@ | du | | dv | |
| \ o \ / | | o o o o |
| \ / \ / | | \ \ / / |
| o---------o o---------o \ | | \ o / |
| \ | | \ / \ / |
| \ | | o-----o o-----o |
| \ | | |
o----------------------------------------\----o o-----------------------------o
\
\ o-----------------------------o
\ |`````````````````````````````|
\ |````` o-----o```o-----o``````|
\ |`````/```````\`/```````\`````|
\ |````/`````````o`````````\````|
\ |```/`````````/`\`````````\```|
\|``o`````````o```o`````````o``|
@``|```du````|```|````dv```|``|
|``o`````````o```o`````````o``|
|```\`````````\`/`````````/```|
|````\`````````o`````````/````|
|`````\```````/`\```````/`````|
|``````o-----o```o-----o``````|
|`````````````````````````````|
o-----------------------------o
Figure 37-b. Tacit Extension of J (Bundle)
</pre>

===Figure 37-c. Tacit Extension of J (Compact)===

<pre>
o---------------------------------------------------------------------o
| |
| |
| o-------------------o o-------------------o |
| / \ / \ |
| / o \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| o o (du).(dv) o o |
| | | -->-- | | |
| | | \ / | | |
| | dv .(du) | \ / | du .(dv) | |
| | u <---------------@---------------> v | |
| | | | | | |
| | | | | | |
| | | | | | |
| o o | o o |
| \ \ | / / |
| \ \ | / / |
| \ \ | / / |
| \ \ | / / |
| \ \|/ / |
| \ | / |
| \ /|\ / |
| o-------------------o | o-------------------o |
| | |
| du . dv |
| | |
| V |
| |
o---------------------------------------------------------------------o
Figure 37-c. Tacit Extension of J (Compact)
</pre>

===Figure 37-d. Tacit Extension of J (Digraph)===

<pre>
o-----------------------------------------------------------o
| |
| (du).(dv) |
| --->--- |
| \ / |
| \ / |
| \ / |
| u @ v |
| /|\ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| (du) dv / | \ du (dv) |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| v | v |
| @ | @ |
| u (v) | (u) v |
| | |
| | |
| | |
| | |
| du . dv |
| | |
| | |
| | |
| | |
| v |
| @ |
| |
| (u).(v) |
| |
o-----------------------------------------------------------o
Figure 37-d. Tacit Extension of J (Digraph)
</pre>

===Table 38. Computation of EJ (Method 1)===

<pre>
Table 38. Computation of EJ (Method 1)
o-------------------------------------------------------------------------------o
| |
| EJ = J<u + du, v + dv> |
| |
| = (u, du)(v, dv) |
| |
| = u v J<1 + du, 1 + dv> + |
| |
| u (v) J<1 + du, 0 + dv> + |
| |
| (u) v J<0 + du, 1 + dv> + |
| |
| (u)(v) J<0 + du, 0 + dv> |
| |
| = u v J<(du), (dv)> + |
| |
| u (v) J<(du), dv > + |
| |
| (u) v J< du , (dv)> + |
| |
| (u)(v) J< du , dv > |
| |
o-------------------------------------------------------------------------------o
| |
| EJ = u v (du)(dv) |
| + u (v)(du) dv |
| + (u) v du (dv) |
| + (u)(v) du dv |
| |
o-------------------------------------------------------------------------------o
</pre>

===Table 39. Computation of EJ (Method 2)===

<pre>
Table 39. Computation of EJ (Method 2)
o-------------------------------------------------------------------------------o
| |
| EJ = <u + du> <v + dv> |
| |
| = u v + u dv + v du + du dv |
| |
| EJ = u v (du)(dv) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv |
| |
o-------------------------------------------------------------------------------o
</pre>

===Figure 40-a. Enlargement of J (Areal)===

<pre>
o---------------------------------------o
| |
| o |
| /%\ |
| /%%%\ |
| /%%%%%\ |
| o%%%%%%%o |
| / \%%%%%/ \ |
| / \%%%/ \ |
| / \%/ \ |
| o o o |
| /%\ / \ /%\ |
| /%%%\ / \ /%%%\ |
| /%%%%%\ / \ /%%%%%\ |
| o%%%%%%%o o%%%%%%%o |
| / \%%%%%/ \ / \%%%%%/ \ |
| / \%%%/ \ / \%%%/ \ |
| / \%/ \ / \%/ \ |
| o o o o o |
| |\ / \ /%\ / \ /| |
| | \ / \ /%%%\ / \ / | |
| | \ / \ /%%%%%\ / \ / | |
| | o o%%%%%%%o o | |
| | |\ / \%%%%%/ \ /| | |
| | | \ / \%%%/ \ / | | |
| | u | \ / \%/ \ / | v | |
| o---+---o o o---+---o |
| | \ / \ / | |
| | \ / \ / | |
| | du \ / \ / dv | |
| o-------o o-------o |
| \ / |
| \ / |
| \ / |
| o |
| |
o---------------------------------------o
Figure 40-a. Enlargement of J (Areal)
</pre>

===Figure 40-b. Enlargement of J (Bundle)===

<pre>
o-----------------------------o
| |
| o-----o o-----o |
| / \ / \ |
| / o \ |
| / /%\ \ |
| o o%%%o o |
@ | du |%%%| dv | |
/| o o%%%o o |
/ | \ \%/ / |
/ | \ o / |
/ | \ / \ / |
/ | o-----o o-----o |
/ | |
/ o-----------------------------o
/
o----------------------------------------/----o o-----------------------------o
| / | | |
| @ | | o-----o o-----o |
| | | /%%%%%%%\ / \ |
| o---------o o---------o | | /%%%%%%%%%o \ |
| / \ / \ | | /%%%%%%%%%/ \ \ |
| / o \ | | o%%%%%%%%%o o o |
| / /`\ @------\-----------@ |%% du %%%| | dv | |
| / /```\ \ | | o%%%%%%%%%o o o |
| / /`````\ \ | | \%%%%%%%%%\ / / |
| / /```````\ \ | | \%%%%%%%%%o / |
| o o`````````o o | | \%%%%%%%/ \ / |
| | |````@````| | | | o-----o o-----o |
| | |`````\```| | | | |
| | |``````\``| | | o-----------------------------o
| | u |```````\`| v | |
| | |````````\| | | o-----------------------------o
| | |`````````| | | | |
| | |`````````|\ | | | o-----o o-----o |
| o o`````````o \ o | | / \ /%%%%%%%\ |
| \ \```````/ \ / | | / o%%%%%%%%%\ |
| \ \`````/ \ / | | / / \%%%%%%%%%\ |
| \ \```/ \ / | | o o o%%%%%%%%%o |
| \ @------\-/---------\---------------@ | du | |%%% dv %%| |
| \ o \ / | | o o o%%%%%%%%%o |
| \ / \ / | | \ \ /%%%%%%%%%/ |
| o---------o o---------o \ | | \ o%%%%%%%%%/ |
| \ | | \ / \%%%%%%%/ |
| \ | | o-----o o-----o |
| \ | | |
o----------------------------------------\----o o-----------------------------o
\
\ o-----------------------------o
\ |%%%%%%%%%%%%%%%%%%%%%%%%%%%%%|
\ |%%%%%%o-----o%%%o-----o%%%%%%|
\ |%%%%%/ \%/ \%%%%%|
\ |%%%%/ o \%%%%|
\ |%%%/ / \ \%%%|
\|%%o o o o%%|
@%%| du | | dv |%%|
|%%o o o o%%|
|%%%\ \ / /%%%|
|%%%%\ o /%%%%|
|%%%%%\ /%\ /%%%%%|
|%%%%%%o-----o%%%o-----o%%%%%%|
|%%%%%%%%%%%%%%%%%%%%%%%%%%%%%|
o-----------------------------o
Figure 40-b. Enlargement of J (Bundle)
</pre>

===Figure 40-c. Enlargement of J (Compact)===

<pre>
o---------------------------------------------------------------------o
| |
| |
| o-------------------o o-------------------o |
| / \ / \ |
| / o \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| o o (du).(dv) o o |
| | | -->-- | | |
| | | \ / | | |
| | dv .(du) | \ / | du .(dv) | |
| | u o---------------->@<----------------o v | |
| | | ^ | | |
| | | | | | |
| | | | | | |
| o o | o o |
| \ \ | / / |
| \ \ | / / |
| \ \ | / / |
| \ \ | / / |
| \ \|/ / |
| \ | / |
| \ /|\ / |
| o-------------------o | o-------------------o |
| | |
| du . dv |
| | |
| o |
| |
o---------------------------------------------------------------------o
Figure 40-c. Enlargement of J (Compact)
</pre>

===Figure 40-d. Enlargement of J (Digraph)===

<pre>
o-----------------------------------------------------------o
| |
| (du).(dv) |
| --->--- |
| \ / |
| \ / |
| \ / |
| u @ v |
| ^^^ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| (du) dv / | \ du (dv) |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| @ | @ |
| u (v) | (u) v |
| | |
| | |
| | |
| | |
| du . dv |
| | |
| | |
| | |
| | |
| | |
| @ |
| |
| (u).(v) |
| |
o-----------------------------------------------------------o
Figure 40-d. Enlargement of J (Digraph)
</pre>

===Table 41. Computation of DJ (Method 1)===

<pre>
Table 41. Computation of DJ (Method 1)
o-------------------------------------------------------------------------------o
| |
| DJ = EJ + !e!J |
| |
| = J<u + du, v + dv> + J<u, v> |
| |
| = (u, du)(v, dv) + u v |
| |
o-------------------------------------------------------------------------------o
| |
| DJ = 0 |
| |
| + u v (du) dv + u (v)(du) dv |
| |
| + u v du (dv) + (u) v du (dv) |
| |
| + u v du dv + (u)(v) du dv |
| |
o-------------------------------------------------------------------------------o
| |
| DJ = u v ((du)(dv)) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv |
| |
o-------------------------------------------------------------------------------o
</pre>

===Table 42. Computation of DJ (Method 2)===

<pre>
Table 42. Computation of DJ (Method 2)
o-------------------------------------------------------------------------------o
| |
| DJ = !e!J + EJ |
| |
| = J<u, v> + J<u + du, v + dv> |
| |
| = u v + (u, du)(v, dv) |
| |
| = 0 + u dv + v du + du dv |
| |
| = 0 + u (du) dv + v du (dv) + ((u, v)) du dv |
| |
o-------------------------------------------------------------------------------o
</pre>

===Table 43. Computation of DJ (Method 3)===

<pre>
Table 43. Computation of DJ (Method 3)
o-------------------------------------------------------------------------------o
| |
| DJ = !e!J + EJ |
| |
o-------------------------------------------------------------------------------o
| |
| !e!J = u v (du)(dv) + u v (du) dv + u v du (dv) + u v du dv |
| |
| EJ = u v (du)(dv) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv |
| |
o-------------------------------------------------------------------------------o
| |
| DJ = 0 . (du)(dv) + u . (du) dv + v . du (dv) + ((u, v)) du dv |
| |
o-------------------------------------------------------------------------------o
</pre>

===Formula Display 8===

<pre>
o-------------------------------------------------------------------------------o
| |
| !e!J = {Dispositions from J to J } + {Dispositions from J to (J)} |
| |
| EJ = {Dispositions from J to J } + {Dispositions from (J) to J } |
| |
| DJ = (!e!J, EJ) |
| |
| DJ = {Dispositions from J to (J)} + {Dispositions from (J) to J } |
| |
o-------------------------------------------------------------------------------o
</pre>

===Figure 44-a. Difference Map of J (Areal)===

<pre>
o---------------------------------------o
| |
| o |
| / \ |
| / \ |
| / \ |
| o o |
| /%\ /%\ |
| /%%%\ /%%%\ |
| /%%%%%\ /%%%%%\ |
| o%%%%%%%o%%%%%%%o |
| /%\%%%%%/%\%%%%%/%\ |
| /%%%\%%%/%%%\%%%/%%%\ |
| /%%%%%\%/%%%%%\%/%%%%%\ |
| o%%%%%%%o%%%%%%%o%%%%%%%o |
| / \%%%%%/ \%%%%%/ \%%%%%/ \ |
| / \%%%/ \%%%/ \%%%/ \ |
| / \%/ \%/ \%/ \ |
| o o o o o |
| |\ / \ /%\ / \ /| |
| | \ / \ /%%%\ / \ / | |
| | \ / \ /%%%%%\ / \ / | |
| | o o%%%%%%%o o | |
| | |\ / \%%%%%/ \ /| | |
| | | \ / \%%%/ \ / | | |
| | u | \ / \%/ \ / | v | |
| o---+---o o o---+---o |
| | \ / \ / | |
| | \ / \ / | |
| | du \ / \ / dv | |
| o-------o o-------o |
| \ / |
| \ / |
| \ / |
| o |
| |
o---------------------------------------o
Figure 44-a. Difference Map of J (Areal)
</pre>

===Figure 44-b. Difference Map of J (Bundle)===

<pre>
o-----------------------------o
| |
| o-----o o-----o |
| / \ / \ |
| / o \ |
| / /%\ \ |
| o o%%%o o |
@ | du |%%%| dv | |
/| o o%%%o o |
/ | \ \%/ / |
/ | \ o / |
/ | \ / \ / |
/ | o-----o o-----o |
/ | |
/ o-----------------------------o
/
o----------------------------------------/----o o-----------------------------o
| / | | |
| @ | | o-----o o-----o |
| | | /%%%%%%%\ / \ |
| o---------o o---------o | | /%%%%%%%%%o \ |
| / \ / \ | | /%%%%%%%%%/ \ \ |
| / o \ | | o%%%%%%%%%o o o |
| / /`\ @------\-----------@ |%% du %%%| | dv | |
| / /```\ \ | | o%%%%%%%%%o o o |
| / /`````\ \ | | \%%%%%%%%%\ / / |
| / /```````\ \ | | \%%%%%%%%%o / |
| o o`````````o o | | \%%%%%%%/ \ / |
| | |````@````| | | | o-----o o-----o |
| | |`````\```| | | | |
| | |``````\``| | | o-----------------------------o
| | u |```````\`| v | |
| | |````````\| | | o-----------------------------o
| | |`````````| | | | |
| | |`````````|\ | | | o-----o o-----o |
| o o`````````o \ o | | / \ /%%%%%%%\ |
| \ \```````/ \ / | | / o%%%%%%%%%\ |
| \ \`````/ \ / | | / / \%%%%%%%%%\ |
| \ \```/ \ / | | o o o%%%%%%%%%o |
| \ @------\-/---------\---------------@ | du | |%%% dv %%| |
| \ o \ / | | o o o%%%%%%%%%o |
| \ / \ / | | \ \ /%%%%%%%%%/ |
| o---------o o---------o \ | | \ o%%%%%%%%%/ |
| \ | | \ / \%%%%%%%/ |
| \ | | o-----o o-----o |
| \ | | |
o----------------------------------------\----o o-----------------------------o
\
\ o-----------------------------o
\ | |
\ | o-----o o-----o |
\ | /%%%%%%%\ /%%%%%%%\ |
\ | /%%%%%%%%%o%%%%%%%%%\ |
\ | /%%%%%%%%%/%\%%%%%%%%%\ |
\| o%%%%%%%%%o%%%o%%%%%%%%%o |
@ |%% du %%%|%%%|%%% dv %%| |
| o%%%%%%%%%o%%%o%%%%%%%%%o |
| \%%%%%%%%%\%/%%%%%%%%%/ |
| \%%%%%%%%%o%%%%%%%%%/ |
| \%%%%%%%/ \%%%%%%%/ |
| o-----o o-----o |
| |
o-----------------------------o
Figure 44-b. Difference Map of J (Bundle)
</pre>

===Figure 44-c. Difference Map of J (Compact)===

<pre>
o---------------------------------------------------------------------o
| |
| |
| o-------------------o o-------------------o |
| / \ / \ |
| / o \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| o o o o |
| | | | | |
| | | | | |
| | dv .(du) | | du .(dv) | |
| | u @<--------------->@<--------------->@ v | |
| | | ^ | | |
| | | | | | |
| | | | | | |
| o o | o o |
| \ \ | / / |
| \ \ | / / |
| \ \ | / / |
| \ \ | / / |
| \ \|/ / |
| \ | / |
| \ /|\ / |
| o-------------------o | o-------------------o |
| | |
| du . dv |
| | |
| v |
| @ |
| |
o---------------------------------------------------------------------o
Figure 44-c. Difference Map of J (Compact)
</pre>

===Figure 44-d. Difference Map of J (Digraph)===

<pre>
o-----------------------------------------------------------o
| |
| u v |
| |
| @ |
| ^^^ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| (du) dv / | \ du (dv) |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| v | v |
| @ | @ |
| u (v) | (u) v |
| | |
| | |
| | |
| | |
| du | dv |
| | |
| | |
| | |
| | |
| v |
| @ |
| |
| (u) (v) |
| |
o-----------------------------------------------------------o
Figure 44-d. Difference Map of J (Digraph)
</pre>

===Table 45. Computation of dJ===

<pre>
Table 45. Computation of dJ
o-------------------------------------------------------------------------------o
| |
| DJ = u v ((du)(dv)) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv |
| |
| => |
| |
| dj = u v (du, dv) + u (v) dv + (u) v du + (u)(v) . 0 |
| |
o-------------------------------------------------------------------------------o
</pre>

===Figure 46-a. Differential of J (Areal)===

<pre>
o---------------------------------------o
| |
| o |
| / \ |
| / \ |
| / \ |
| o o |
| /%\ /%\ |
| /%%%\ /%%%\ |
| /%%%%%\ /%%%%%\ |
| o%%%%%%%o%%%%%%%o |
| /%\%%%%%/ \%%%%%/%\ |
| /%%%\%%%/ \%%%/%%%\ |
| /%%%%%\%/ \%/%%%%%\ |
| o%%%%%%%o o%%%%%%%o |
| / \%%%%%/%\ /%\%%%%%/ \ |
| / \%%%/%%%\ /%%%\%%%/ \ |
| / \%/%%%%%\ /%%%%%\%/ \ |
| o o%%%%%%%o%%%%%%%o o |
| |\ / \%%%%%/ \%%%%%/ \ /| |
| | \ / \%%%/ \%%%/ \ / | |
| | \ / \%/ \%/ \ / | |
| | o o o o | |
| | |\ / \ / \ /| | |
| | | \ / \ / \ / | | |
| | u | \ / \ / \ / | v | |
| o---+---o o o---+---o |
| | \ / \ / | |
| | \ / \ / | |
| | du \ / \ / dv | |
| o-------o o-------o |
| \ / |
| \ / |
| \ / |
| o |
| |
o---------------------------------------o
Figure 46-a. Differential of J (Areal)
</pre>

===Figure 46-b. Differential of J (Bundle)===

<pre>
o-----------------------------o
| |
| o-----o o-----o |
| / \ / \ |
| / o \ |
| / / \ \ |
| o o o o |
@ | du | | dv | |
/| o o o o |
/ | \ \ / / |
/ | \ o / |
/ | \ / \ / |
/ | o-----o o-----o |
/ | |
/ o-----------------------------o
/
o----------------------------------------/----o o-----------------------------o
| / | | |
| @ | | o-----o o-----o |
| | | /%%%%%%%\ / \ |
| o---------o o---------o | | /%%%%%%%%%o \ |
| / \ / \ | | /%%%%%%%%%/%\ \ |
| / o \ | | o%%%%%%%%%o%%%o o |
| / /`\ @------\-----------@ |%% du %%%|%%%| dv | |
| / /```\ \ | | o%%%%%%%%%o%%%o o |
| / /`````\ \ | | \%%%%%%%%%\%/ / |
| / /```````\ \ | | \%%%%%%%%%o / |
| o o`````````o o | | \%%%%%%%/ \ / |
| | |````@````| | | | o-----o o-----o |
| | |`````\```| | | | |
| | |``````\``| | | o-----------------------------o
| | u |```````\`| v | |
| | |````````\| | | o-----------------------------o
| | |`````````| | | | |
| | |`````````|\ | | | o-----o o-----o |
| o o`````````o \ o | | / \ /%%%%%%%\ |
| \ \```````/ \ / | | / o%%%%%%%%%\ |
| \ \`````/ \ / | | / /%\%%%%%%%%%\ |
| \ \```/ \ / | | o o%%%o%%%%%%%%%o |
| \ @------\-/---------\---------------@ | du |%%%|%%% dv %%| |
| \ o \ / | | o o%%%o%%%%%%%%%o |
| \ / \ / | | \ \%/%%%%%%%%%/ |
| o---------o o---------o \ | | \ o%%%%%%%%%/ |
| \ | | \ / \%%%%%%%/ |
| \ | | o-----o o-----o |
| \ | | |
o----------------------------------------\----o o-----------------------------o
\
\ o-----------------------------o
\ | |
\ | o-----o o-----o |
\ | /%%%%%%%\ /%%%%%%%\ |
\ | /%%%%%%%%%o%%%%%%%%%\ |
\ | /%%%%%%%%%/ \%%%%%%%%%\ |
\| o%%%%%%%%%o o%%%%%%%%%o |
@ |%% du %%%| |%%% dv %%| |
| o%%%%%%%%%o o%%%%%%%%%o |
| \%%%%%%%%%\ /%%%%%%%%%/ |
| \%%%%%%%%%o%%%%%%%%%/ |
| \%%%%%%%/ \%%%%%%%/ |
| o-----o o-----o |
| |
o-----------------------------o
Figure 46-b. Differential of J (Bundle)
</pre>

===Figure 46-c. Differential of J (Compact)===

<pre>
o---------------------------------------------------------------------o
| |
| |
| o-------------------o o-------------------o |
| / \ / \ |
| / o \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / @ \ \ |
| / / ^ ^ \ \ |
| o o / \ o o |
| | | / \ | | |
| | | / \ | | |
| | |/ \| | |
| | u (du)/ dv du \(dv) v | |
| | /| |\ | |
| | / | | \ | |
| | / | | \ | |
| o / o o \ o |
| \ / \ / \ / |
| \ v \ du dv / v / |
| \ @<----------------------->@ / |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| o-------------------o o-------------------o |
| |
| |
o---------------------------------------------------------------------o
Figure 46-c. Differential of J (Compact)
</pre>

===Figure 46-d. Differential of J (Digraph)===

<pre>
o-----------------------------------------------------------o
| |
| u v |
| @ |
| ^ ^ |
| / \ |
| / \ |
| / \ |
| / \ |
| (du) dv / \ du (dv) |
| / \ |
| / \ |
| / \ |
| / \ |
| v v |
| u (v) @<--------------------->@ (u) v |
| du dv |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| @ |
| (u) (v) |
| |
o-----------------------------------------------------------o
Figure 46-d. Differential of J (Digraph)
</pre>

===Table 47. Computation of rJ===

<pre>
Table 47. Computation of rJ
o-------------------------------------------------------------------------------o
| |
| rJ = DJ + dJ |
| |
o-------------------------------------------------------------------------------o
| |
| DJ = u v ((du)(dv)) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv |
| |
| dJ = u v (du, dv) + u (v) dv + (u) v du + (u)(v) . 0 |
| |
o-------------------------------------------------------------------------------o
| |
| rJ = u v du dv + u (v) du dv + (u) v du dv + (u)(v) du dv |
| |
o-------------------------------------------------------------------------------o
</pre>

===Figure 48-a. Remainder of J (Areal)===

<pre>
o---------------------------------------o
| |
| o |
| / \ |
| / \ |
| / \ |
| o o |
| / \ / \ |
| / \ / \ |
| / \ / \ |
| o o o |
| / \ /%\ / \ |
| / \ /%%%\ / \ |
| / \ /%%%%%\ / \ |
| o o%%%%%%%o o |
| / \ /%\%%%%%/%\ / \ |
| / \ /%%%\%%%/%%%\ / \ |
| / \ /%%%%%\%/%%%%%\ / \ |
| o o%%%%%%%o%%%%%%%o o |
| |\ / \%%%%%/%\%%%%%/ \ /| |
| | \ / \%%%/%%%\%%%/ \ / | |
| | \ / \%/%%%%%\%/ \ / | |
| | o o%%%%%%%o o | |
| | |\ / \%%%%%/ \ /| | |
| | | \ / \%%%/ \ / | | |
| | u | \ / \%/ \ / | v | |
| o---+---o o o---+---o |
| | \ / \ / | |
| | \ / \ / | |
| | du \ / \ / dv | |
| o-------o o-------o |
| \ / |
| \ / |
| \ / |
| o |
| |
o---------------------------------------o
Figure 48-a. Remainder of J (Areal)
</pre>

===Figure 48-b. Remainder of J (Bundle)===

<pre>
o-----------------------------o
| |
| o-----o o-----o |
| / \ / \ |
| / o \ |
| / /%\ \ |
| o o%%%o o |
@ | du |%%%| dv | |
/| o o%%%o o |
/ | \ \%/ / |
/ | \ o / |
/ | \ / \ / |
/ | o-----o o-----o |
/ | |
/ o-----------------------------o
/
o----------------------------------------/----o o-----------------------------o
| / | | |
| @ | | o-----o o-----o |
| | | / \ / \ |
| o---------o o---------o | | / o \ |
| / \ / \ | | / /%\ \ |
| / o \ | | o o%%%o o |
| / /`\ @------\-----------@ | du |%%%| dv | |
| / /```\ \ | | o o%%%o o |
| / /`````\ \ | | \ \%/ / |
| / /```````\ \ | | \ o / |
| o o`````````o o | | \ / \ / |
| | |````@````| | | | o-----o o-----o |
| | |`````\```| | | | |
| | |``````\``| | | o-----------------------------o
| | u |```````\`| v | |
| | |````````\| | | o-----------------------------o
| | |`````````| | | | |
| | |`````````|\ | | | o-----o o-----o |
| o o`````````o \ o | | / \ / \ |
| \ \```````/ \ / | | / o \ |
| \ \`````/ \ / | | / /%\ \ |
| \ \```/ \ / | | o o%%%o o |
| \ @------\-/---------\---------------@ | du |%%%| dv | |
| \ o \ / | | o o%%%o o |
| \ / \ / | | \ \%/ / |
| o---------o o---------o \ | | \ o / |
| \ | | \ / \ / |
| \ | | o-----o o-----o |
| \ | | |
o----------------------------------------\----o o-----------------------------o
\
\ o-----------------------------o
\ | |
\ | o-----o o-----o |
\ | / \ / \ |
\ | / o \ |
\ | / /%\ \ |
\| o o%%%o o |
@ | du |%%%| dv | |
| o o%%%o o |
| \ \%/ / |
| \ o / |
| \ / \ / |
| o-----o o-----o |
| |
o-----------------------------o
Figure 48-b. Remainder of J (Bundle)
</pre>

===Figure 48-c. Remainder of J (Compact)===

<pre>
o---------------------------------------------------------------------o
| |
| |
| o-------------------o o-------------------o |
| / \ / \ |
| / o \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| o o o o |
| | | | | |
| | | | | |
| | | du dv | | |
| | u @<------------------------->@ v | |
| | | | | |
| | | | | |
| | | | | |
| o o @ o o |
| \ \ ^ / / |
| \ \ | / / |
| \ \ | / / |
| \ \ | / / |
| \ \|/ / |
| \ du | dv / |
| \ /|\ / |
| o-------------------o | o-------------------o |
| | |
| | |
| v |
| @ |
| |
o---------------------------------------------------------------------o
Figure 48-c. Remainder of J (Compact)
</pre>

===Figure 48-d. Remainder of J (Digraph)===

<pre>
o-----------------------------------------------------------o
| |
| u v |
| @ |
| ^ |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| du | dv |
| u (v) @<----------|---------->@ (u) v |
| du | dv |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| v |
| @ |
| (u) (v) |
| |
o-----------------------------------------------------------o
Figure 48-d. Remainder of J (Digraph)
</pre>

===Table 49. Computation Summary for J===

<pre>
Table 49. Computation Summary for J
o-------------------------------------------------------------------------------o
| |
| !e!J = uv . 1 + u(v) . 0 + (u)v . 0 + (u)(v) . 0 |
| |
| EJ = uv . (du)(dv) + u(v) . (du)dv + (u)v . du(dv) + (u)(v) . du dv |
| |
| DJ = uv . ((du)(dv)) + u(v) . (du)dv + (u)v . du(dv) + (u)(v) . du dv |
| |
| dJ = uv . (du, dv) + u(v) . dv + (u)v . du + (u)(v) . 0 |
| |
| rJ = uv . du dv + u(v) . du dv + (u)v . du dv + (u)(v) . du dv |
| |
o-------------------------------------------------------------------------------o
</pre>

===Table 50. Computation of an Analytic Series in Terms of Coordinates===

<pre>
Table 50. Computation of an Analytic Series in Terms of Coordinates
o-----------o-------------o-------------oo-------------o---------o-------------o
| u v | du dv | u' v' || !e!J EJ | DJ | dJ d^2.J |
o-----------o-------------o-------------oo-------------o---------o-------------o
| | | || | | |
| 0 0 | 0 0 | 0 0 || 0 0 | 0 | 0 0 |
| | | || | | |
| | 0 1 | 0 1 || 0 | 0 | 0 0 |
| | | || | | |
| | 1 0 | 1 0 || 0 | 0 | 0 0 |
| | | || | | |
| | 1 1 | 1 1 || 1 | 1 | 0 1 |
| | | || | | |
o-----------o-------------o-------------oo-------------o---------o-------------o
| | | || | | |
| 0 1 | 0 0 | 0 1 || 0 0 | 0 | 0 0 |
| | | || | | |
| | 0 1 | 0 0 || 0 | 0 | 0 0 |
| | | || | | |
| | 1 0 | 1 1 || 1 | 1 | 1 0 |
| | | || | | |
| | 1 1 | 1 0 || 0 | 0 | 1 1 |
| | | || | | |
o-----------o-------------o-------------oo-------------o---------o-------------o
| | | || | | |
| 1 0 | 0 0 | 1 0 || 0 0 | 0 | 0 0 |
| | | || | | |
| | 0 1 | 1 1 || 1 | 1 | 1 0 |
| | | || | | |
| | 1 0 | 0 0 || 0 | 0 | 0 0 |
| | | || | | |
| | 1 1 | 0 1 || 0 | 0 | 1 1 |
| | | || | | |
o-----------o-------------o-------------oo-------------o---------o-------------o
| | | || | | |
| 1 1 | 0 0 | 1 1 || 1 1 | 0 | 0 0 |
| | | || | | |
| | 0 1 | 1 0 || 0 | 1 | 1 0 |
| | | || | | |
| | 1 0 | 0 1 || 0 | 1 | 1 0 |
| | | || | | |
| | 1 1 | 0 0 || 0 | 1 | 0 1 |
| | | || | | |
o-----------o-------------o-------------oo-------------o---------o-------------o
</pre>

===Formula Display 9===

<pre>
o-------------------------------------------------o
| |
| u' = u + du = (u, du) |
| |
| v' = v + du = (v, dv) |
| |
o-------------------------------------------------o
</pre>

===Formula Display 10===

<pre>
o--------------------------------------------------------------o
| |
| EJ<u, v, du, dv> = J<u + du, v + dv> = J<u', v'> |
| |
o--------------------------------------------------------------o
</pre>

===Table 51. Computation of an Analytic Series in Symbolic Terms===

<pre>
Table 51. Computation of an Analytic Series in Symbolic Terms
o-----------o---------o------------o------------o------------o-----------o
| u v | J | EJ | DJ | dJ | d^2.J |
o-----------o---------o------------o------------o------------o-----------o
| | | | | | |
| 0 0 | 0 | du dv | du dv | () | du dv |
| | | | | | |
| 0 1 | 0 | du (dv) | du (dv) | du | du dv |
| | | | | | |
| 1 0 | 0 | (du) dv | (du) dv | dv | du dv |
| | | | | | |
| 1 1 | 1 | (du)(dv) | ((du)(dv)) | (du, dv) | du dv |
| | | | | | |
o-----------o---------o------------o------------o------------o-----------o
</pre>

===Figure 52. Decomposition of the Enlarged Conjunction EJ = (J, DJ)===

<pre>
o o o
/%\ /%\ / \
/%%%\ /%%%\ / \
o%%%%%o o%%%%%o o o
/ \%%%/ \ /%\%%%/%\ /%\ /%\
/ \%/ \ /%%%\%/%%%\ /%%%\ /%%%\
o o o o%%%%%o%%%%%o o%%%%%o%%%%%o
/%\ / \ /%\ / \%%%/%\%%%/ \ /%\%%%/%\%%%/%\
/%%%\ / \ /%%%\ / \%/%%%\%/ \ /%%%\%/%%%\%/%%%\
o%%%%%o o%%%%%o o o%%%%%o o o%%%%%o%%%%%o%%%%%o
/ \%%%/ \ / \%%%/ \ / \ / \%%%/ \ / \ / \%%%/ \%%%/ \%%%/ \
/ \%/ \ / \%/ \ / \ / \%/ \ / \ / \%/ \%/ \%/ \
o o o o o o o o o o o o o o o
|\ / \ /%\ / \ /| |\ / \ / \ / \ /| |\ / \ /%\ / \ /|
| \ / \ /%%%\ / \ / | | \ / \ / \ / \ / | | \ / \ /%%%\ / \ / |
| o o%%%%%o o | | o o o o | | o o%%%%%o o |
| |\ / \%%%/ \ /| | | |\ / \ / \ /| | | |\ / \%%%/ \ /| |
|u | \ / \%/ \ / | v| |u | \ / \ / \ / | v| |u | \ / \%/ \ / | v|
o--+--o o o--+--o o--+--o o o--+--o o--+--o o o--+--o
| \ / \ / | | \ / \ / | | \ / \ / |
| du \ / \ / dv | | du \ / \ / dv | | du \ / \ / dv |
o-----o o-----o o-----o o-----o o-----o o-----o
\ / \ / \ /
\ / \ / \ /
o o o

EJ = J + DJ

o-----------------------o o-----------------------o o-----------------------o
| | | | | |
| o--o o--o | | o--o o--o | | o--o o--o |
| / \ / \ | | / \ / \ | | / \ / \ |
| / o \ | | / o \ | | / o \ |
| / u / \ v \ | | / u / \ v \ | | / u / \ v \ |
| o /->-\ o | | o /->-\ o | | o / \ o |
| | o \ / o | | | | o \ / o | | | | o o | |
| | @--|->@<-|--@ | | | | @<-|--@--|->@ | | | | @<-|->@<-|->@ | |
| | o ^ o | | | | o | o | | | | o ^ o | |
| o \ | / o | | o \ | / o | | o \ | / o |
| \ \|/ / | | \ \|/ / | | \ \|/ / |
| \ | / | | \ | / | | \ | / |
| \ /|\ / | | \ /|\ / | | \ /|\ / |
| o--o | o--o | | o--o v o--o | | o--o v o--o |
| @ | | @ | | @ |
o-----------------------o o-----------------------o o-----------------------o
Figure 52. Decomposition of the Enlarged Conjunction EJ = (J, DJ)
</pre>

===Figure 53. Decomposition of the Differed Conjunction DJ = (dJ, ddJ)===

<pre>
o o o
/ \ / \ / \
/ \ / \ / \
o o o o o o
/%\ /%\ /%\ /%\ / \ / \
/%%%\ /%%%\ /%%%\%/%%%\ / \ / \
o%%%%%o%%%%%o o%%%%%o%%%%%o o o o
/%\%%%/%\%%%/%\ /%\%%%/ \%%%/%\ / \ /%\ / \
/%%%\%/%%%\%/%%%\ /%%%\%/ \%/%%%\ / \ /%%%\ / \
o%%%%%o%%%%%o%%%%%o o%%%%%o o%%%%%o o o%%%%%o o
/ \%%%/ \%%%/ \%%%/ \ / \%%%/%\ /%\%%%/ \ / \ /%\%%%/%\ / \
/ \%/ \%/ \%/ \ / \%/%%%\ /%%%\%/ \ / \ /%%%\%/%%%\ / \
o o o o o o o%%%%%o%%%%%o o o o%%%%%o%%%%%o o
|\ / \ /%\ / \ /| |\ / \%%%/ \%%%/ \ /| |\ / \%%%/%\%%%/ \ /|
| \ / \ /%%%\ / \ / | | \ / \%/ \%/ \ / | | \ / \%/%%%\%/ \ / |
| o o%%%%%o o | | o o o o | | o o%%%%%o o |
| |\ / \%%%/ \ /| | | |\ / \ / \ /| | | |\ / \%%%/ \ /| |
|u | \ / \%/ \ / | v| |u | \ / \ / \ / | v| |u | \ / \%/ \ / | v|
o--+--o o o--+--o o--+--o o o--+--o o--+--o o o--+--o
| \ / \ / | | \ / \ / | | \ / \ / |
| du \ / \ / dv | | du \ / \ / dv | | du \ / \ / dv |
o-----o o-----o o-----o o-----o o-----o o-----o
\ / \ / \ /
\ / \ / \ /
o o o

DJ = dJ + ddJ

o-----------------------o o-----------------------o o-----------------------o
| | | | | |
| o--o o--o | | o--o o--o | | o--o o--o |
| / \ / \ | | / \ / \ | | / \ / \ |
| / o \ | | / o \ | | / o \ |
| / u / \ v \ | | / u / \ v \ | | / u / \ v \ |
| o / \ o | | o / \ o | | o / \ o |
| | o o | | | | o o | | | | o o | |
| | @<-|->@<-|->@ | | | | @<-|->@<-|->@ | | | | @<-|-----|->@ | |
| | o ^ o | | | | ^ o o ^ | | | | o @ o | |
| o \ | / o | | o \ \ / / o | | o \ ^ / o |
| \ \|/ / | | \ --\-/-- / | | \ \|/ / |
| \ | / | | \ o / | | \ | / |
| \ /|\ / | | \ / \ / | | \ /|\ / |
| o--o v o--o | | o--o o--o | | o--o v o--o |
| @ | | @ | | @ |
o-----------------------o o-----------------------o o-----------------------o
Figure 53. Decomposition of the Differed Conjunction DJ = (dJ, ddJ)
</pre>

===Table 54. Cast of Characters: Expansive Subtypes of Objects and Operators===

<pre>
Table 54. Cast of Characters: Expansive Subtypes of Objects and Operators
o------o-------------------------o------------------o----------------------------o
| Item | Notation | Description | Type |
o------o-------------------------o------------------o----------------------------o
| | | | |
| U% | = [u, v] | Source Universe | [B^2] |
| | | | |
o------o-------------------------o------------------o----------------------------o
| | | | |
| X% | = [x] | Target Universe | [B^1] |
| | | | |
o------o-------------------------o------------------o----------------------------o
| | | | |
| EU% | = [u, v, du, dv] | Extended | [B^2 x D^2] |
| | | Source Universe | |
| | | | |
o------o-------------------------o------------------o----------------------------o
| | | | |
| EX% | = [x, dx] | Extended | [B^1 x D^1] |
| | | Target Universe | |
| | | | |
o------o-------------------------o------------------o----------------------------o
| | | | |
| J | J : U -> B | Proposition | (B^2 -> B) c [B^2] |
| | | | |
o------o-------------------------o------------------o----------------------------o
| | | | |
| J | J : U% -> X% | Transformation, | [B^2] -> [B^1] |
| | | or Mapping | |
| | | | |
o------o-------------------------o------------------o----------------------------o
| | | | |
| W | W : | Operator | |
| | U% -> EU%, | | [B^2] -> [B^2 x D^2], |
| | X% -> EX%, | | [B^1] -> [B^1 x D^1], |
| | (U%->X%)->(EU%->EX%), | | ([B^2] -> [B^1]) |
| | for each W among: | | -> |
| | e!, !h!, E, D, d | | ([B^2 x D^2]->[B^1 x D^1]) |
| | | | |
o------o-------------------------o------------------o----------------------------o
| | | |
| !e! | | Tacit Extension Operator !e! |
| !h! | | Trope Extension Operator !h! |
| E | | Enlargement Operator E |
| D | | Difference Operator D |
| d | | Differential Operator d |
| | | |
o------o-------------------------o------------------o----------------------------o
| | | | |
| $W$ | $W$ : | Operator | |
| | U% -> $T$U% = EU%, | | [B^2] -> [B^2 x D^2], |
| | X% -> $T$X% = EX%, | | [B^1] -> [B^1 x D^1], |
| | (U%->X%)->($T$U%->$T$X%)| | ([B^2] -> [B^1]) |
| | for each $W$ among: | | -> |
| | $e$, $E$, $D$, $T$ | | ([B^2 x D^2]->[B^1 x D^1]) |
| | | | |
o------o-------------------------o------------------o----------------------------o
| | | |
| $e$ | | Radius Operator $e$ = <!e!, !h!> |
| $E$ | | Secant Operator $E$ = <!e!, E > |
| $D$ | | Chord Operator $D$ = <!e!, D > |
| $T$ | | Tangent Functor $T$ = <!e!, d > |
| | | |
o------o-------------------------o-----------------------------------------------o
</pre>

===Table 55. Synopsis of Terminology: Restrictive and Alternative Subtypes===

<pre>
Table 55. Synopsis of Terminology: Restrictive and Alternative Subtypes
o--------------o----------------------o--------------------o----------------------o
| | Operator | Proposition | Map |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Tacit | !e! : | !e!J : | !e!J : |
| Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> B | [u,v,du,dv]->[x] |
| | (U%->X%)->(EU%->X%) | B^2 x D^2 -> B | [B^2 x D^2]->[B^1] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Trope | !h! : | !h!J : | !h!J : |
| Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] |
| | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Enlargement | E : | EJ : | EJ : |
| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] |
| | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Difference | D : | DJ : | DJ : |
| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] |
| | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Differential | d : | dJ : | dJ : |
| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] |
| | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Remainder | r : | rJ : | rJ : |
| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] |
| | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Radius | $e$ = <!e!, !h!> : | | $e$J : |
| Operator | U%->EU%, X%->EX%, | | [u,v,du,dv]->[x, dx] |
| | (U%->X%)->(EU%->EX%) | | [B^2 x D^2]->[B x D] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Secant | $E$ = <!e!, E> : | | $E$J : |
| Operator | U%->EU%, X%->EX%, | | [u,v,du,dv]->[x, dx] |
| | (U%->X%)->(EU%->EX%) | | [B^2 x D^2]->[B x D] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Chord | $D$ = <!e!, D> : | | $D$J : |
| Operator | U%->EU%, X%->EX%, | | [u,v,du,dv]->[x, dx] |
| | (U%->X%)->(EU%->EX%) | | [B^2 x D^2]->[B x D] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Tangent | $T$ = <!e!, d> : | dJ : | $T$J : |
| Functor | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[x, dx] |
| | (U%->X%)->(EU%->EX%) | B^2 x D^2 -> D | [B^2 x D^2]->[B x D] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
</pre>

===Figure 56-a1. Radius Map of the Conjunction J = uv===

<pre>
o
/X\
/XXX\
oXXXXXo
/X\XXX/X\
/XXX\X/XXX\
oXXXXXoXXXXXo
/ \XXX/X\XXX/ \
/ \X/XXX\X/ \
o oXXXXXo o
/ \ / \XXX/ \ / \
/ \ / \X/ \ / \
o o o o o
=|\ / \ / \ / \ /|=
= | \ / \ / \ / \ / | =
= | o o o o | =
= | |\ / \ / \ /| | =
= |u | \ / \ / \ / | v| =
o o--+--o o o--+--o o
//\ | \ / \ / | /\\
////\ | du \ / \ / dv | /\\\\
o/////o o-----o o-----o o\\\\\o
//\/////\ \ / /\\\\\/\\
////\/////\ \ / /\\\\\/\\\\
o/////o/////o o o\\\\\o\\\\\o
/ \/////\//// \ = = / \\\\/\\\\\/ \
/ \/////\// \ = = / \\/\\\\\/ \
o o/////o o = = o o\\\\\o o
/ \ / \//// \ / \ = = / \ / \\\\/ \ / \
/ \ / \// \ / \ = = / \ / \\/ \ / \
o o o o o o o o o o
|\ / \ / \ / \ /| |\ / \ / \ / \ /|
| \ / \ / \ / \ / | | \ / \ / \ / \ / |
| o o o o | | o o o o |
| |\ / \ / \ /| | | |\ / \ / \ /| |
|u | \ / \ / \ / | v| |u | \ / \ / \ / | v|
o--+--o o o--+--o o o--+--o o o--+--o
. | \ / \ / | /X\ | \ / \ / | .
.| du \ / \ / dv | /XXX\ | du \ / \ / dv |.
o-----o o-----o /XXXXX\ o-----o o-----o
. \ / /XXXXXXX\ \ / .
. \ / /XXXXXXXXX\ \ / .
. o oXXXXXXXXXXXo o .
. //\XXXXXXXXX/\\ .
. ////\XXXXXXX/\\\\ .
!e!J //////\XXXXX/\\\\\\ !h!J
. ////////\XXX/\\\\\\\\ .
. //////////\X/\\\\\\\\\\ .
. o///////////o\\\\\\\\\\\o .
. |\////////// \\\\\\\\\\/| .
. | \//////// \\\\\\\\/ | .
. | \////// \\\\\\/ | .
. | \//// \\\\/ | .
.| x \// \\/ dx |.
o-----o o-----o
\ /
\ /
x = uv \ / dx = uv
\ /
\ /
o

Figure 56-a1. Radius Map of the Conjunction J = uv
</pre>

===Figure 56-a2. Secant Map of the Conjunction J = uv===

<pre>
o
/X\
/XXX\
oXXXXXo
//\XXX//\
////\X////\
o/////o/////o
/\\/////\////\\
/\\\\/////\//\\\\
o\\\\\o/////o\\\\\o
/ \\\\/ \//// \\\\/ \
/ \\/ \// \\/ \
o o o o o
=|\ / \ /\\ / \ /|=
= | \ / \ /\\\\ / \ / | =
= | o o\\\\\o o | =
= | |\ / \\\\/ \ /| | =
= |u | \ / \\/ \ / | v| =
o o--+--o o o--+--o o
//\ | \ / \ / | /\\
////\ | du \ / \ / dv | /\\\\
o/////o o-----o o-----o o\\\\\o
//\/////\ \ / / \\\\/ \
////\/////\ \ / / \\/ \
o/////o/////o o o o o
/ \/////\//// \ = = /\\ / \ /\\
/ \/////\// \ = = /\\\\ / \ /\\\\
o o/////o o = = o\\\\\o o\\\\\o
/ \ / \//// \ / \ = = / \\\\/ \ / \\\\/ \
/ \ / \// \ / \ = = / \\/ \ / \\/ \
o o o o o o o o o o
|\ / \ / \ / \ /| |\ / \ /\\ / \ /|
| \ / \ / \ / \ / | | \ / \ /\\\\ / \ / |
| o o o o | | o o\\\\\o o |
| |\ / \ / \ /| | | |\ / \\\\/ \ /| |
|u | \ / \ / \ / | v| |u | \ / \\/ \ / | v|
o--+--o o o--+--o o o--+--o o o--+--o
. | \ / \ / | /X\ | \ / \ / | .
.| du \ / \ / dv | /XXX\ | du \ / \ / dv |.
o-----o o-----o /XXXXX\ o-----o o-----o
. \ / /XXXXXXX\ \ / .
. \ / /XXXXXXXXX\ \ / .
. o oXXXXXXXXXXXo o .
. //\XXXXXXXXX/\\ .
. ////\XXXXXXX/\\\\ .
!e!J //////\XXXXX/\\\\\\ EJ
. ////////\XXX/\\\\\\\\ .
. //////////\X/\\\\\\\\\\ .
. o///////////o\\\\\\\\\\\o .
. |\////////// \\\\\\\\\\/| .
. | \//////// \\\\\\\\/ | .
. | \////// \\\\\\/ | .
. | \//// \\\\/ | .
.| x \// \\/ dx |.
o-----o o-----o
\ /
\ / dx = (u, du)(v, dv)
x = uv \ /
\ / dx = uv + u dv + v du + du dv
\ /
o

Figure 56-a2. Secant Map of the Conjunction J = uv
</pre>

===Figure 56-a3. Chord Map of the Conjunction J = uv===

<pre>
o
//\
////\
o/////o
/X\////X\
/XXX\//XXX\
oXXXXXoXXXXXo
/\\XXX/X\XXX/\\
/\\\\X/XXX\X/\\\\
o\\\\\oXXXXXo\\\\\o
/ \\\\/ \XXX/ \\\\/ \
/ \\/ \X/ \\/ \
o o o o o
=|\ / \ /\\ / \ /|=
= | \ / \ /\\\\ / \ / | =
= | o o\\\\\o o | =
= | |\ / \\\\/ \ /| | =
= |u | \ / \\/ \ / | v| =
o o--+--o o o--+--o o
//\ | \ / \ / | / \
////\ | du \ / \ / dv | / \
o/////o o-----o o-----o o o
//\/////\ \ / /\\ /\\
////\/////\ \ / /\\\\ /\\\\
o/////o/////o o o\\\\\o\\\\\o
/ \/////\//// \ = = /\\\\\/\\\\\/\\
/ \/////\// \ = = /\\\\\/\\\\\/\\\\
o o/////o o = = o\\\\\o\\\\\o\\\\\o
/ \ / \//// \ / \ = = / \\\\/ \\\\/ \\\\/ \
/ \ / \// \ / \ = = / \\/ \\/ \\/ \
o o o o o o o o o o
|\ / \ / \ / \ /| |\ / \ /\\ / \ /|
| \ / \ / \ / \ / | | \ / \ /\\\\ / \ / |
| o o o o | | o o\\\\\o o |
| |\ / \ / \ /| | | |\ / \\\\/ \ /| |
|u | \ / \ / \ / | v| |u | \ / \\/ \ / | v|
o--+--o o o--+--o o o--+--o o o--+--o
. | \ / \ / | /X\ | \ / \ / | .
.| du \ / \ / dv | /XXX\ | du \ / \ / dv |.
o-----o o-----o /XXXXX\ o-----o o-----o
. \ / /XXXXXXX\ \ / .
. \ / /XXXXXXXXX\ \ / .
. o oXXXXXXXXXXXo o .
. //\XXXXXXXXX/\\ .
. ////\XXXXXXX/\\\\ .
!e!J //////\XXXXX/\\\\\\ DJ
. ////////\XXX/\\\\\\\\ .
. //////////\X/\\\\\\\\\\ .
. o///////////o\\\\\\\\\\\o .
. |\////////// \\\\\\\\\\/| .
. | \//////// \\\\\\\\/ | .
. | \////// \\\\\\/ | .
. | \//// \\\\/ | .
.| x \// \\/ dx |.
o-----o o-----o
\ /
\ / dx = (u, du)(v, dv) - uv
x = uv \ /
\ / dx = u dv + v du + du dv
\ /
o

Figure 56-a3. Chord Map of the Conjunction J = uv
</pre>

===Figure 56-a4. Tangent Map of the Conjunction J = uv===

<pre>
o
//\
////\
o/////o
/X\////X\
/XXX\//XXX\
oXXXXXoXXXXXo
/\\XXX//\XXX/\\
/\\\\X////\X/\\\\
o\\\\\o/////o\\\\\o
/ \\\\/\\////\\\\\/ \
/ \\/\\\\//\\\\\/ \
o o\\\\\o\\\\\o o
=|\ / \\\\/ \\\\/ \ /|=
= | \ / \\/ \\/ \ / | =
= | o o o o | =
= | |\ / \ / \ /| | =
= |u | \ / \ / \ / | v| =
o o--+--o o o--+--o o
//\ | \ / \ / | / \
////\ | du \ / \ / dv | / \
o/////o o-----o o-----o o o
//\/////\ \ / /\\ /\\
////\/////\ \ / /\\\\ /\\\\
o/////o/////o o o\\\\\o\\\\\o
/ \/////\//// \ = = /\\\\\/ \\\\/\\
/ \/////\// \ = = /\\\\\/ \\/\\\\
o o/////o o = = o\\\\\o o\\\\\o
/ \ / \//// \ / \ = = / \\\\/\\ /\\\\\/ \
/ \ / \// \ / \ = = / \\/\\\\ /\\\\\/ \
o o o o o o o\\\\\o\\\\\o o
|\ / \ / \ / \ /| |\ / \\\\/ \\\\/ \ /|
| \ / \ / \ / \ / | | \ / \\/ \\/ \ / |
| o o o o | | o o o o |
| |\ / \ / \ /| | | |\ / \ / \ /| |
|u | \ / \ / \ / | v| |u | \ / \ / \ / | v|
o--+--o o o--+--o o o--+--o o o--+--o
. | \ / \ / | /X\ | \ / \ / | .
.| du \ / \ / dv | /XXX\ | du \ / \ / dv |.
o-----o o-----o /XXXXX\ o-----o o-----o
. \ / /XXXXXXX\ \ / .
. \ / /XXXXXXXXX\ \ / .
. o oXXXXXXXXXXXo o .
. //\XXXXXXXXX/\\ .
. ////\XXXXXXX/\\\\ .
!e!J //////\XXXXX/\\\\\\ dJ
. ////////\XXX/\\\\\\\\ .
. //////////\X/\\\\\\\\\\ .
. o///////////o\\\\\\\\\\\o .
. |\////////// \\\\\\\\\\/| .
. | \//////// \\\\\\\\/ | .
. | \////// \\\\\\/ | .
. | \//// \\\\/ | .
.| x \// \\/ dx |.
o-----o o-----o
\ /
\ /
x = uv \ / dx = u dv + v du
\ /
\ /
o

Figure 56-a4. Tangent Map of the Conjunction J = uv
</pre>

===Figure 56-b1. Radius Map of the Conjunction J = uv===

<pre>
o-----------------------o
| |
| |
| |
| o--o o--o |
| / \ / \ |
| / o \ |
| / du / \ dv \ |
| o / \ o |
| | o o | |
| | | | | |
| | o o | |
| o \ / o |
| \ \ / / |
| \ o / |
| \ / \ / |
| o--o o--o |
| |
| |
| |
o-----------------------@
\
o-----------------------o \
| | \
| | \
| | \
| o--o o--o | \
| / \ / \ | \
| / o \ | \
| / du / \ dv \ | \
| o / \ o | \
| | o o | @ \
| | | | | |\ \
| | o o | | \ \
| o \ / o | \ \
| \ \ / / | \ \
| \ o / | \ \
| \ / \ / | \ \
| o--o o--o | \ \
| | \ \
| | \ \
| | \ \
o-----------------------o \ \
\ \
o-----------------------@ o--------\----------\---o o-----------------------o
| |\ | \ \ | |```````````````````````|
| | \ | \ @ | |```````````````````````|
| | \| \ | |```````````````````````|
| o--o o--o | \ o--o \o--o | |``````o--o```o--o``````|
| / \ / \ | |\ / \ /\ \ | |`````/````\`/````\`````|
| / o \ | | \ / o @ \ | |````/``````o``````\````|
| / du / \ dv \ | | \/ du /`\ dv \ | |```/``du``/`\``dv``\```|
| o / \ o | | o\ /```\ o | |``o``````/```\``````o``|
| | o o | | | | \ o`````o | | |``|`````o`````o`````|``|
| | | | | | | | @ |``@--|-----|------@``|`````|`````|`````|``|
| | o o | | | | o`````o | | |``|`````o`````o`````|``|
| o \ / o | | o \```/ o | |``o``````\```/``````o``|
| \ \ / / | | \ \`/ / | |```\``````\`/``````/```|
| \ o / | | \ o / | |````\``````o``````/````|
| \ / \ / | | \ / \ / | |`````\````/`\````/`````|
| o--o o--o | | o--o o--o | |``````o--o```o--o``````|
| | | | |```````````````````````|
| | | | |```````````````````````|
| | | | |```````````````````````|
o-----------------------o o-----------------------o o-----------------------o
\ / \ / \ /
\ !h!J / \ J / \ !h!J /
\ / \ / \ /
\ / o----------\---------/----------o \ /
\ / | \ / | \ /
\ / | \ / | \ /
\ / | o-----o-----o | \ /
\ / | /`````````````\ | \ /
\ / | /```````````````\ | \ /
o------\---/------o | /`````````````````\ | o------\---/------o
| \ / | | /```````````````````\ | | \ / |
| o--o--o | | /`````````````````````\ | | o--o--o |
| /```````\ | | o```````````````````````o | | /```````\ |
| /`````````\ | | |```````````````````````| | | /`````````\ |
| o```````````o | | |```````````````````````| | | o```````````o |
| |````dx`````| @----@ |```````````x`````@-----|------@ |``` dx ````| |
| o```````````o | | |```````````````````````| | | o```````````o |
| \`````````/ | | |```````````````````````| | | \`````````/ |
| \```````/ | | o```````````````````````o | | \```````/ |
| o-----o | | \`````````````````````/ | | o-----o |
| | | \```````````````````/ | | |
o-----------------o | \`````````````````/ | o-----------------o
| \```````````````/ |
| \`````````````/ |
| o-----------o |
| |
| |
o-------------------------------o

Figure 56-b1. Radius Map of the Conjunction J = uv
</pre>

===Figure 56-b2. Secant Map of the Conjunction J = uv===

<pre>
o-----------------------o
| |
| |
| |
| o--o o--o |
| / \ / \ |
| / o \ |
| / du /`\ dv \ |
| o /```\ o |
| | o`````o | |
| | |`````| | |
| | o`````o | |
| o \```/ o |
| \ \`/ / |
| \ o / |
| \ / \ / |
| o--o o--o |
| |
| |
| |
o-----------------------@
\
o-----------------------o \
| | \
| | \
| | \
| o--o o--o | \
| /````\ / \ | \
| /``````o \ | \
| /``du``/ \ dv \ | \
| o``````/ \ o | \
| |`````o o | @ \
| |`````| | | |\ \
| |`````o o | | \ \
| o``````\ / o | \ \
| \``````\ / / | \ \
| \``````o / | \ \
| \````/ \ / | \ \
| o--o o--o | \ \
| | \ \
| | \ \
| | \ \
o-----------------------o \ \
\ \
o-----------------------@ o--------\----------\---o o-----------------------o
| |\ | \ \ | |```````````````````````|
| | \ | \ @ | |```````````````````````|
| | \| \ | |```````````````````````|
| o--o o--o | \ o--o \o--o | |``````o--o```o--o``````|
| / \ /````\ | |\ / \ /\ \ | |`````/ \`/ \`````|
| / o``````\ | | \ / o @ \ | |````/ o \````|
| / du / \``dv``\ | | \/ du /`\ dv \ | |```/ du / \ dv \```|
| o / \``````o | | o\ /```\ o | |``o / \ o``|
| | o o`````| | | | \ o`````o | | |``| o o |``|
| | | |`````| | | | @ |``@--|-----|------@``| | | |``|
| | o o`````| | | | o`````o | | |``| o o |``|
| o \ /``````o | | o \```/ o | |``o \ / o``|
| \ \ /``````/ | | \ \`/ / | |```\ \ / /```|
| \ o``````/ | | \ o / | |````\ o /````|
| \ / \````/ | | \ / \ / | |`````\ /`\ /`````|
| o--o o--o | | o--o o--o | |``````o--o```o--o``````|
| | | | |```````````````````````|
| | | | |```````````````````````|
| | | | |```````````````````````|
o-----------------------o o-----------------------o o-----------------------o
\ / \ / \ /
\ EJ / \ J / \ EJ /
\ / \ / \ /
\ / o----------\---------/----------o \ /
\ / | \ / | \ /
\ / | \ / | \ /
\ / | o-----o-----o | \ /
\ / | /`````````````\ | \ /
\ / | /```````````````\ | \ /
o------\---/------o | /`````````````````\ | o------\---/------o
| \ / | | /```````````````````\ | | \ / |
| o--o--o | | /`````````````````````\ | | o--o--o |
| /```````\ | | o```````````````````````o | | /```````\ |
| /`````````\ | | |```````````````````````| | | /`````````\ |
| o```````````o | | |```````````````````````| | | o```````````o |
| |````dx`````| @----@ |```````````x`````@-----|------@ |``` dx ````| |
| o```````````o | | |```````````````````````| | | o```````````o |
| \`````````/ | | |```````````````````````| | | \`````````/ |
| \```````/ | | o```````````````````````o | | \```````/ |
| o-----o | | \`````````````````````/ | | o-----o |
| | | \```````````````````/ | | |
o-----------------o | \`````````````````/ | o-----------------o
| \```````````````/ |
| \`````````````/ |
| o-----------o |
| |
| |
o-------------------------------o

Figure 56-b2. Secant Map of the Conjunction J = uv
</pre>

===Figure 56-b3. Chord Map of the Conjunction J = uv===

<pre>
o-----------------------o
| |
| |
| |
| o--o o--o |
| / \ / \ |
| / o \ |
| / du /`\ dv \ |
| o /```\ o |
| | o`````o | |
| | |`````| | |
| | o`````o | |
| o \```/ o |
| \ \`/ / |
| \ o / |
| \ / \ / |
| o--o o--o |
| |
| |
| |
o-----------------------@
\
o-----------------------o \
| | \
| | \
| | \
| o--o o--o | \
| /````\ / \ | \
| /``````o \ | \
| /``du``/ \ dv \ | \
| o``````/ \ o | \
| |`````o o | @ \
| |`````| | | |\ \
| |`````o o | | \ \
| o``````\ / o | \ \
| \``````\ / / | \ \
| \``````o / | \ \
| \````/ \ / | \ \
| o--o o--o | \ \
| | \ \
| | \ \
| | \ \
o-----------------------o \ \
\ \
o-----------------------@ o--------\----------\---o o-----------------------o
| |\ | \ \ | | |
| | \ | \ @ | | |
| | \| \ | | |
| o--o o--o | \ o--o \o--o | | o--o o--o |
| / \ /````\ | |\ / \ /\ \ | | /````\ /````\ |
| / o``````\ | | \ / o @ \ | | /``````o``````\ |
| / du / \``dv``\ | | \/ du /`\ dv \ | | /``du``/`\``dv``\ |
| o / \``````o | | o\ /```\ o | | o``````/```\``````o |
| | o o`````| | | | \ o`````o | | | |`````o`````o`````| |
| | | |`````| | | | @ |``@--|-----|------@ |`````|`````|`````| |
| | o o`````| | | | o`````o | | | |`````o`````o`````| |
| o \ /``````o | | o \```/ o | | o``````\```/``````o |
| \ \ /``````/ | | \ \`/ / | | \``````\`/``````/ |
| \ o``````/ | | \ o / | | \``````o``````/ |
| \ / \````/ | | \ / \ / | | \````/ \````/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
| | | | | |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
\ / \ / \ /
\ DJ / \ J / \ DJ /
\ / \ / \ /
\ / o----------\---------/----------o \ /
\ / | \ / | \ /
\ / | \ / | \ /
\ / | o-----o-----o | \ /
\ / | /`````````````\ | \ /
\ / | /```````````````\ | \ /
o------\---/------o | /`````````````````\ | o------\---/------o
| \ / | | /```````````````````\ | | \ / |
| o--o--o | | /`````````````````````\ | | o--o--o |
| /```````\ | | o```````````````````````o | | /```````\ |
| /`````````\ | | |```````````````````````| | | /`````````\ |
| o```````````o | | |```````````````````````| | | o```````````o |
| |````dx`````| @----@ |```````````x`````@-----|------@ |``` dx ````| |
| o```````````o | | |```````````````````````| | | o```````````o |
| \`````````/ | | |```````````````````````| | | \`````````/ |
| \```````/ | | o```````````````````````o | | \```````/ |
| o-----o | | \`````````````````````/ | | o-----o |
| | | \```````````````````/ | | |
o-----------------o | \`````````````````/ | o-----------------o
| \```````````````/ |
| \`````````````/ |
| o-----------o |
| |
| |
o-------------------------------o

Figure 56-b3. Chord Map of the Conjunction J = uv
</pre>

===Figure 56-b4. Tangent Map of the Conjunction J = uv===

<pre>
o-----------------------o
| |
| |
| |
| o--o o--o |
| / \ / \ |
| / o \ |
| / du / \ dv \ |
| o / \ o |
| | o o | |
| | | | | |
| | o o | |
| o \ / o |
| \ \ / / |
| \ o / |
| \ / \ / |
| o--o o--o |
| |
| |
| |
o-----------------------@
\
o-----------------------o \
| | \
| | \
| | \
| o--o o--o | \
| /````\ / \ | \
| /``````o \ | \
| /``du``/`\ dv \ | \
| o``````/```\ o | \
| |`````o`````o | @ \
| |`````|`````| | |\ \
| |`````o`````o | | \ \
| o``````\```/ o | \ \
| \``````\`/ / | \ \
| \``````o / | \ \
| \````/ \ / | \ \
| o--o o--o | \ \
| | \ \
| | \ \
| | \ \
o-----------------------o \ \
\ \
o-----------------------@ o--------\----------\---o o-----------------------o
| |\ | \ \ | | |
| | \ | \ @ | | |
| | \| \ | | |
| o--o o--o | \ o--o \o--o | | o--o o--o |
| / \ /````\ | |\ / \ /\ \ | | /````\ /````\ |
| / o``````\ | | \ / o @ \ | | /``````o``````\ |
| / du /`\``dv``\ | | \/ du /`\ dv \ | | /``du``/ \``dv``\ |
| o /```\``````o | | o\ /```\ o | | o``````/ \``````o |
| | o`````o`````| | | | \ o`````o | | | |`````o o`````| |
| | |`````|`````| | | | @ |``@--|-----|------@ |`````| |`````| |
| | o`````o`````| | | | o`````o | | | |`````o o`````| |
| o \```/``````o | | o \```/ o | | o``````\ /``````o |
| \ \`/``````/ | | \ \`/ / | | \``````\ /``````/ |
| \ o``````/ | | \ o / | | \``````o``````/ |
| \ / \````/ | | \ / \ / | | \````/ \````/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
| | | | | |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
\ / \ / \ /
\ dJ / \ J / \ dJ /
\ / \ / \ /
\ / o----------\---------/----------o \ /
\ / | \ / | \ /
\ / | \ / | \ /
\ / | o-----o-----o | \ /
\ / | /`````````````\ | \ /
\ / | /```````````````\ | \ /
o------\---/------o | /`````````````````\ | o------\---/------o
| \ / | | /```````````````````\ | | \ / |
| o--o--o | | /`````````````````````\ | | o--o--o |
| /```````\ | | o```````````````````````o | | /```````\ |
| /`````````\ | | |```````````````````````| | | /`````````\ |
| o```````````o | | |```````````````````````| | | o```````````o |
| |````dx`````| @----@ |```````````x`````@-----|------@ |``` dx ````| |
| o```````````o | | |```````````````````````| | | o```````````o |
| \`````````/ | | |```````````````````````| | | \`````````/ |
| \```````/ | | o```````````````````````o | | \```````/ |
| o-----o | | \`````````````````````/ | | o-----o |
| | | \```````````````````/ | | |
o-----------------o | \`````````````````/ | o-----------------o
| \```````````````/ |
| \`````````````/ |
| o-----------o |
| |
| |
o-------------------------------o

Figure 56-b4. Tangent Map of the Conjunction J = uv
</pre>

===Figure 57-1. Radius Operator Diagram for the Conjunction J = uv===

<pre>
o o
//\ /X\
////\ /XXX\
//////\ oXXXXXo
////////\ /X\XXX/X\
//////////\ /XXX\X/XXX\
o///////////o oXXXXXoXXXXXo
/ \////////// \ / \XXX/X\XXX/ \
/ \//////// \ / \X/XXX\X/ \
/ \////// \ o oXXXXXo o
/ \//// \ / \ / \XXX/ \ / \
/ \// \ / \ / \X/ \ / \
o o o o o o o o
|\ / \ /| |\ / \ / \ / \ /|
| \ / \ / | | \ / \ / \ / \ / |
| \ / \ / | | o o o o |
| \ / \ / | | |\ / \ / \ /| |
| u \ / \ / v | |u | \ / \ / \ / | v|
o-----o o-----o o--+--o o o--+--o
\ / | \ / \ / |
\ / | du \ / \ / dv |
\ / o-----o o-----o
\ / \ /
\ / \ /
o o
U% $e$ $E$U%
o------------------>o
| |
| |
| |
| |
J | | $e$J
| |
| |
| |
v v
o------------------>o
X% $e$ $E$X%
o o
//\ /X\
////\ /XXX\
//////\ /XXXXX\
////////\ /XXXXXXX\
//////////\ /XXXXXXXXX\
////////////o oXXXXXXXXXXXo
///////////// \ //\XXXXXXXXX/\\
///////////// \ ////\XXXXXXX/\\\\
///////////// \ //////\XXXXX/\\\\\\
///////////// \ ////////\XXX/\\\\\\\\
///////////// \ //////////\X/\\\\\\\\\\
o//////////// o o///////////o\\\\\\\\\\\o
|\////////// / |\////////// \\\\\\\\\\/|
| \//////// / | \//////// \\\\\\\\/ |
| \////// / | \////// \\\\\\/ |
| \//// / | \//// \\\\/ |
| x \// / | x \// \\/ dx |
o-----o / o-----o o-----o
\ / \ /
\ / \ /
\ / \ /
\ / \ /
\ / \ /
o o

Figure 57-1. Radius Operator Diagram for the Conjunction J = uv
</pre>

===Figure 57-2. Secant Operator Diagram for the Conjunction J = uv===

<pre>
o o
//\ /X\
////\ /XXX\
//////\ oXXXXXo
////////\ //\XXX//\
//////////\ ////\X////\
o///////////o o/////o/////o
/ \////////// \ /\\/////\////\\
/ \//////// \ /\\\\/////\//\\\\
/ \////// \ o\\\\\o/////o\\\\\o
/ \//// \ / \\\\/ \//// \\\\/ \
/ \// \ / \\/ \// \\/ \
o o o o o o o o
|\ / \ /| |\ / \ /\\ / \ /|
| \ / \ / | | \ / \ /\\\\ / \ / |
| \ / \ / | | o o\\\\\o o |
| \ / \ / | | |\ / \\\\/ \ /| |
| u \ / \ / v | |u | \ / \\/ \ / | v|
o-----o o-----o o--+--o o o--+--o
\ / | \ / \ / |
\ / | du \ / \ / dv |
\ / o-----o o-----o
\ / \ /
\ / \ /
o o
U% $E$ $E$U%
o------------------>o
| |
| |
| |
| |
J | | $E$J
| |
| |
| |
v v
o------------------>o
X% $E$ $E$X%
o o
//\ /X\
////\ /XXX\
//////\ /XXXXX\
////////\ /XXXXXXX\
//////////\ /XXXXXXXXX\
////////////o oXXXXXXXXXXXo
///////////// \ //\XXXXXXXXX/\\
///////////// \ ////\XXXXXXX/\\\\
///////////// \ //////\XXXXX/\\\\\\
///////////// \ ////////\XXX/\\\\\\\\
///////////// \ //////////\X/\\\\\\\\\\
o//////////// o o///////////o\\\\\\\\\\\o
|\////////// / |\////////// \\\\\\\\\\/|
| \//////// / | \//////// \\\\\\\\/ |
| \////// / | \////// \\\\\\/ |
| \//// / | \//// \\\\/ |
| x \// / | x \// \\/ dx |
o-----o / o-----o o-----o
\ / \ /
\ / \ /
\ / \ /
\ / \ /
\ / \ /
o o

Figure 57-2. Secant Operator Diagram for the Conjunction J = uv
</pre>

===Figure 57-3. Chord Operator Diagram for the Conjunction J = uv===

<pre>
o o
//\ //\
////\ ////\
//////\ o/////o
////////\ /X\////X\
//////////\ /XXX\//XXX\
o///////////o oXXXXXoXXXXXo
/ \////////// \ /\\XXX/X\XXX/\\
/ \//////// \ /\\\\X/XXX\X/\\\\
/ \////// \ o\\\\\oXXXXXo\\\\\o
/ \//// \ / \\\\/ \XXX/ \\\\/ \
/ \// \ / \\/ \X/ \\/ \
o o o o o o o o
|\ / \ /| |\ / \ /\\ / \ /|
| \ / \ / | | \ / \ /\\\\ / \ / |
| \ / \ / | | o o\\\\\o o |
| \ / \ / | | |\ / \\\\/ \ /| |
| u \ / \ / v | |u | \ / \\/ \ / | v|
o-----o o-----o o--+--o o o--+--o
\ / | \ / \ / |
\ / | du \ / \ / dv |
\ / o-----o o-----o
\ / \ /
\ / \ /
o o
U% $D$ $E$U%
o------------------>o
| |
| |
| |
| |
J | | $D$J
| |
| |
| |
v v
o------------------>o
X% $D$ $E$X%
o o
//\ /X\
////\ /XXX\
//////\ /XXXXX\
////////\ /XXXXXXX\
//////////\ /XXXXXXXXX\
////////////o oXXXXXXXXXXXo
///////////// \ //\XXXXXXXXX/\\
///////////// \ ////\XXXXXXX/\\\\
///////////// \ //////\XXXXX/\\\\\\
///////////// \ ////////\XXX/\\\\\\\\
///////////// \ //////////\X/\\\\\\\\\\
o//////////// o o///////////o\\\\\\\\\\\o
|\////////// / |\////////// \\\\\\\\\\/|
| \//////// / | \//////// \\\\\\\\/ |
| \////// / | \////// \\\\\\/ |
| \//// / | \//// \\\\/ |
| x \// / | x \// \\/ dx |
o-----o / o-----o o-----o
\ / \ /
\ / \ /
\ / \ /
\ / \ /
\ / \ /
o o

Figure 57-3. Chord Operator Diagram for the Conjunction J = uv
</pre>

===Figure 57-4. Tangent Functor Diagram for the Conjunction J = uv===

<pre>
o o
//\ //\
////\ ////\
//////\ o/////o
////////\ /X\////X\
//////////\ /XXX\//XXX\
o///////////o oXXXXXoXXXXXo
/ \////////// \ /\\XXX//\XXX/\\
/ \//////// \ /\\\\X////\X/\\\\
/ \////// \ o\\\\\o/////o\\\\\o
/ \//// \ / \\\\/\\////\\\\\/ \
/ \// \ / \\/\\\\//\\\\\/ \
o o o o o\\\\\o\\\\\o o
|\ / \ /| |\ / \\\\/ \\\\/ \ /|
| \ / \ / | | \ / \\/ \\/ \ / |
| \ / \ / | | o o o o |
| \ / \ / | | |\ / \ / \ /| |
| u \ / \ / v | |u | \ / \ / \ / | v|
o-----o o-----o o--+--o o o--+--o
\ / | \ / \ / |
\ / | du \ / \ / dv |
\ / o-----o o-----o
\ / \ /
\ / \ /
o o
U% $T$ $E$U%
o------------------>o
| |
| |
| |
| |
J | | $T$J
| |
| |
| |
v v
o------------------>o
X% $T$ $E$X%
o o
//\ /X\
////\ /XXX\
//////\ /XXXXX\
////////\ /XXXXXXX\
//////////\ /XXXXXXXXX\
////////////o oXXXXXXXXXXXo
///////////// \ //\XXXXXXXXX/\\
///////////// \ ////\XXXXXXX/\\\\
///////////// \ //////\XXXXX/\\\\\\
///////////// \ ////////\XXX/\\\\\\\\
///////////// \ //////////\X/\\\\\\\\\\
o//////////// o o///////////o\\\\\\\\\\\o
|\////////// / |\////////// \\\\\\\\\\/|
| \//////// / | \//////// \\\\\\\\/ |
| \////// / | \////// \\\\\\/ |
| \//// / | \//// \\\\/ |
| x \// / | x \// \\/ dx |
o-----o / o-----o o-----o
\ / \ /
\ / \ /
\ / \ /
\ / \ /
\ / \ /
o o

Figure 57-4. Tangent Functor Diagram for the Conjunction J = uv
</pre>

===Formula Display 11===

<pre>
o-----------------------------------------------------------o
| |
| F = <f, g> = <F_1, F_2> : [u, v] -> [x, y] |
| |
| where f = F_1 : [u, v] -> [x] |
| |
| and g = F_2 : [u, v] -> [y] |
| |
o-----------------------------------------------------------o
</pre>

===Table 58. Cast of Characters: Expansive Subtypes of Objects and Operators===

<pre>
Table 58. Cast of Characters: Expansive Subtypes of Objects and Operators
o------o-------------------------o------------------o----------------------------o
| Item | Notation | Description | Type |
o------o-------------------------o------------------o----------------------------o
| | | | |
| U% | = [u, v] | Source Universe | [B^n] |
| | | | |
o------o-------------------------o------------------o----------------------------o
| | | | |
| X% | = [x, y] | Target Universe | [B^k] |
| | = [f, g] | | |
| | | | |
o------o-------------------------o------------------o----------------------------o
| | | | |
| EU% | = [u, v, du, dv] | Extended | [B^n x D^n] |
| | | Source Universe | |
| | | | |
o------o-------------------------o------------------o----------------------------o
| | | | |
| EX% | = [x, y, dx, dy] | Extended | [B^k x D^k] |
| | = [f, g, df, dg] | Target Universe | |
| | | | |
o------o-------------------------o------------------o----------------------------o
| | | | |
| F | F = <f, g> : U% -> X% | Transformation, | [B^n] -> [B^k] |
| | | or Mapping | |
| | | | |
o------o-------------------------o------------------o----------------------------o
| | | | |
| | f, g : U -> B | Proposition, | B^n -> B |
| | | special case | |
| f | f : U -> [x] c X% | of a mapping, | c (B^n, B^n -> B) |
| | | or component | |
| g | g : U -> [y] c X% | of a mapping. | = (B^n +-> B) = [B^n] |
| | | | |
o------o-------------------------o------------------o----------------------------o
| | | | |
| W | W : | Operator | |
| | U% -> EU%, | | [B^n] -> [B^n x D^n], |
| | X% -> EX%, | | [B^k] -> [B^k x D^k], |
| | (U%->X%)->(EU%->EX%), | | ([B^n] -> [B^k]) |
| | for each W among: | | -> |
| | !e!, !h!, E, D, d | | ([B^n x D^n]->[B^k x D^k]) |
| | | | |
o------o-------------------------o------------------o----------------------------o
| | | |
| !e! | | Tacit Extension Operator !e! |
| !h! | | Trope Extension Operator !h! |
| E | | Enlargement Operator E |
| D | | Difference Operator D |
| d | | Differential Operator d |
| | | |
o------o-------------------------o------------------o----------------------------o
| | | | |
| $W$ | $W$ : | Operator | |
| | U% -> $T$U% = EU%, | | [B^n] -> [B^n x D^n], |
| | X% -> $T$X% = EX%, | | [B^k] -> [B^k x D^k], |
| | (U%->X%)->($T$U%->$T$X%)| | ([B^n] -> [B^k]) |
| | for each $W$ among: | | -> |
| | $e$, $E$, $D$, $T$ | | ([B^n x D^n]->[B^k x D^k]) |
| | | | |
o------o-------------------------o------------------o----------------------------o
| | | |
| $e$ | | Radius Operator $e$ = <!e!, !h!> |
| $E$ | | Secant Operator $E$ = <!e!, E > |
| $D$ | | Chord Operator $D$ = <!e!, D > |
| $T$ | | Tangent Functor $T$ = <!e!, d > |
| | | |
o------o-------------------------o-----------------------------------------------o
</pre>

===Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes===

<pre>
Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes
o--------------o----------------------o--------------------o----------------------o
| | Operator | Proposition | Transformation |
| | or | or | or |
| | Operand | Component | Mapping |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Operand | F = <F_1, F_2> | F_i : <|u,v|> -> B | F : [u, v] -> [x, y] |
| | | | |
| | F = <f, g> : U -> X | F_i : B^n -> B | F : B^n -> B^k |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Tacit | !e! : | !e!F_i : | !e!F : |
| Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> B | [u,v,du,dv]->[x, y] |
| | (U%->X%)->(EU%->X%) | B^n x D^n -> B | [B^n x D^n]->[B^k] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Trope | !h! : | !h!F_i : | !h!F : |
| Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
| | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Enlargement | E : | EF_i : | EF : |
| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
| | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Difference | D : | DF_i : | DF : |
| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
| | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Differential | d : | dF_i : | dF : |
| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
| | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Remainder | r : | rF_i : | rF : |
| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
| | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Radius | $e$ = <!e!, !h!> : | | $e$F : |
| Operator | | | |
| | U%->EU%, X%->EX%, | | [u, v, du, dv] -> |
| | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], |
| | | | |
| | | | [B^n x D^n] -> |
| | | | [B^k x D^k] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Secant | $E$ = <!e!, E> : | | $E$F : |
| Operator | | | |
| | U%->EU%, X%->EX%, | | [u, v, du, dv] -> |
| | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], |
| | | | |
| | | | [B^n x D^n] -> |
| | | | [B^k x D^k] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Chord | $D$ = <!e!, D> : | | $D$F : |
| Operator | | | |
| | U%->EU%, X%->EX%, | | [u, v, du, dv] -> |
| | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], |
| | | | |
| | | | [B^n x D^n] -> |
| | | | [B^k x D^k] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Tangent | $T$ = <!e!, d> : | dF_i : | $T$F : |
| Functor | | | |
| | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u, v, du, dv] -> |
| | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], |
| | | | |
| | | B^n x D^n -> D | [B^n x D^n] -> |
| | | | [B^k x D^k] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
</pre>

===Formula Display 12===

<pre>
o-----------------------------------------------------------o
| |
| x = f(u, v) = ((u)(v)) |
| |
| y = g(u, v) = ((u, v)) |
| |
o-----------------------------------------------------------o
</pre>

===Formula Display 13===

<pre>
o-----------------------------------------------------------o
| |
| <x, y> = F<u, v> = <((u)(v)), ((u, v))> |
| |
o-----------------------------------------------------------o
</pre>

===Table 60. Propositional Transformation===

<pre>
Table 60. Propositional Transformation
o-------------o-------------o-------------o-------------o
| u | v | f | g |
o-------------o-------------o-------------o-------------o
| | | | |
| 0 | 0 | 0 | 1 |
| | | | |
| 0 | 1 | 1 | 0 |
| | | | |
| 1 | 0 | 1 | 0 |
| | | | |
| 1 | 1 | 1 | 1 |
| | | | |
o-------------o-------------o-------------o-------------o
| | | ((u)(v)) | ((u, v)) |
o-------------o-------------o-------------o-------------o
</pre>

===Figure 61. Propositional Transformation===

<pre>
o-----------------------------------------------------o
| U |
| |
| o-----------o o-----------o |
| / \ / \ |
| / o \ |
| / / \ \ |
| / / \ \ |
| o o o o |
| | | | | |
| | u | | v | |
| | | | | |
| o o o o |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| o-----------o o-----------o |
| |
| |
o-----------------------------------------------------o
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
o-------------------------o o-------------------------o
| U | |\U \\\\\\\\\\\\\\\\\\\\\\|
| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|
| //////\ //////\ | |\\\\\/ \\/ \\\\\\|
| ////////o///////\ | |\\\\/ o \\\\\|
| //////////\///////\ | |\\\/ /\\ \\\\|
| o///////o///o///////o | |\\o o\\\o o\\|
| |// u //|///|// v //| | |\\| u |\\\| v |\\|
| o///////o///o///////o | |\\o o\\\o o\\|
| \///////\////////// | |\\\\ \\/ /\\\|
| \///////o//////// | |\\\\\ o /\\\\|
| \////// \////// | |\\\\\\ /\\ /\\\\\|
| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|
| | |\\\\\\\\\\\\\\\\\\\\\\\\\|
o-------------------------o o-------------------------o
\ | | /
\ | | /
\ | | /
\ f | | g /
\ | | /
\ | | /
\ | | /
\ | | /
\ | | /
\ | | /
o-------\----|---------------------------|----/-------o
| X \ | | / |
| \| |/ |
| o-----------o o-----------o |
| //////////////\ /\\\\\\\\\\\\\\ |
| ////////////////o\\\\\\\\\\\\\\\\ |
| /////////////////X\\\\\\\\\\\\\\\\\ |
| /////////////////XXX\\\\\\\\\\\\\\\\\ |
| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |
| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |
| |////// x //////|XXXXX|\\\\\\ y \\\\\\| |
| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |
| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |
| \///////////////\XXX/\\\\\\\\\\\\\\\/ |
| \///////////////\X/\\\\\\\\\\\\\\\/ |
| \///////////////o\\\\\\\\\\\\\\\/ |
| \////////////// \\\\\\\\\\\\\\/ |
| o-----------o o-----------o |
| |
| |
o-----------------------------------------------------o
Figure 61. Propositional Transformation
</pre>

===Figure 62. Propositional Transformation (Short Form)===

<pre>
o-------------------------o o-------------------------o
| U | |\U \\\\\\\\\\\\\\\\\\\\\\|
| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|
| //////\ //////\ | |\\\\\/ \\/ \\\\\\|
| ////////o///////\ | |\\\\/ o \\\\\|
| //////////\///////\ | |\\\/ /\\ \\\\|
| o///////o///o///////o | |\\o o\\\o o\\|
| |// u //|///|// v //| | |\\| u |\\\| v |\\|
| o///////o///o///////o | |\\o o\\\o o\\|
| \///////\////////// | |\\\\ \\/ /\\\|
| \///////o//////// | |\\\\\ o /\\\\|
| \////// \////// | |\\\\\\ /\\ /\\\\\|
| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|
| | |\\\\\\\\\\\\\\\\\\\\\\\\\|
o-------------------------o o-------------------------o
\ / \ /
\ / \ /
\ / \ /
\ f / \ g /
\ / \ /
\ / \ /
\ / \ /
\ / \ /
\ / \ /
o---------\-----/---------------------\-----/---------o
| X \ / \ / |
| \ / \ / |
| o-----------o o-----------o |
| //////////////\ /\\\\\\\\\\\\\\ |
| ////////////////o\\\\\\\\\\\\\\\\ |
| /////////////////X\\\\\\\\\\\\\\\\\ |
| /////////////////XXX\\\\\\\\\\\\\\\\\ |
| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |
| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |
| |////// x //////|XXXXX|\\\\\\ y \\\\\\| |
| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |
| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |
| \///////////////\XXX/\\\\\\\\\\\\\\\/ |
| \///////////////\X/\\\\\\\\\\\\\\\/ |
| \///////////////o\\\\\\\\\\\\\\\/ |
| \////////////// \\\\\\\\\\\\\\/ |
| o-----------o o-----------o |
| |
| |
o-----------------------------------------------------o
Figure 62. Propositional Transformation (Short Form)
</pre>

===Figure 63. Transformation of Positions===

<pre>
o-----------------------------------------------------o
|`U` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
|` ` ` ` ` ` o-----------o ` o-----------o ` ` ` ` ` `|
|` ` ` ` ` `/' ' ' ' ' ' '\`/' ' ' ' ' ' '\` ` ` ` ` `|
|` ` ` ` ` / ' ' ' ' ' ' ' o ' ' ' ' ' ' ' \ ` ` ` ` `|
|` ` ` ` `/' ' ' ' ' ' ' '/^\' ' ' ' ' ' ' '\` ` ` ` `|
|` ` ` ` / ' ' ' ' ' ' ' /^^^\ ' ' ' ' ' ' ' \ ` ` ` `|
|` ` ` `o' ' ' ' ' ' ' 'o^^^^^o' ' ' ' ' ' ' 'o` ` ` `|
|` ` ` `|' ' ' ' ' ' ' '|^^^^^|' ' ' ' ' ' ' '|` ` ` `|
|` ` ` `|' ' ' ' u ' ' '|^^^^^|' ' ' v ' ' ' '|` ` ` `|
|` ` ` `|' ' ' ' ' ' ' '|^^^^^|' ' ' ' ' ' ' '|` ` ` `|
|` `@` `o' ' ' ' @ ' ' 'o^^@^^o' ' ' @ ' ' ' 'o` ` ` `|
|` ` \ ` \ ' ' ' | ' ' ' \^|^/ ' ' ' | ' ' ' / ` ` ` `|
|` ` `\` `\' ' ' | ' ' ' '\|/' ' ' ' | ' ' '/` ` ` ` `|
|` ` ` \ ` \ ' ' | ' ' ' ' | ' ' ' ' | ' ' / ` ` ` ` `|
|` ` ` `\` `\' ' | ' ' ' '/|\' ' ' ' | ' '/` ` ` ` ` `|
|` ` ` ` \ ` o---|-------o | o-------|---o ` ` ` ` ` `|
|` ` ` ` `\` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` `|
|` ` ` ` ` \ ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` `|
o-----------\----|---------|---------|----------------o
" " \ | | | " "
" " \ | | | " "
" " \ | | | " "
" " \| | | " "
o-------------------------o \ | | o-------------------------o
| U | |\ | | |`U```````````````````````|
| o---o o---o | | \ | | |``````o---o```o---o``````|
| /'''''\ /'''''\ | | \ | | |`````/ \`/ \`````|
| /'''''''o'''''''\ | | \ | | |````/ o \````|
| /'''''''/'\'''''''\ | | \ | | |```/ /`\ \```|
| o'''''''o'''o'''''''o | | \ | | |``o o```o o``|
| |'''u'''|'''|'''v'''| | | \ | | |``| u |```| v |``|
| o'''''''o'''o'''''''o | | \ | | |``o o```o o``|
| \'''''''\'/'''''''/ | | \| | |```\ \`/ /```|
| \'''''''o'''''''/ | | \ | |````\ o /````|
| \'''''/ \'''''/ | | |\ | |`````\ /`\ /`````|
| o---o o---o | | | \ | |``````o---o```o---o``````|
| | | | \ * |`````````````````````````|
o-------------------------o | | \ / o-------------------------o
\ | | | \ / | /
\ ((u)(v)) | | | \/ | ((u, v)) /
\ | | | /\ | /
\ | | | / \ | /
\ | | | / \ | /
\ | | | / * | /
\ | | | / | | /
\ | | |/ | | /
\ | | / | | /
\ | | /| | | /
o-------\----|---|-------/-|---------|---|----/-------o
| X \ | | / | | | / |
| \| | / | | |/ |
| o---|----/--o | o-------|---o |
| /' ' | ' / ' '\|/` ` ` ` | ` `\ |
| / ' ' | '/' ' ' | ` ` ` ` | ` ` \ |
| /' ' ' | / ' ' '/|\` ` ` ` | ` ` `\ |
| / ' ' ' |/' ' ' /^|^\ ` ` ` | ` ` ` \ |
| @ o' ' ' ' @ ' ' 'o^^@^^o` ` ` @ ` ` ` `o |
| |' ' ' ' ' ' ' '|^^^^^|` ` ` ` ` ` ` `| |
| |' ' ' ' f ' ' '|^^^^^|` ` ` g ` ` ` `| |
| |' ' ' ' ' ' ' '|^^^^^|` ` ` ` ` ` ` `| |
| o' ' ' ' ' ' ' 'o^^^^^o` ` ` ` ` ` ` `o |
| \ ' ' ' ' ' ' ' \^^^/ ` ` ` ` ` ` ` / |
| \' ' ' ' ' ' ' '\^/` ` ` ` ` ` ` `/ |
| \ ' ' ' ' ' ' ' o ` ` ` ` ` ` ` / |
| \' ' ' ' ' ' '/ \` ` ` ` ` ` `/ |
| o-----------o o-----------o |
| |
| |
o-----------------------------------------------------o
Figure 63. Transformation of Positions
</pre>

===Table 64. Transformation of Positions===

<pre>
Table 64. Transformation of Positions
o-----o----------o----------o-------o-------o--------o--------o-------------o
| u v | x | y | x y | x(y) | (x)y | (x)(y) | X% = [x, y] |
o-----o----------o----------o-------o-------o--------o--------o-------------o
| | | | | | | | ^ |
| 0 0 | 0 | 1 | 0 | 0 | 1 | 0 | | |
| | | | | | | | |
| 0 1 | 1 | 0 | 0 | 1 | 0 | 0 | F |
| | | | | | | | = |
| 1 0 | 1 | 0 | 0 | 1 | 0 | 0 | <f , g> |
| | | | | | | | |
| 1 1 | 1 | 1 | 1 | 0 | 0 | 0 | ^ |
| | | | | | | | | |
o-----o----------o----------o-------o-------o--------o--------o-------------o
| | ((u)(v)) | ((u, v)) | u v | (u,v) | (u)(v) | 0 | U% = [u, v] |
o-----o----------o----------o-------o-------o--------o--------o-------------o
</pre>

===Table 65. Induced Transformation on Propositions===

<pre>
Table 65. Induced Transformation on Propositions
o------------o---------------------------------o------------o
| X% | <--- F = <f , g> <--- | U% |
o------------o----------o-----------o----------o------------o
| | u = | 1 1 0 0 | = u | |
| | v = | 1 0 1 0 | = v | |
| f_i <x, y> o----------o-----------o----------o f_j <u, v> |
| | x = | 1 1 1 0 | = f<u,v> | |
| | y = | 1 0 0 1 | = g<u,v> | |
o------------o----------o-----------o----------o------------o
| | | | | |
| f_0 | () | 0 0 0 0 | () | f_0 |
| | | | | |
| f_1 | (x)(y) | 0 0 0 1 | () | f_0 |
| | | | | |
| f_2 | (x) y | 0 0 1 0 | (u)(v) | f_1 |
| | | | | |
| f_3 | (x) | 0 0 1 1 | (u)(v) | f_1 |
| | | | | |
| f_4 | x (y) | 0 1 0 0 | (u, v) | f_6 |
| | | | | |
| f_5 | (y) | 0 1 0 1 | (u, v) | f_6 |
| | | | | |
| f_6 | (x, y) | 0 1 1 0 | (u v) | f_7 |
| | | | | |
| f_7 | (x y) | 0 1 1 1 | (u v) | f_7 |
| | | | | |
o------------o----------o-----------o----------o------------o
| | | | | |
| f_8 | x y | 1 0 0 0 | u v | f_8 |
| | | | | |
| f_9 | ((x, y)) | 1 0 0 1 | u v | f_8 |
| | | | | |
| f_10 | y | 1 0 1 0 | ((u, v)) | f_9 |
| | | | | |
| f_11 | (x (y)) | 1 0 1 1 | ((u, v)) | f_9 |
| | | | | |
| f_12 | x | 1 1 0 0 | ((u)(v)) | f_14 |
| | | | | |
| f_13 | ((x) y) | 1 1 0 1 | ((u)(v)) | f_14 |
| | | | | |
| f_14 | ((x)(y)) | 1 1 1 0 | (()) | f_15 |
| | | | | |
| f_15 | (()) | 1 1 1 1 | (()) | f_15 |
| | | | | |
o------------o----------o-----------o----------o------------o
</pre>

===Formula Display 14===

<pre>
o-------------------------------------------------o
| |
| EG_i = G_i <u + du, v + dv> |
| |
o-------------------------------------------------o
</pre>

===Formula Display 15===

<pre>
o-------------------------------------------------o
| |
| DG_i = G_i <u, v> + EG_i <u, v, du, dv> |
| |
| = G_i <u, v> + G_i <u + du, v + dv> |
| |
o-------------------------------------------------o
</pre>

===Formula Display 16===

<pre>
o-------------------------------------------------o
| |
| Ef = ((u + du)(v + dv)) |
| |
| Eg = ((u + du, v + dv)) |
| |
o-------------------------------------------------o
</pre>

===Formula Display 17===

<pre>
o-------------------------------------------------o
| |
| Df = ((u)(v)) + ((u + du)(v + dv)) |
| |
| Dg = ((u, v)) + ((u + du, v + dv)) |
| |
o-------------------------------------------------o
</pre>

===Table 66-i. Computation Summary for f‹u, v› = ((u)(v))===

<pre>
Table 66-i. Computation Summary for f<u, v> = ((u)(v))
o--------------------------------------------------------------------------------o
| |
| !e!f = uv. 1 + u(v). 1 + (u)v. 1 + (u)(v). 0 |
| |
| Ef = uv. (du dv) + u(v). (du (dv)) + (u)v.((du) dv) + (u)(v).((du)(dv)) |
| |
| Df = uv. du dv + u(v). du (dv) + (u)v. (du) dv + (u)(v).((du)(dv)) |
| |
| df = uv. 0 + u(v). du + (u)v. dv + (u)(v). (du, dv) |
| |
| rf = uv. du dv + u(v). du dv + (u)v. du dv + (u)(v). du dv |
| |
o--------------------------------------------------------------------------------o
</pre>

===Table 66-ii. Computation Summary for g‹u, v› = ((u, v))===

<pre>
Table 66-ii. Computation Summary for g<u, v> = ((u, v))
o--------------------------------------------------------------------------------o
| |
| !e!g = uv. 1 + u(v). 0 + (u)v. 0 + (u)(v). 1 |
| |
| Eg = uv.((du, dv)) + u(v). (du, dv) + (u)v. (du, dv) + (u)(v).((du, dv)) |
| |
| Dg = uv. (du, dv) + u(v). (du, dv) + (u)v. (du, dv) + (u)(v). (du, dv) |
| |
| dg = uv. (du, dv) + u(v). (du, dv) + (u)v. (du, dv) + (u)(v). (du, dv) |
| |
| rg = uv. 0 + u(v). 0 + (u)v. 0 + (u)(v). 0 |
| |
o--------------------------------------------------------------------------------o
</pre>

===Table 67. Computation of an Analytic Series in Terms of Coordinates===

<pre>
Table 67. Computation of an Analytic Series in Terms of Coordinates
o--------o-------o-------o--------o-------o-------o-------o-------o
| u v | du dv | u' v' | f g | Ef Eg | Df Dg | df dg | rf rg |
o--------o-------o-------o--------o-------o-------o-------o-------o
| | | | | | | | |
| 0 0 | 0 0 | 0 0 | 0 1 | 0 1 | 0 0 | 0 0 | 0 0 |
| | | | | | | | |
| | 0 1 | 0 1 | | 1 0 | 1 1 | 1 1 | 0 0 |
| | | | | | | | |
| | 1 0 | 1 0 | | 1 0 | 1 1 | 1 1 | 0 0 |
| | | | | | | | |
| | 1 1 | 1 1 | | 1 1 | 1 0 | 0 0 | 1 0 |
| | | | | | | | |
o--------o-------o-------o--------o-------o-------o-------o-------o
| | | | | | | | |
| 0 1 | 0 0 | 0 1 | 1 0 | 1 0 | 0 0 | 0 0 | 0 0 |
| | | | | | | | |
| | 0 1 | 0 0 | | 0 1 | 1 1 | 1 1 | 0 0 |
| | | | | | | | |
| | 1 0 | 1 1 | | 1 1 | 0 1 | 0 1 | 0 0 |
| | | | | | | | |
| | 1 1 | 1 0 | | 1 0 | 0 0 | 1 0 | 1 0 |
| | | | | | | | |
o--------o-------o-------o--------o-------o-------o-------o-------o
| | | | | | | | |
| 1 0 | 0 0 | 1 0 | 1 0 | 1 0 | 0 0 | 0 0 | 0 0 |
| | | | | | | | |
| | 0 1 | 1 1 | | 1 1 | 0 1 | 0 1 | 0 0 |
| | | | | | | | |
| | 1 0 | 0 0 | | 0 1 | 1 1 | 1 1 | 0 0 |
| | | | | | | | |
| | 1 1 | 0 1 | | 1 0 | 0 0 | 1 0 | 1 0 |
| | | | | | | | |
o--------o-------o-------o--------o-------o-------o-------o-------o
| | | | | | | | |
| 1 1 | 0 0 | 1 1 | 1 1 | 1 1 | 0 0 | 0 0 | 0 0 |
| | | | | | | | |
| | 0 1 | 1 0 | | 1 0 | 0 1 | 0 1 | 0 0 |
| | | | | | | | |
| | 1 0 | 0 1 | | 1 0 | 0 1 | 0 1 | 0 0 |
| | | | | | | | |
| | 1 1 | 0 0 | | 0 1 | 1 0 | 0 0 | 1 0 |
| | | | | | | | |
o--------o-------o-------o--------o-------o-------o-------o-------o
</pre>

===Table 68. Computation of an Analytic Series in Symbolic Terms===

<pre>
Table 68. Computation of an Analytic Series in Symbolic Terms
o-----o-----o------------o----------o----------o----------o----------o----------o
| u v | f g | Df | Dg | df | dg | rf | rf |
o-----o-----o------------o----------o----------o----------o----------o----------o
| | | | | | | | |
| 0 0 | 0 1 | ((du)(dv)) | (du, dv) | (du, dv) | (du, dv) | du dv | () |
| | | | | | | | |
| 0 1 | 1 0 | (du) dv | (du, dv) | dv | (du, dv) | du dv | () |
| | | | | | | | |
| 1 0 | 1 0 | du (dv) | (du, dv) | du | (du, dv) | du dv | () |
| | | | | | | | |
| 1 1 | 1 1 | du dv | (du, dv) | () | (du, dv) | du dv | () |
| | | | | | | | |
o-----o-----o------------o----------o----------o----------o----------o----------o
</pre>

===Formula Display 18===

<pre>
o-------------------------------------------------------------------------o
| |
| Df = uv. du dv + u(v). du (dv) + (u)v.(du) dv + (u)(v).((du)(dv)) |
| |
| Dg = uv.(du, dv) + u(v).(du, dv) + (u)v.(du, dv) + (u)(v). (du, dv) |
| |
o-------------------------------------------------------------------------o

===Figure 69. Difference Map of F = <f, g> = <((u)(v)), ((u, v))>===

o-----------------------------------o o-----------------------------------o
| U | |`U`````````````````````````````````|
| | |```````````````````````````````````|
| ^ | |```````````````````````````````````|
| | | |```````````````````````````````````|
| o-------o | o-------o | |```````o-------o```o-------o```````|
| ^ /`````````\|/`````````\ ^ | | ^ ```/ ^ \`/ ^ \``` ^ |
| \ /```````````|```````````\ / | |``\``/ \ o / \``/``|
| \/`````u`````/|\`````v`````\/ | |```\/ u \/`\/ v \/```|
| /\``````````/`|`\``````````/\ | |```/\ /\`/\ /\```|
| o``\````````o``@``o````````/``o | |``o \ o``@``o / o``|
| |```\```````|`````|```````/```| | |``| \ |`````| / |``|
| |````@``````|`````|``````@````| | |``| @-------->`<--------@ |``|
| |```````````|`````|```````````| | |``| |`````| |``|
| o```````````o` ^ `o```````````o | |``o o`````o o``|
| \```````````\`|`/```````````/ | |```\ \```/ /```|
| \```` ^ ````\|/```` ^ ````/ | |````\ ^ \`/ ^ /````|
| \`````\`````|`````/`````/ | |`````\ \ o / /`````|
| \`````\```/|\```/`````/ | |``````\ \ /`\ / /``````|
| o-----\-o | o-/-----o | |```````o-----\-o```o-/-----o```````|
| \ | / | |``````````````\`````/``````````````|
| \ | / | |```````````````\```/```````````````|
| \|/ | |````````````````\`/````````````````|
| @ | |`````````````````@`````````````````|
o-----------------------------------o o-----------------------------------o
\ / \ /
\ / \ /
\ ((u)(v)) / \ ((u, v)) /
\ / \ /
\ / \ /
o----------\-------------/-----------------------\-------------/----------o
| X \ / \ / |
| \ / \ / |
| \ / \ / |
| o----------------o o----------------o |
| / \ / \ |
| / o \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| o o o o |
| | | | | |
| | | | | |
| | f | | g | |
| | | | | |
| | | | | |
| o o o o |
| \ \ / / |
| \ \ / / |
| \ \ / / |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| o----------------o o----------------o |
| |
| |
| |
o-------------------------------------------------------------------------o
Figure 69. Difference Map of F = <f, g> = <((u)(v)), ((u, v))>
</pre>

===Figure 70-a. Tangent Functor Diagram for F‹u, v› = ‹((u)(v)), ((u, v))›===

<pre>
o-------------------------------------------------------------------------------o
| |
| df = uv. 0 + u(v). du + (u)v. dv + (u)(v).(du, dv) |
| |
| dg = uv.(du, dv) + u(v).(du, dv) + (u)v.(du, dv) + (u)(v).(du, dv) |
| |
o-------------------------------------------------------------------------------o

o o
/ \ / \
/ \ / \
/ \ / O \
/ \ o /@\ o
/ \ / \ / \
/ \ / \ / \
/ O \ / O \ / O \
o /@\ o o /@\ o /@\ o
/ \ / \ / \ \ / \ \ / \
/ \ / \ / \ / \ / \
/ \ / \ / O \ / O \ / O \
/ \ / \ o /@ o /@\ o /@ o
/ \ / \ / \ \ / \ / \ \ / \
/ \ / \ / \ / \ / \ / \
/ O \ / O \ / O \ / O \ / O \ / O \
o /@ o /@ o o /@ o /@ o /@ o /@ o
|\ / \ /| |\ / \ / / \ / / \ /|
| \ / \ / | | \ / \ / \ / \ / |
| \ / \ / | | \ / O \ / O \ / O \ / |
| \ / \ / | | o /@ o @\ o /@ o |
| \ / \ / | | |\ / \ / \ / \ / \ /| |
| \ / \ / | | | \ / \ / \ / | |
| u \ / O \ / v | | u | \ / O \ / O \ / | v |
o-------o @\ o-------o o---+---o @\ o @\ o---+---o
\ / | \ / \ / \ / \ / |
\ / | \ / \ / |
\ / | du \ / O \ / dv |
\ / o-------o @\ o-------o
\ / \ /
\ / \ /
\ / \ /
o o
U% $T$ $E$U%
o------------------>o
| |
| |
| |
| |
F | | $T$F
| |
| |
| |
v v
o------------------>o
X% $T$ $E$X%
o o
/ \ / \
/ \ / \
/ \ / O \
/ \ o /@\ o
/ \ / \ / \
/ \ / \ / \
/ O \ / O \ / O \
o /@\ o o /@\ o /@\ o
/ \ / \ / \ \ / \ / / \
/ \ / \ / \ / \ / \
/ \ / \ / O \ / O \ / O \
/ \ / \ o /@ o /@\ o @\ o
/ \ / \ / \ \ / \ / \ / \ / / \
/ \ / \ / \ / \ / \ / \
/ O \ / O \ / O \ / O \ / O \ / O \
o /@ o @\ o o /@ o /@ o @\ o @\ o
|\ / \ /| |\ / \ / \ / \ / \ / \ /|
| \ / \ / | | \ / \ / \ / \ / |
| \ / \ / | | \ / O \ / O \ / O \ / |
| \ / \ / | | o /@ o @ o @\ o |
| \ / \ / | | |\ / / \ / \ / \ \ /| |
| \ / \ / | | | \ / \ / \ / | |
| x \ / O \ / y | | x | \ / O \ / O \ / | y |
o-------o @ o-------o o---+---o @ o @ o---+---o
\ / | \ / / \ \ / |
\ / | \ / \ / |
\ / | dx \ / O \ / dy |
\ / o-------o @ o-------o
\ / \ /
\ / \ /
\ / \ /
o o

Figure 70-a. Tangent Functor Diagram for F‹u, v› = <((u)(v)), ((u, v))>
</pre>

===Figure 70-b. Tangent Functor Ferris Wheel for F‹u, v› = ‹((u)(v)), ((u, v))›===

<pre>
o-----------------------o o-----------------------o o-----------------------o
| dU | | dU | | dU |
| o--o o--o | | o--o o--o | | o--o o--o |
| /////\ /////\ | | /XXXX\ /XXXX\ | | /\\\\\ /\\\\\ |
| ///////o//////\ | | /XXXXXXoXXXXXX\ | | /\\\\\\o\\\\\\\ |
| //////// \//////\ | | /XXXXXX/ \XXXXXX\ | | /\\\\\\/ \\\\\\\\ |
| o/////// \//////o | | oXXXXXX/ \XXXXXXo | | o\\\\\\/ \\\\\\\o |
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |
| |/du//| |//dv/| | | |XXXXX| |XXXXX| | | |\du\\| |\\dv\| |
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |
| o//////\ ///////o | | oXXXXXX\ /XXXXXXo | | o\\\\\\\ /\\\\\\o |
| \//////\ //////// | | \XXXXXX\ /XXXXXX/ | | \\\\\\\\ /\\\\\\/ |
| \//////o/////// | | \XXXXXXoXXXXXX/ | | \\\\\\\o\\\\\\/ |
| \///// \///// | | \XXXX/ \XXXX/ | | \\\\\/ \\\\\/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
= du' @ (u)(v) o-----------------------o dv' @ (u)(v) =
= | dU' | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////o\\\\\\/ |
| \///// \\\\\/ |
| o--o o--o |
| |
o-----------------------o

o-----------------------o o-----------------------o o-----------------------o
| dU | | dU | | dU |
| o--o o--o | | o--o o--o | | o--o o--o |
| / \ /////\ | | /\\\\\ /XXXX\ | | /\\\\\ /\\\\\ |
| / o//////\ | | /\\\\\\oXXXXXX\ | | /\\\\\\o\\\\\\\ |
| / //\//////\ | | /\\\\\\//\XXXXXX\ | | /\\\\\\/ \\\\\\\\ |
| o ////\//////o | | o\\\\\\////\XXXXXXo | | o\\\\\\/ \\\\\\\o |
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |
| | du |/////|//dv/| | | |\\\\\|/////|XXXXX| | | |\du\\| |\\dv\| |
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |
| o \//////////o | | o\\\\\\\////XXXXXXo | | o\\\\\\\ /\\\\\\o |
| \ \///////// | | \\\\\\\\//XXXXXX/ | | \\\\\\\\ /\\\\\\/ |
| \ o/////// | | \\\\\\\oXXXXXX/ | | \\\\\\\o\\\\\\/ |
| \ / \///// | | \\\\\/ \XXXX/ | | \\\\\/ \\\\\/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
= du' @ (u) v o-----------------------o dv' @ (u) v =
= | dU' | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////o\\\\\\/ |
| \///// \\\\\/ |
| o--o o--o |
| |
o-----------------------o

o-----------------------o o-----------------------o o-----------------------o
| dU | | dU | | dU |
| o--o o--o | | o--o o--o | | o--o o--o |
| /////\ / \ | | /XXXX\ /\\\\\ | | /\\\\\ /\\\\\ |
| ///////o \ | | /XXXXXXo\\\\\\\ | | /\\\\\\o\\\\\\\ |
| /////////\ \ | | /XXXXXX//\\\\\\\\ | | /\\\\\\/ \\\\\\\\ |
| o//////////\ o | | oXXXXXX////\\\\\\\o | | o\\\\\\/ \\\\\\\o |
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |
| |/du//|/////| dv | | | |XXXXX|/////|\\\\\| | | |\du\\| |\\dv\| |
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |
| o//////\//// o | | oXXXXXX\////\\\\\\o | | o\\\\\\\ /\\\\\\o |
| \//////\// / | | \XXXXXX\//\\\\\\/ | | \\\\\\\\ /\\\\\\/ |
| \//////o / | | \XXXXXXo\\\\\\/ | | \\\\\\\o\\\\\\/ |
| \///// \ / | | \XXXX/ \\\\\/ | | \\\\\/ \\\\\/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
= du' @ u (v) o-----------------------o dv' @ u (v) =
= | dU' | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////o\\\\\\/ |
| \///// \\\\\/ |
| o--o o--o |
| |
o-----------------------o

o-----------------------o o-----------------------o o-----------------------o
| dU | | dU | | dU |
| o--o o--o | | o--o o--o | | o--o o--o |
| / \ / \ | | /\\\\\ /\\\\\ | | /\\\\\ /\\\\\ |
| / o \ | | /\\\\\\o\\\\\\\ | | /\\\\\\o\\\\\\\ |
| / / \ \ | | /\\\\\\/ \\\\\\\\ | | /\\\\\\/ \\\\\\\\ |
| o / \ o | | o\\\\\\/ \\\\\\\o | | o\\\\\\/ \\\\\\\o |
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |
| | du | | dv | | | |\\\\\| |\\\\\| | | |\du\\| |\\dv\| |
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |
| o \ / o | | o\\\\\\\ /\\\\\\o | | o\\\\\\\ /\\\\\\o |
| \ \ / / | | \\\\\\\\ /\\\\\\/ | | \\\\\\\\ /\\\\\\/ |
| \ o / | | \\\\\\\o\\\\\\/ | | \\\\\\\o\\\\\\/ |
| \ / \ / | | \\\\\/ \\\\\/ | | \\\\\/ \\\\\/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
= du' @ u v o-----------------------o dv' @ u v =
= | dU' | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////o\\\\\\/ |
| \///// \\\\\/ |
| o--o o--o |
| |
o-----------------------o

o-----------------------o o-----------------------o o-----------------------o
| U | |\U\\\\\\\\\\\\\\\\\\\\\| |\U\\\\\\\\\\\\\\\\\\\\\|
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|
| /////\ /////\ | |\\\\\/////\\/////\\\\\\| |\\\\\/ \\/ \\\\\\|
| ///////o//////\ | |\\\\///////o//////\\\\\| |\\\\/ o \\\\\|
| /////////\//////\ | |\\\////////X\//////\\\\| |\\\/ /\\ \\\\|
| o//////////\//////o | |\\o///////XXX\//////o\\| |\\o /\\\\ o\\|
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|
| |//u//|/////|//v//| | |\\|//u//|XXXXX|//v//|\\| |\\| u |\\\\\| v |\\|
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|
| o//////\//////////o | |\\o//////\XXX///////o\\| |\\o \\\\/ o\\|
| \//////\///////// | |\\\\//////\X////////\\\| |\\\\ \\/ /\\\|
| \//////o/////// | |\\\\\//////o///////\\\\| |\\\\\ o /\\\\|
| \///// \///// | |\\\\\\/////\\/////\\\\\| |\\\\\\ /\\ /\\\\\|
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|
| | |\\\\\\\\\\\\\\\\\\\\\\\| |\\\\\\\\\\\\\\\\\\\\\\\|
o-----------------------o o-----------------------o o-----------------------o
= u' o-----------------------o v' =
= | U' | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/u'//|XXXXX|\\v'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////o\\\\\\/ |
| \///// \\\\\/ |
| o--o o--o |
| |
o-----------------------o

Figure 70-b. Tangent Functor Ferris Wheel for F<u, v> = <((u)(v)), ((u, v))>
</pre>

==Logical Tables==

===Higher Order Propositions===

{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ '''Table 7. Higher Order Propositions (n = 1)'''
|- style="background:paleturquoise"
| \ ''x'' || 1 0 || ''F''
|''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m''
|''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m''
|- style="background:paleturquoise"
| ''F'' \ || &nbsp; || &nbsp;
|00||01||02||03||04||05||06||07||08||09||10||11||12||13||14||15
|-
| ''F<sub>0</sub> || 0 0 || 0 ||0||1||0||1||0||1||0||1||0||1||0||1||0||1||0||1
|-
| ''F<sub>1</sub> || 0 1 || (x) ||0||0||1||1||0||0||1||1||0||0||1||1||0||0||1||1
|-
| ''F<sub>2</sub> || 1 0 || x ||0||0||0||0||1||1||1||1||0||0||0||0||1||1||1||1
|-
| ''F<sub>3</sub> || 1 1 || 1 ||0||0||0||0||0||0||0||0||1||1||1||1||1||1||1||1
|}
<br>

{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ '''Table 8. Interpretive Categories for Higher Order Propositions (n = 1)'''
|- style="background:paleturquoise"
|Measure||Happening||Exactness||Existence||Linearity||Uniformity||Information
|-
|''m''<sub>0</sub>||nothing happens||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|-
|''m''<sub>1</sub>||&nbsp;||just false||nothing exists||&nbsp;||&nbsp;||&nbsp;
|-
|''m''<sub>2</sub>||&nbsp;||just not x||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|-
|''m''<sub>3</sub>||&nbsp;||&nbsp;||nothing is x||&nbsp;||&nbsp;||&nbsp;
|-
|''m''<sub>4</sub>||&nbsp;||just x||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|-
|''m''<sub>5</sub>||&nbsp;||&nbsp;||everything is x||F is linear||&nbsp;||&nbsp;
|-
|''m''<sub>6</sub>||&nbsp;||&nbsp;||&nbsp;||&nbsp;||F is not uniform||F is informed
|-
|''m''<sub>7</sub>||&nbsp;||not just true||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|-
|''m''<sub>8</sub>||&nbsp;||just true||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|-
|''m''<sub>9</sub>||&nbsp;||&nbsp;||&nbsp;||&nbsp;||F is uniform||F is not informed
|-
|''m''<sub>10</sub>||&nbsp;||&nbsp;||something is not x||F is not linear||&nbsp;||&nbsp;
|-
|''m''<sub>11</sub>||&nbsp;||not just x||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|-
|''m''<sub>12</sub>||&nbsp;||&nbsp;||something is x||&nbsp;||&nbsp;||&nbsp;
|-
|''m''<sub>13</sub>||&nbsp;||not just not x||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|-
|''m''<sub>14</sub>||&nbsp;||not just false||something exists||&nbsp;||&nbsp;||&nbsp;
|-
|''m''<sub>15</sub>||anything happens||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|}
<br>

{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ '''Table 9. Higher Order Propositions (n = 2)'''
|- style="background:paleturquoise"
| align=right | ''x'' : || 1100 || ''f''
|''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m''
|''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m''
|''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m''
|- style="background:paleturquoise"
| align=right | ''y'' : || 1010 || &nbsp;
|0||1||2||3||4||5||6||7||8||9||10||11||12
|13||14||15||16||17||18||19||20||21||22||23
|-
| ''f<sub>0</sub> || 0000 || ( )
| 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1
| 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1
| 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1
|-
| ''f<sub>1</sub> || 0001 || (x)(y)
|&nbsp;||&nbsp;|| 1 || 1 || 0 || 0 || 1 || 1
| 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1
| 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1
|-
| ''f<sub>2</sub> || 0010 || (x) y
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1 || 1 || 1 || 1
| 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1
| 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1
|-
| ''f<sub>3</sub> || 0011 || (x)
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
| 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
| ''f<sub>4</sub> || 0100 || x (y)
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
| 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
|-
| ''f<sub>5</sub> || 0101 || (y)
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|-
| ''f<sub>6</sub> || 0110 || (x, y)
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|-
| ''f<sub>7</sub> || 0111 || (x y)
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|-
| ''f<sub>8</sub> || 1000 || x y
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|-
| ''f<sub>9</sub> || 1001 || ((x, y))
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|-
| ''f<sub>10</sub> || 1010 || y
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|-
| ''f<sub>11</sub> || 1011 || (x (y))
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|-
| ''f<sub>12</sub> || 1100 || x
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|-
| ''f<sub>13</sub> || 1101 || ((x) y)
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|-
| ''f<sub>14</sub> || 1110 || ((x)(y))
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|-
| ''f<sub>15</sub> || 1111 || (( ))
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|}
<br>

{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ '''Table 10. Qualifiers of Implication Ordering: &alpha;<sub>''i''&nbsp;</sub>''f'' = &Upsilon;(''f''<sub>''i''</sub> &rArr; ''f'')'''
|- style="background:paleturquoise"
| align=right | ''x'' : || 1100 || ''f''
|&alpha;||&alpha;||&alpha;||&alpha;||&alpha;||&alpha;||&alpha;||&alpha;
|&alpha;||&alpha;||&alpha;||&alpha;||&alpha;||&alpha;||&alpha;||&alpha;
|- style="background:paleturquoise"
| align=right | ''y'' : || 1010 || &nbsp;
|15||14||13||12||11||10||9||8||7||6||5||4||3||2||1||0
|-
| ''f<sub>0</sub> || 0000 || ( )
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
|-
| ''f<sub>1</sub> || 0001 || (x)(y)
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1 || 1
|-
| ''f<sub>2</sub> || 0010 || (x) y
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1 ||&nbsp;|| 1
|-
| ''f<sub>3</sub> || 0011 || (x)
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1 || 1 || 1 || 1
|-
| ''f<sub>4</sub> || 0100 || x (y)
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;|| 1 ||&nbsp;||&nbsp;||&nbsp;|| 1
|-
| ''f<sub>5</sub> || 0101 || (y)
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;|| 1 || 1 ||&nbsp;||&nbsp;|| 1 || 1
|-
| ''f<sub>6</sub> || 0110 || (x, y)
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;|| 1 ||&nbsp;|| 1 ||&nbsp;|| 1 ||&nbsp;|| 1
|-
| ''f<sub>7</sub> || 0111 || (x y)
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
| 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
|-
| ''f<sub>8</sub> || 1000 || x y
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
|-
| ''f<sub>9</sub> || 1001 || ((x, y))
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1 || 1
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1 || 1
|-
| ''f<sub>10</sub> || 1010 || y
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1 ||&nbsp;|| 1
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1 ||&nbsp;|| 1
|-
| ''f<sub>11</sub> || 1011 || (x (y))
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1 || 1 || 1 || 1
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1 || 1 || 1 || 1
|-
| ''f<sub>12</sub> || 1100 || x
|&nbsp;||&nbsp;||&nbsp;|| 1 ||&nbsp;||&nbsp;||&nbsp;|| 1
|&nbsp;||&nbsp;||&nbsp;|| 1 ||&nbsp;||&nbsp;||&nbsp;|| 1
|-
| ''f<sub>13</sub> || 1101 || ((x) y)
|&nbsp;||&nbsp;|| 1 || 1 ||&nbsp;||&nbsp;|| 1 || 1
|&nbsp;||&nbsp;|| 1 || 1 ||&nbsp;||&nbsp;|| 1 || 1
|-
| ''f<sub>14</sub> || 1110 || ((x)(y))
|&nbsp;|| 1 ||&nbsp;|| 1 ||&nbsp;|| 1 ||&nbsp;|| 1
|&nbsp;|| 1 ||&nbsp;|| 1 ||&nbsp;|| 1 ||&nbsp;|| 1
|-
| ''f<sub>15</sub> || 1111 || (( ))
| 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
| 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
|}
<br>

{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ '''Table 11. Qualifiers of Implication Ordering: &beta;<sub>''i''&nbsp;</sub>''f'' = &Upsilon;(''f'' &rArr; ''f''<sub>''i''</sub>)'''
|- style="background:paleturquoise"
| align=right | ''x'' : || 1100 || ''f''
|&beta;||&beta;||&beta;||&beta;||&beta;||&beta;||&beta;||&beta;
|&beta;||&beta;||&beta;||&beta;||&beta;||&beta;||&beta;||&beta;
|- style="background:paleturquoise"
| align=right | ''y'' : || 1010 || &nbsp;
|0||1||2||3||4||5||6||7||8||9||10||11||12||13||14||15
|-
| ''f<sub>0</sub> || 0000 || ( )
| 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
| 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
|-
| ''f<sub>1</sub> || 0001 || (x)(y)
|&nbsp;|| 1 ||&nbsp;|| 1 ||&nbsp;|| 1 ||&nbsp;|| 1
|&nbsp;|| 1 ||&nbsp;|| 1 ||&nbsp;|| 1 ||&nbsp;|| 1
|-
| ''f<sub>2</sub> || 0010 || (x) y
|&nbsp;||&nbsp;|| 1 || 1 ||&nbsp;||&nbsp;|| 1 || 1
|&nbsp;||&nbsp;|| 1 || 1 ||&nbsp;||&nbsp;|| 1 || 1
|-
| ''f<sub>3</sub> || 0011 || (x)
|&nbsp;||&nbsp;||&nbsp;|| 1 ||&nbsp;||&nbsp;||&nbsp;|| 1
|&nbsp;||&nbsp;||&nbsp;|| 1 ||&nbsp;||&nbsp;||&nbsp;|| 1
|-
| ''f<sub>4</sub> || 0100 || x (y)
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1 || 1 || 1 || 1
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1 || 1 || 1 || 1
|-
| ''f<sub>5</sub> || 0101 || (y)
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1 ||&nbsp;|| 1
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1 ||&nbsp;|| 1
|-
| ''f<sub>6</sub> || 0110 || (x, y)
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1 || 1
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1 || 1
|-
| ''f<sub>7</sub> || 0111 || (x y)
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
|-
| ''f<sub>8</sub> || 1000 || x y
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
| 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
|-
| ''f<sub>9</sub> || 1001 || ((x, y))
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;|| 1 ||&nbsp;|| 1 ||&nbsp;|| 1 ||&nbsp;|| 1
|-
| ''f<sub>10</sub> || 1010 || y
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;|| 1 || 1 ||&nbsp;||&nbsp;|| 1 || 1
|-
| ''f<sub>11</sub> || 1011 || (x (y))
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;|| 1 ||&nbsp;||&nbsp;||&nbsp;|| 1
|-
| ''f<sub>12</sub> || 1100 || x
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1 || 1 || 1 || 1
|-
| ''f<sub>13</sub> || 1101 || ((x) y)
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1 ||&nbsp;|| 1
|-
| ''f<sub>14</sub> || 1110 || ((x)(y))
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1 || 1
|-
| ''f<sub>15</sub> || 1111 || (( ))
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
|}
<br>

{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ '''Table 13. Syllogistic Premisses as Higher Order Indicator Functions'''
| A
| align=left | Universal Affirmative
| align=left | All
| x || is || y
| align=left | Indicator of " x (y)" = 0
|-
| E
| align=left | Universal Negative
| align=left | All
| x || is || (y)
| align=left | Indicator of " x y " = 0
|-
| I
| align=left | Particular Affirmative
| align=left | Some
| x || is || y
| align=left | Indicator of " x y " = 1
|-
| O
| align=left | Particular Negative
| align=left | Some
| x || is || (y)
| align=left | Indicator of " x (y)" = 1
|}
<br>

{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ '''Table 14. Relation of Quantifiers to Higher Order Propositions'''
|- style="background:paleturquoise"
|Mnemonic||Category||Classical Form||Alternate Form||Symmetric Form||Operator
|-
| E<br>Exclusive
| Universal<br>Negative
| align=left | All x is (y)
| align=left | &nbsp;
| align=left | No x is y
| (''L''<sub>11</sub>)
|-
| A<br>Absolute
| Universal<br>Affirmative
| align=left | All x is y
| align=left | &nbsp;
| align=left | No x is (y)
| (''L''<sub>10</sub>)
|-
| &nbsp;
| &nbsp;
| align=left | All y is x
| align=left | No y is (x)
| align=left | No (x) is y
| (''L''<sub>01</sub>)
|-
| &nbsp;
| &nbsp;
| align=left | All (y) is x
| align=left | No (y) is (x)
| align=left | No (x) is (y)
| (''L''<sub>00</sub>)
|-
| &nbsp;
| &nbsp;
| align=left | Some (x) is (y)
| align=left | &nbsp;
| align=left | Some (x) is (y)
| ''L''<sub>00</sub>
|-
| &nbsp;
| &nbsp;
| align=left | Some (x) is y
| align=left | &nbsp;
| align=left | Some (x) is y
| ''L''<sub>01</sub>
|-
| O<br>Obtrusive
| Particular<br>Negative
| align=left | Some x is (y)
| align=left | &nbsp;
| align=left | Some x is (y)
| ''L''<sub>10</sub>
|-
| I<br>Indefinite
| Particular<br>Affirmative
| align=left | Some x is y
| align=left | &nbsp;
| align=left | Some x is y
| ''L''<sub>11</sub>
|}
<br>

{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ '''Table 15. Simple Qualifiers of Propositions (n = 2)'''
|- style="background:paleturquoise"
| align=right | ''x'' : || 1100 || ''f''
| (''L''<sub>11</sub>)
| (''L''<sub>10</sub>)
| (''L''<sub>01</sub>)
| (''L''<sub>00</sub>)
| ''L''<sub>00</sub>
| ''L''<sub>01</sub>
| ''L''<sub>10</sub>
| ''L''<sub>11</sub>
|- style="background:paleturquoise"
| align=right | ''y'' : || 1010 || &nbsp;
| align=left | no x <br> is y
| align=left | no x <br> is (y)
| align=left | no (x) <br> is y
| align=left | no (x) <br> is (y)
| align=left | some (x) <br> is (y)
| align=left | some (x) <br> is y
| align=left | some x <br> is (y)
| align=left | some x <br> is y
|-
| ''f<sub>0</sub> || 0000 || ( )
| 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0
|-
| ''f<sub>1</sub> || 0001 || (x)(y)
| 1 || 1 || 1 || 0 || 1 || 0 || 0 || 0
|-
| ''f<sub>2</sub> || 0010 || (x) y
| 1 || 1 || 0 || 1 || 0 || 1 || 0 || 0
|-
| ''f<sub>3</sub> || 0011 || (x)
| 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0
|-
| ''f<sub>4</sub> || 0100 || x (y)
| 1 || 0 || 1 || 1 || 0 || 0 || 1 || 0
|-
| ''f<sub>5</sub> || 0101 || (y)
| 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0
|-
| ''f<sub>6</sub> || 0110 || (x, y)
| 1 || 0 || 0 || 1 || 0 || 1 || 1 || 0
|-
| ''f<sub>7</sub> || 0111 || (x y)
| 1 || 0 || 0 || 0 || 1 || 1 || 1 || 0
|-
| ''f<sub>8</sub> || 1000 || x y
| 0 || 1 || 1 || 1 || 0 || 0 || 0 || 1
|-
| ''f<sub>9</sub> || 1001 || ((x, y))
| 0 || 1 || 1 || 0 || 1 || 0 || 0 || 1
|-
| ''f<sub>10</sub> || 1010 || y
| 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1
|-
| ''f<sub>11</sub> || 1011 || (x (y))
| 0 || 1 || 0 || 0 || 1 || 1 || 0 || 1
|-
| ''f<sub>12</sub> || 1100 || x
| 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1
|-
| ''f<sub>13</sub> || 1101 || ((x) y)
| 0 || 0 || 1 || 0 || 1 || 0 || 1 || 1
|-
| ''f<sub>14</sub> || 1110 || ((x)(y))
| 0 || 0 || 0 || 1 || 0 || 1 || 1 || 1
|-
| ''f<sub>15</sub> || 1111 || (( ))
| 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1
|}
<br>

Table 7. Higher Order Propositions (n = 1)
o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o
| \ x | 1 0 | F |m |m |m |m |m |m |m |m |m |m |m |m |m |m |m |m |
| F \ | | |00|01|02|03|04|05|06|07|08|09|10|11|12|13|14|15 |
o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o
| | | | |
| F_0 | 0 0 | 0 | 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 |
| | | | |
| F_1 | 0 1 | (x) | 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 |
| | | | |
| F_2 | 1 0 | x | 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 |
| | | | |
| F_3 | 1 1 | 1 | 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 |
| | | | |
o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o
<br>

Table 8. Interpretive Categories for Higher Order Propositions (n = 1)
o-------o----------o------------o------------o----------o----------o-----------o
|Measure| Happening| Exactness | Existence | Linearity|Uniformity|Information|
o-------o----------o------------o------------o----------o----------o-----------o
| m_0 | nothing | | | | | |
| | happens | | | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_1 | | | nothing | | | |
| | | just false | exists | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_2 | | | | | | |
| | | just not x | | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_3 | | | nothing | | | |
| | | | is x | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_4 | | | | | | |
| | | just x | | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_5 | | | everything | F is | | |
| | | | is x | linear | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_6 | | | | | F is not | F is |
| | | | | | uniform | informed |
o-------o----------o------------o------------o----------o----------o-----------o
| m_7 | | not | | | | |
| | | just true | | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_8 | | | | | | |
| | | just true | | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_9 | | | | | F is | F is not |
| | | | | | uniform | informed |
o-------o----------o------------o------------o----------o----------o-----------o
| m_10 | | | something | F is not | | |
| | | | is not x | linear | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_11 | | not | | | | |
| | | just x | | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_12 | | | something | | | |
| | | | is x | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_13 | | not | | | | |
| | | just not x | | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_14 | | not | something | | | |
| | | just false | exists | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_15 | anything | | | | | |
| | happens | | | | | |
o-------o----------o------------o------------o----------o----------o-----------o
<br>

Table 9. Higher Order Propositions (n = 2)
o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
| | x | 1100 | f |m|m|m|m|m|m|m|m|m|m|m|m|m|m|m|m|.|
| | y | 1010 | |0|0|0|0|0|0|0|0|0|0|1|1|1|1|1|1|.|
| f \ | | |0|1|2|3|4|5|6|7|8|9|0|1|2|3|4|5|.|
o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
| | | | |
| f_0 | 0000 | () |0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 |
| | | | |
| f_1 | 0001 | (x)(y) | 1 1 0 0 1 1 0 0 1 1 0 0 1 1 |
| | | | |
| f_2 | 0010 | (x) y | 1 1 1 1 0 0 0 0 1 1 1 1 |
| | | | |
| f_3 | 0011 | (x) | 1 1 1 1 1 1 1 1 |
| | | | |
| f_4 | 0100 | x (y) | |
| | | | |
| f_5 | 0101 | (y) | |
| | | | |
| f_6 | 0110 | (x, y) | |
| | | | |
| f_7 | 0111 | (x y) | |
| | | | |
o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
| | | | |
| f_8 | 1000 | x y | |
| | | | |
| f_9 | 1001 | ((x, y)) | |
| | | | |
| f_10 | 1010 | y | |
| | | | |
| f_11 | 1011 | (x (y)) | |
| | | | |
| f_12 | 1100 | x | |
| | | | |
| f_13 | 1101 | ((x) y) | |
| | | | |
| f_14 | 1110 | ((x)(y)) | |
| | | | |
| f_15 | 1111 | (()) | |
| | | | |
o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
<br>

Table 10. Qualifiers of Implication Ordering: !a!_i f = !Y!(f_i => f)
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
| | x | 1100 | f |a |a |a |a |a |a |a |a |a |a |a |a |a |a |a |a |
| | y | 1010 | |1 |1 |1 |1 |1 |1 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |
| f \ | | |5 |4 |3 |2 |1 |0 |9 |8 |7 |6 |5 |4 |3 |2 |1 |0 |
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
| | | | |
| f_0 | 0000 | () | 1 |
| | | | |
| f_1 | 0001 | (x)(y) | 1 1 |
| | | | |
| f_2 | 0010 | (x) y | 1 1 |
| | | | |
| f_3 | 0011 | (x) | 1 1 1 1 |
| | | | |
| f_4 | 0100 | x (y) | 1 1 |
| | | | |
| f_5 | 0101 | (y) | 1 1 1 1 |
| | | | |
| f_6 | 0110 | (x, y) | 1 1 1 1 |
| | | | |
| f_7 | 0111 | (x y) | 1 1 1 1 1 1 1 1 |
| | | | |
| f_8 | 1000 | x y | 1 1 |
| | | | |
| f_9 | 1001 | ((x, y)) | 1 1 1 1 |
| | | | |
| f_10 | 1010 | y | 1 1 1 1 |
| | | | |
| f_11 | 1011 | (x (y)) | 1 1 1 1 1 1 1 1 |
| | | | |
| f_12 | 1100 | x | 1 1 1 1 |
| | | | |
| f_13 | 1101 | ((x) y) | 1 1 1 1 1 1 1 1 |
| | | | |
| f_14 | 1110 | ((x)(y)) | 1 1 1 1 1 1 1 1 |
| | | | |
| f_15 | 1111 | (()) |1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
| | | | |
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
<br>

Table 11. Qualifiers of Implication Ordering: !b!_i f = !Y!(f => f_i)
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
| | x | 1100 | f |b |b |b |b |b |b |b |b |b |b |b |b |b |b |b |b |
| | y | 1010 | |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |1 |1 |1 |1 |1 |1 |
| f \ | | |0 |1 |2 |3 |4 |5 |6 |7 |8 |9 |0 |1 |2 |3 |4 |5 |
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
| | | | |
| f_0 | 0000 | () |1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
| | | | |
| f_1 | 0001 | (x)(y) | 1 1 1 1 1 1 1 1 |
| | | | |
| f_2 | 0010 | (x) y | 1 1 1 1 1 1 1 1 |
| | | | |
| f_3 | 0011 | (x) | 1 1 1 1 |
| | | | |
| f_4 | 0100 | x (y) | 1 1 1 1 1 1 1 1 |
| | | | |
| f_5 | 0101 | (y) | 1 1 1 1 |
| | | | |
| f_6 | 0110 | (x, y) | 1 1 1 1 |
| | | | |
| f_7 | 0111 | (x y) | 1 1 |
| | | | |
| f_8 | 1000 | x y | 1 1 1 1 1 1 1 1 |
| | | | |
| f_9 | 1001 | ((x, y)) | 1 1 1 1 |
| | | | |
| f_10 | 1010 | y | 1 1 1 1 |
| | | | |
| f_11 | 1011 | (x (y)) | 1 1 |
| | | | |
| f_12 | 1100 | x | 1 1 1 1 |
| | | | |
| f_13 | 1101 | ((x) y) | 1 1 |
| | | | |
| f_14 | 1110 | ((x)(y)) | 1 1 |
| | | | |
| f_15 | 1111 | (()) | 1 |
| | | | |
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
<br>

Table 13. Syllogistic Premisses as Higher Order Indicator Functions
o---o------------------------o-----------------o---------------------------o
| | | | |
| A | Universal Affirmative | All x is y | Indicator of " x (y)" = 0 |
| | | | |
| E | Universal Negative | All x is (y) | Indicator of " x y " = 0 |
| | | | |
| I | Particular Affirmative | Some x is y | Indicator of " x y " = 1 |
| | | | |
| O | Particular Negative | Some x is (y) | Indicator of " x (y)" = 1 |
| | | | |
o---o------------------------o-----------------o---------------------------o
<br>

Table 14. Relation of Quantifiers to Higher Order Propositions
o------------o------------o-----------o-----------o-----------o-----------o
| Mnemonic | Category | Classical | Alternate | Symmetric | Operator |
| | | Form | Form | Form | |
o============o============o===========o===========o===========o===========o
| E | Universal | All x | | No x | (L_11) |
| Exclusive | Negative | is (y) | | is y | |
o------------o------------o-----------o-----------o-----------o-----------o
| A | Universal | All x | | No x | (L_10) |
| Absolute | Affrmtve | is y | | is (y) | |
o------------o------------o-----------o-----------o-----------o-----------o
| | | All y | No y | No (x) | (L_01) |
| | | is x | is (x) | is y | |
o------------o------------o-----------o-----------o-----------o-----------o
| | | All (y) | No (y) | No (x) | (L_00) |
| | | is x | is (x) | is (y) | |
o------------o------------o-----------o-----------o-----------o-----------o
| | | Some (x) | | Some (x) | L_00 |
| | | is (y) | | is (y) | |
o------------o------------o-----------o-----------o-----------o-----------o
| | | Some (x) | | Some (x) | L_01 |
| | | is y | | is y | |
o------------o------------o-----------o-----------o-----------o-----------o
| O | Particular | Some x | | Some x | L_10 |
| Obtrusive | Negative | is (y) | | is (y) | |
o------------o------------o-----------o-----------o-----------o-----------o
| I | Particular | Some x | | Some x | L_11 |
| Indefinite | Affrmtve | is y | | is y | |
o------------o------------o-----------o-----------o-----------o-----------o
<br>

Table 15. Simple Qualifiers of Propositions (n = 2)
o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o
| | x | 1100 | f |(L11)|(L10)|(L01)|(L00)| L00 | L01 | L10 | L11 |
| | y | 1010 | |no x|no x|no ~x|no ~x|sm ~x|sm ~x|sm x|sm x|
| f \ | | |is y|is ~y|is y|is ~y|is ~y|is y|is ~y|is y|
o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o
| | | | |
| f_0 | 0000 | () | 1 1 1 1 0 0 0 0 |
| | | | |
| f_1 | 0001 | (x)(y) | 1 1 1 0 1 0 0 0 |
| | | | |
| f_2 | 0010 | (x) y | 1 1 0 1 0 1 0 0 |
| | | | |
| f_3 | 0011 | (x) | 1 1 0 0 1 1 0 0 |
| | | | |
| f_4 | 0100 | x (y) | 1 0 1 1 0 0 1 0 |
| | | | |
| f_5 | 0101 | (y) | 1 0 1 0 1 0 1 0 |
| | | | |
| f_6 | 0110 | (x, y) | 1 0 0 1 0 1 1 0 |
| | | | |
| f_7 | 0111 | (x y) | 1 0 0 0 1 1 1 0 |
| | | | |
| f_8 | 1000 | x y | 0 1 1 1 0 0 0 1 |
| | | | |
| f_9 | 1001 | ((x, y)) | 0 1 1 0 1 0 0 1 |
| | | | |
| f_10 | 1010 | y | 0 1 0 1 0 1 0 1 |
| | | | |
| f_11 | 1011 | (x (y)) | 0 1 0 0 1 1 0 1 |
| | | | |
| f_12 | 1100 | x | 0 0 1 1 0 0 1 1 |
| | | | |
| f_13 | 1101 | ((x) y) | 0 0 1 0 1 0 1 1 |
| | | | |
| f_14 | 1110 | ((x)(y)) | 0 0 0 1 0 1 1 1 |
| | | | |
| f_15 | 1111 | (()) | 0 0 0 0 1 1 1 1 |
| | | | |
o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o
<br>

===[[Zeroth Order Logic]]===

{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ '''Table 1. Propositional Forms on Two Variables'''
|- style="background:paleturquoise"
! style="width:15%" | L<sub>1</sub>
! style="width:15%" | L<sub>2</sub>
! style="width:15%" | L<sub>3</sub>
! style="width:15%" | L<sub>4</sub>
! style="width:15%" | L<sub>5</sub>
! style="width:15%" | L<sub>6</sub>
|- style="background:paleturquoise"
| &nbsp;
| align="right" | x :
| 1 1 0 0
| &nbsp;
| &nbsp;
| &nbsp;
|- style="background:paleturquoise"
| &nbsp;
| align="right" | y :
| 1 0 1 0
| &nbsp;
| &nbsp;
| &nbsp;
|-
| f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || (&nbsp;) || false || 0
|-
| f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || &not;x &and; &not;y
|-
| f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || &not;x &and; y
|-
| f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || &not;x
|-
| f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x &and; &not;y
|-
| f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || &not;y
|-
| f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x &ne; y
|-
| f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x&nbsp;y) || not both x and y || &not;x &or; &not;y
|-
| f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x&nbsp;y || x and y || x &and; y
|-
| f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y
|-
| f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y
|-
| f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x &rarr; y
|-
| f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x
|-
| f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x &larr; y
|-
| f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y || x &or; y
|-
| f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || ((&nbsp;)) || true || 1
|}
<br>

{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:90%"
|+ '''Table 1. Propositional Forms on Two Variables'''
|- style="background:aliceblue"
! style="width:15%" | L<sub>1</sub>
! style="width:15%" | L<sub>2</sub>
! style="width:15%" | L<sub>3</sub>
! style="width:15%" | L<sub>4</sub>
! style="width:15%" | L<sub>5</sub>
! style="width:15%" | L<sub>6</sub>
|- style="background:aliceblue"
| &nbsp;
| align="right" | x :
| 1 1 0 0
| &nbsp;
| &nbsp;
| &nbsp;
|- style="background:aliceblue"
| &nbsp;
| align="right" | y :
| 1 0 1 0
| &nbsp;
| &nbsp;
| &nbsp;
|-
| f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || (&nbsp;) || false || 0
|-
| f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || &not;x &and; &not;y
|-
| f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || &not;x &and; y
|-
| f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || &not;x
|-
| f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x &and; &not;y
|-
| f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || &not;y
|-
| f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x &ne; y
|-
| f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x&nbsp;y) || not both x and y || &not;x &or; &not;y
|-
| f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x&nbsp;y || x and y || x &and; y
|-
| f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y
|-
| f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y
|-
| f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x &rarr; y
|-
| f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x
|-
| f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x &larr; y
|-
| f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y || x &or; y
|-
| f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || ((&nbsp;)) || true || 1
|}
<br>

===Template Draft===

{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:98%"
|+ '''Propositional Forms on Two Variables'''
|- style="background:aliceblue"
! style="width:14%" | L<sub>1</sub>
! style="width:14%" | L<sub>2</sub>
! style="width:14%" | L<sub>3</sub>
! style="width:14%" | L<sub>4</sub>
! style="width:14%" | L<sub>5</sub>
! style="width:14%" | L<sub>6</sub>
! style="width:14%" | Name
|- style="background:aliceblue"
| &nbsp;
| align="right" | x :
| 1 1 0 0
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
|- style="background:aliceblue"
| &nbsp;
| align="right" | y :
| 1 0 1 0
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
|-
| f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || (&nbsp;) || false || 0 || Falsity
|-
| f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || &not;x &and; &not;y || [[NNOR]]
|-
| f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || &not;x &and; y || Insuccede
|-
| f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || &not;x || Not One
|-
| f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x &and; &not;y || Imprecede
|-
| f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || &not;y || Not Two
|-
| f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x &ne; y || Inequality
|-
| f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x&nbsp;y) || not both x and y || &not;x &or; &not;y || NAND
|-
| f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x&nbsp;y || x and y || x &and; y || [[Conjunction]]
|-
| f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y || Equality
|-
| f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y || Two
|-
| f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x &rarr; y || [[Logical implcation|Implication]]
|-
| f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x || One
|-
| f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x &larr; y || [[Logical involution|Involution]]
|-
| f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y || x &or; y || [[Disjunction]]
|-
| f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || ((&nbsp;)) || true || 1 || Tautology
|}
<br>

===[[Truth Tables]]===

====[[Logical negation]]====

'''Logical negation''' is an [[logical operation|operation]] on one [[logical value]], typically the value of a [[proposition]], that produces a value of ''true'' when its operand is false and a value of ''false'' when its operand is true.

The [[truth table]] of '''NOT p''' (also written as '''~p''' or '''&not;p''') is as follows:

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:40%"
|+ '''Logical Negation'''
|- style="background:aliceblue"
! style="width:20%" | p
! style="width:20%" | &not;p
|-
| F || T
|-
| T || F
|}
<br>

The logical negation of a proposition '''p''' is notated in different ways in various contexts of discussion and fields of application. Among these variants are the following:

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; width:40%"
|+ '''Variant Notations'''
|- style="background:aliceblue"
! style="text-align:center" | Notation
! Vocalization
|-
| style="text-align:center" | <math>\bar{p}</math>
| bar ''p''
|-
| style="text-align:center" | <math>p'\!</math>
| ''p'' prime,<p> ''p'' complement
|-
| style="text-align:center" | <math>!p\!</math>
| bang ''p''
|}
<br>

No matter how it is notated or symbolized, the logical negation &not;''p'' is read as "it is not the case that ''p''", or usually more simply as "not ''p''".

* Within a system of [[classical logic]], double negation, that is, the negation of the negation of a proposition ''p'', is [[logically equivalent]] to the initial proposition ''p''. Expressed in symbolic terms, &not;(&not;''p'') &hArr; ''p''.

* Within a system of [[intuitionistic logic]], however, &not;&not;''p'' is a weaker statement than ''p''. On the other hand, the logical equivalence &not;&not;&not;''p'' &hArr; &not;''p'' remains valid.

Logical negation can be defined in terms of other logical operations. For example, ~''p'' can be defined as ''p'' &rarr; ''F'', where &rarr; is [[material implication]] and ''F'' is absolute falsehood. Conversely, one can define ''F'' as ''p'' &amp; ~''p'' for any proposition ''p'', where &amp; is [[logical conjunction]]. The idea here is that any [[contradiction]] is false. While these ideas work in both classical and intuitionistic logic, they don't work in [[Brazilian logic]], where contradictions are not necessarily false. But in classical logic, we get a further identity: ''p'' &rarr; ''q'' can be defined as ~''p'' &or; ''q'', where &or; is [[logical disjunction]].

Algebraically, logical negation corresponds to the ''complement'' in a [[Boolean algebra]] (for classical logic) or a [[Heyting algebra]] (for intuitionistic logic).

====[[Logical conjunction]]====

'''Logical conjunction''' is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' if and only if both of its operands are true.

The [[truth table]] of '''p AND q''' (also written as '''p &and; q''', '''p & q''', or '''p<math>\cdot</math>q''') is as follows:

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%"
|+ '''Logical Conjunction'''
|- style="background:aliceblue"
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | p &and; q
|-
| F || F || F
|-
| F || T || F
|-
| T || F || F
|-
| T || T || T
|}
<br>

====[[Logical disjunction]]====

'''Logical disjunction''' is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if both of its operands are false.

The [[truth table]] of '''p OR q''' (also written as '''p &or; q''') is as follows:

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%"
|+ '''Logical Disjunction'''
|- style="background:aliceblue"
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | p &or; q
|-
| F || F || F
|-
| F || T || T
|-
| T || F || T
|-
| T || T || T
|}
<br>

====[[Logical equality]]====

'''Logical equality''' is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' if and only if both operands are false or both operands are true.

The [[truth table]] of '''p EQ q''' (also written as '''p = q''', '''p &harr; q''', or '''p &equiv; q''') is as follows:

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%"
|+ '''Logical Equality'''
|- style="background:aliceblue"
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | p = q
|-
| F || F || T
|-
| F || T || F
|-
| T || F || F
|-
| T || T || T
|}
<br>

====[[Exclusive disjunction]]====

'''Exclusive disjunction''' is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' just in case exactly one of its operands is true.

The [[truth table]] of '''p XOR q''' (also written as '''p + q''', '''p &oplus; q''', or '''p &ne; q''') is as follows:

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%"
|+ '''Exclusive Disjunction'''
|- style="background:aliceblue"
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | p XOR q
|-
| F || F || F
|-
| F || T || T
|-
| T || F || T
|-
| T || T || F
|}
<br>

The following equivalents can then be deduced:

: <math>\begin{matrix}
p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\
\\
& = & (p \lor q) & \land & (\lnot p \lor \lnot q) \\
\\
& = & (p \lor q) & \land & \lnot (p \land q)
\end{matrix}</math>

'''Generalized''' or '''n-ary''' XOR is true when the number of 1-bits is odd.

====[[Logical implication]]====

The '''material conditional''' and '''logical implication''' are both associated with an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if the first operand is true and the second operand is false.

The [[truth table]] associated with the material conditional '''if p then q''' (symbolized as '''p&nbsp;&rarr;&nbsp;q''') and the logical implication '''p implies q''' (symbolized as '''p&nbsp;&rArr;&nbsp;q''') is as follows:

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%"
|+ '''Logical Implication'''
|- style="background:aliceblue"
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | p &rArr; q
|-
| F || F || T
|-
| F || T || T
|-
| T || F || F
|-
| T || T || T
|}
<br>

====[[Logical NAND]]====

The '''NAND operation''' is a [[logical operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if both of its operands are true. In other words, it produces a value of ''true'' if and only if at least one of its operands is false.

The [[truth table]] of '''p NAND q''' (also written as '''p&nbsp;|&nbsp;q''' or '''p&nbsp;&uarr;&nbsp;q''') is as follows:

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%"
|+ '''Logical NAND'''
|- style="background:aliceblue"
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | p &uarr; q
|-
| F || F || T
|-
| F || T || T
|-
| T || F || T
|-
| T || T || F
|}
<br>

====[[Logical NNOR]]====

The '''NNOR operation''' is a [[logical operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' if and only if both of its operands are false. In other words, it produces a value of ''false'' if and only if at least one of its operands is true.

The [[truth table]] of '''p NNOR q''' (also written as '''p&nbsp;&perp;&nbsp;q''' or '''p&nbsp;&darr;&nbsp;q''') is as follows:

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%"
|+ '''Logical NOR'''
|- style="background:aliceblue"
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | p &darr; q
|-
| F || F || T
|-
| F || T || F
|-
| T || F || F
|-
| T || T || F
|}
<br>

===Exclusive Disjunction===

A + B = (A &#8743; !B) &#8744; (!A &#8743; B)
= {(A &#8743; !B) &#8744; !A} &#8743; {(A &#8743; !B) &#8744; B}
= {(A &#8744; !A) &#8743; (!B &#8744; !A)} &#8743; {(A &#8744; B) &#8743; (!B &#8744; B)}
= (!A &#8744; !B) &#8743; (A &#8744; B)
= !(A &#8743; B) &#8743; (A &#8744; B)


p + q = (p &#8743; !q) &#8744; (!p &#8743; B)

= {(p &#8743; !q) &#8744; !p} &#8743; {(p &#8743; !q) &#8744; q}

= {(p &#8744; !q) &#8743; (!q &#8744; !p)} &#8743; {(p &#8744; q) &#8743; (!q &#8744; q)}

= (!p &#8744; !q) &#8743; (p &#8744; q)

= !(p &#8743; q) &#8743; (p &#8744; q)


p + q = (p &#8743; ~q) &#8744; (~p &#8743; q)

= ((p &#8743; ~q) &#8744; ~p) &#8743; ((p &#8743; ~q) &#8744; q)

= ((p &#8744; ~q) &#8743; (~q &#8744; ~p)) &#8743; ((p &#8744; q) &#8743; (~q &#8744; q))

= (~p &#8744; ~q) &#8743; (p &#8744; q)

= ~(p &#8743; q) &#8743; (p &#8744; q)

: <math>\begin{matrix}
p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\
& = & ((p \land \lnot q) \lor \lnot p) & \and & ((p \land \lnot q) \lor q) \\
& = & ((p \lor \lnot q) \land (\lnot q \lor \lnot p)) & \land & ((p \lor q) \land (\lnot q \lor q)) \\
& = & (\lnot p \lor \lnot q) & \land & (p \lor q) \\
& = & \lnot (p \land q) & \land & (p \lor q)
\end{matrix}</math>

==Relational Tables==

===Sign Relations===

{| cellpadding="4"
| width="20px" | &nbsp;
| align="center" | '''O''' || = || Object Domain
|-
| width="20px" | &nbsp;
| align="center" | '''S''' || = || Sign Domain
|-
| width="20px" | &nbsp;
| align="center" | '''I''' || = || Interpretant Domain
|}
<br>

{| cellpadding="4"
| width="20px" | &nbsp;
| align="center" | '''O'''
| =
| {Ann, Bob}
| =
| {A, B}
|-
| width="20px" | &nbsp;
| align="center" | '''S'''
| =
| {"Ann", "Bob", "I", "You"}
| =
| {"A", "B", "i", "u"}
|-
| width="20px" | &nbsp;
| align="center" | '''I'''
| =
| {"Ann", "Bob", "I", "You"}
| =
| {"A", "B", "i", "u"}
|}
<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
|+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A
|- style="background:paleturquoise"
! style="width:20%" | Object
! style="width:20%" | Sign
! style="width:20%" | Interpretant
|-
| '''A''' || '''"A"''' || '''"A"'''
|-
| '''A''' || '''"A"''' || '''"i"'''
|-
| '''A''' || '''"i"''' || '''"A"'''
|-
| '''A''' || '''"i"''' || '''"i"'''
|-
| '''B''' || '''"B"''' || '''"B"'''
|-
| '''B''' || '''"B"''' || '''"u"'''
|-
| '''B''' || '''"u"''' || '''"B"'''
|-
| '''B''' || '''"u"''' || '''"u"'''
|}
<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
|+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B
|- style="background:paleturquoise"
! style="width:20%" | Object
! style="width:20%" | Sign
! style="width:20%" | Interpretant
|-
| '''A''' || '''"A"''' || '''"A"'''
|-
| '''A''' || '''"A"''' || '''"u"'''
|-
| '''A''' || '''"u"''' || '''"A"'''
|-
| '''A''' || '''"u"''' || '''"u"'''
|-
| '''B''' || '''"B"''' || '''"B"'''
|-
| '''B''' || '''"B"''' || '''"i"'''
|-
| '''B''' || '''"i"''' || '''"B"'''
|-
| '''B''' || '''"i"''' || '''"i"'''
|}
<br>

===Triadic Relations===

====Algebraic Examples====

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
|+ '''L'''<sub>0</sub> = {(''x'', ''y'', ''z'') &isin; '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 0}
|- style="background:paleturquoise"
! X !! Y !! Z
|-
| '''0''' || '''0''' || '''0'''
|-
| '''0''' || '''1''' || '''1'''
|-
| '''1''' || '''0''' || '''1'''
|-
| '''1''' || '''1''' || '''0'''
|}
<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
|+ '''L'''<sub>1</sub> = {(''x'', ''y'', ''z'') &isin; '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 1}
|- style="background:paleturquoise"
! X !! Y !! Z
|-
| '''0''' || '''0''' || '''1'''
|-
| '''0''' || '''1''' || '''0'''
|-
| '''1''' || '''0''' || '''0'''
|-
| '''1''' || '''1''' || '''1'''
|}
<br>

====Semiotic Examples====

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
|+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A
|- style="background:paleturquoise"
! style="width:20%" | Object
! style="width:20%" | Sign
! style="width:20%" | Interpretant
|-
| '''A''' || '''"A"''' || '''"A"'''
|-
| '''A''' || '''"A"''' || '''"i"'''
|-
| '''A''' || '''"i"''' || '''"A"'''
|-
| '''A''' || '''"i"''' || '''"i"'''
|-
| '''B''' || '''"B"''' || '''"B"'''
|-
| '''B''' || '''"B"''' || '''"u"'''
|-
| '''B''' || '''"u"''' || '''"B"'''
|-
| '''B''' || '''"u"''' || '''"u"'''
|}
<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
|+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B
|- style="background:paleturquoise"
! style="width:20%" | Object
! style="width:20%" | Sign
! style="width:20%" | Interpretant
|-
| '''A''' || '''"A"''' || '''"A"'''
|-
| '''A''' || '''"A"''' || '''"u"'''
|-
| '''A''' || '''"u"''' || '''"A"'''
|-
| '''A''' || '''"u"''' || '''"u"'''
|-
| '''B''' || '''"B"''' || '''"B"'''
|-
| '''B''' || '''"B"''' || '''"i"'''
|-
| '''B''' || '''"i"''' || '''"B"'''
|-
| '''B''' || '''"i"''' || '''"i"'''
|}
<br>

===Dyadic Projections===

{| cellpadding="4"
| width="20px" | &nbsp;
| '''L'''<sub>OS</sub>
| =
| ''proj''<sub>OS</sub>('''L''')
| =
| { (''o'', ''s'') &isin; '''O''' &times; '''S''' : (''o'', ''s'', ''i'') &isin; '''L''' for some ''i'' &isin; '''I''' }
|-
| width="20px" | &nbsp;
| '''L'''<sub>SO</sub>
| =
| ''proj''<sub>SO</sub>('''L''')
| =
| { (''s'', ''o'') &isin; '''S''' &times; '''O''' : (''o'', ''s'', ''i'') &isin; '''L''' for some ''i'' &isin; '''I''' }
|-
| width="20px" | &nbsp;
| '''L'''<sub>IS</sub>
| =
| ''proj''<sub>IS</sub>('''L''')
| =
| { (''i'', ''s'') &isin; '''I''' &times; '''S''' : (''o'', ''s'', ''i'') &isin; '''L''' for some ''o'' &isin; '''O''' }
|-
| width="20px" | &nbsp;
| '''L'''<sub>SI</sub>
| =
| ''proj''<sub>SI</sub>('''L''')
| =
| { (''s'', ''i'') &isin; '''S''' &times; '''I''' : (''o'', ''s'', ''i'') &isin; '''L''' for some ''o'' &isin; '''O''' }
|-
| width="20px" | &nbsp;
| '''L'''<sub>OI</sub>
| =
| ''proj''<sub>OI</sub>('''L''')
| =
| { (''o'', ''i'') &isin; '''O''' &times; '''I''' : (''o'', ''s'', ''i'') &isin; '''L''' for some ''s'' &isin; '''S''' }
|-
| width="20px" | &nbsp;
| '''L'''<sub>IO</sub>
| =
| ''proj''<sub>IO</sub>('''L''')
| =
| { (''i'', ''o'') &isin; '''I''' &times; '''O''' : (''o'', ''s'', ''i'') &isin; '''L''' for some ''s'' &isin; '''S''' }
|}
<br>

====Method 1 : Subtitles as Captions====

{| align="center" style="width:90%"
|
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ ''proj''<sub>OS</sub>('''L'''<sub>A</sub>)
|- style="background:paleturquoise"
! style="width:50%" | Object
! style="width:50%" | Sign
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"i"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"u"'''
|}
|
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ ''proj''<sub>OS</sub>('''L'''<sub>B</sub>)
|- style="background:paleturquoise"
! style="width:50%" | Object
! style="width:50%" | Sign
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"u"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"i"'''
|}
|}
<br>

{| align="center" style="width:90%"
|
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ ''proj''<sub>SI</sub>('''L'''<sub>A</sub>)
|- style="background:paleturquoise"
! style="width:50%" | Sign
! style="width:50%" | Interpretant
|-
| '''"A"''' || '''"A"'''
|-
| '''"A"''' || '''"i"'''
|-
| '''"i"''' || '''"A"'''
|-
| '''"i"''' || '''"i"'''
|-
| '''"B"''' || '''"B"'''
|-
| '''"B"''' || '''"u"'''
|-
| '''"u"''' || '''"B"'''
|-
| '''"u"''' || '''"u"'''
|}
|
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ ''proj''<sub>SI</sub>('''L'''<sub>B</sub>)
|- style="background:paleturquoise"
! style="width:50%" | Sign
! style="width:50%" | Interpretant
|-
| '''"A"''' || '''"A"'''
|-
| '''"A"''' || '''"u"'''
|-
| '''"u"''' || '''"A"'''
|-
| '''"u"''' || '''"u"'''
|-
| '''"B"''' || '''"B"'''
|-
| '''"B"''' || '''"i"'''
|-
| '''"i"''' || '''"B"'''
|-
| '''"i"''' || '''"i"'''
|}
|}
<br>

{| align="center" style="width:90%"
|
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ ''proj''<sub>OI</sub>('''L'''<sub>A</sub>)
|- style="background:paleturquoise"
! style="width:50%" | Object
! style="width:50%" | Interpretant
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"i"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"u"'''
|}
|
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ ''proj''<sub>OI</sub>('''L'''<sub>B</sub>)
|- style="background:paleturquoise"
! style="width:50%" | Object
! style="width:50%" | Interpretant
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"u"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"i"'''
|}
|}
<br>

====Method 2 : Subtitles as Top Rows====

{| align="center" style="width:90%"
| align="center" style="width:45%" | ''proj''<sub>OS</sub>('''L'''<sub>A</sub>)
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! style="width:50%" | Object
! style="width:50%" | Sign
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"i"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"u"'''
|}
| align="center" style="width:45%" | ''proj''<sub>OS</sub>('''L'''<sub>B</sub>)
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! style="width:50%" | Object
! style="width:50%" | Sign
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"u"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"i"'''
|}
|}
<br>

{| align="center" style="width:90%"
| align="center" style="width:45%" | ''proj''<sub>SI</sub>('''L'''<sub>A</sub>)
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! style="width:50%" | Sign
! style="width:50%" | Interpretant
|-
| '''"A"''' || '''"A"'''
|-
| '''"A"''' || '''"i"'''
|-
| '''"i"''' || '''"A"'''
|-
| '''"i"''' || '''"i"'''
|-
| '''"B"''' || '''"B"'''
|-
| '''"B"''' || '''"u"'''
|-
| '''"u"''' || '''"B"'''
|-
| '''"u"''' || '''"u"'''
|}
| align="center" style="width:45%" | ''proj''<sub>SI</sub>('''L'''<sub>B</sub>)
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! style="width:50%" | Sign
! style="width:50%" | Interpretant
|-
| '''"A"''' || '''"A"'''
|-
| '''"A"''' || '''"u"'''
|-
| '''"u"''' || '''"A"'''
|-
| '''"u"''' || '''"u"'''
|-
| '''"B"''' || '''"B"'''
|-
| '''"B"''' || '''"i"'''
|-
| '''"i"''' || '''"B"'''
|-
| '''"i"''' || '''"i"'''
|}
|}
<br>

{| align="center" style="width:90%"
| align="center" style="width:45%" | ''proj''<sub>OI</sub>('''L'''<sub>A</sub>)
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! style="width:50%" | Object
! style="width:50%" | Interpretant
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"i"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"u"'''
|}
| align="center" style="width:45%" | ''proj''<sub>OI</sub>('''L'''<sub>B</sub>)
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! style="width:50%" | Object
! style="width:50%" | Interpretant
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"u"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"i"'''
|}
|}
<br>

===Relation Reduction===

====Method 1 : Subtitles as Captions====

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
|+ '''L'''<sub>0</sub> = {(''x'', ''y'', ''z'') &#8712; '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 0}
|- style="background:paleturquoise"
! X !! Y !! Z
|-
| '''0''' || '''0''' || '''0'''
|-
| '''0''' || '''1''' || '''1'''
|-
| '''1''' || '''0''' || '''1'''
|-
| '''1''' || '''1''' || '''0'''
|}
<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
|+ '''L'''<sub>1</sub> = {(''x'', ''y'', ''z'') &#8712; '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 1}
|- style="background:paleturquoise"
! X !! Y !! Z
|-
| '''0''' || '''0''' || '''1'''
|-
| '''0''' || '''1''' || '''0'''
|-
| '''1''' || '''0''' || '''0'''
|-
| '''1''' || '''1''' || '''1'''
|}
<br>

{| align="center" style="width:90%"
|
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ proj<sub>''XY''</sub>('''L'''<sub>0</sub>)
|- style="background:paleturquoise"
! X !! Y
|-
| '''0''' || '''0'''
|-
| '''0''' || '''1'''
|-
| '''1''' || '''0'''
|-
| '''1''' || '''1'''
|}
|
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ proj<sub>''XZ''</sub>('''L'''<sub>0</sub>)
|- style="background:paleturquoise"
! X !! Z
|-
| '''0''' || '''0'''
|-
| '''0''' || '''1'''
|-
| '''1''' || '''1'''
|-
| '''1''' || '''0'''
|}
|
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ proj<sub>''YZ''</sub>('''L'''<sub>0</sub>)
|- style="background:paleturquoise"
! Y !! Z
|-
| '''0''' || '''0'''
|-
| '''1''' || '''1'''
|-
| '''0''' || '''1'''
|-
| '''1''' || '''0'''
|}
|}
<br>

{| align="center" style="width:90%"
|
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ proj<sub>''XY''</sub>('''L'''<sub>1</sub>)
|- style="background:paleturquoise"
! X !! Y
|-
| '''0''' || '''0'''
|-
| '''0''' || '''1'''
|-
| '''1''' || '''0'''
|-
| '''1''' || '''1'''
|}
|
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ proj<sub>''XZ''</sub>('''L'''<sub>1</sub>)
|- style="background:paleturquoise"
! X !! Z
|-
| '''0''' || '''1'''
|-
| '''0''' || '''0'''
|-
| '''1''' || '''0'''
|-
| '''1''' || '''1'''
|}
|
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ proj<sub>''YZ''</sub>('''L'''<sub>1</sub>)
|- style="background:paleturquoise"
! Y !! Z
|-
| '''0''' || '''1'''
|-
| '''1''' || '''0'''
|-
| '''0''' || '''0'''
|-
| '''1''' || '''1'''
|}
|}
<br>

{| align="center" cellpadding="4" style="text-align:center; width:90%"
| proj<sub>''XY''</sub>('''L'''<sub>0</sub>) = proj<sub>''XY''</sub>('''L'''<sub>1</sub>)
| proj<sub>''XZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''XZ''</sub>('''L'''<sub>1</sub>)
| proj<sub>''YZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''YZ''</sub>('''L'''<sub>1</sub>)
|}
<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
|+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A
|- style="background:paleturquoise"
! style="width:20%" | Object
! style="width:20%" | Sign
! style="width:20%" | Interpretant
|-
| '''A''' || '''"A"''' || '''"A"'''
|-
| '''A''' || '''"A"''' || '''"i"'''
|-
| '''A''' || '''"i"''' || '''"A"'''
|-
| '''A''' || '''"i"''' || '''"i"'''
|-
| '''B''' || '''"B"''' || '''"B"'''
|-
| '''B''' || '''"B"''' || '''"u"'''
|-
| '''B''' || '''"u"''' || '''"B"'''
|-
| '''B''' || '''"u"''' || '''"u"'''
|}
<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
|+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B
|- style="background:paleturquoise"
! style="width:20%" | Object
! style="width:20%" | Sign
! style="width:20%" | Interpretant
|-
| '''A''' || '''"A"''' || '''"A"'''
|-
| '''A''' || '''"A"''' || '''"u"'''
|-
| '''A''' || '''"u"''' || '''"A"'''
|-
| '''A''' || '''"u"''' || '''"u"'''
|-
| '''B''' || '''"B"''' || '''"B"'''
|-
| '''B''' || '''"B"''' || '''"i"'''
|-
| '''B''' || '''"i"''' || '''"B"'''
|-
| '''B''' || '''"i"''' || '''"i"'''
|}
<br>

{| align="center" style="width:90%"
|
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ proj<sub>''XY''</sub>('''L'''<sub>A</sub>)
|- style="background:paleturquoise"
! style="width:50%" | Object
! style="width:50%" | Sign
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"i"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"u"'''
|}
|
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ proj<sub>''XZ''</sub>('''L'''<sub>A</sub>)
|- style="background:paleturquoise"
! style="width:50%" | Object
! style="width:50%" | Interpretant
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"i"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"u"'''
|}
|
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ proj<sub>''YZ''</sub>('''L'''<sub>A</sub>)
|- style="background:paleturquoise"
! style="width:50%" | Sign
! style="width:50%" | Interpretant
|-
| '''"A"''' || '''"A"'''
|-
| '''"A"''' || '''"i"'''
|-
| '''"i"''' || '''"A"'''
|-
| '''"i"''' || '''"i"'''
|-
| '''"B"''' || '''"B"'''
|-
| '''"B"''' || '''"u"'''
|-
| '''"u"''' || '''"B"'''
|-
| '''"u"''' || '''"u"'''
|}
|}
<br>

{| align="center" style="width:90%"
|
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ proj<sub>''XY''</sub>('''L'''<sub>B</sub>)
|- style="background:paleturquoise"
! style="width:50%" | Object
! style="width:50%" | Sign
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"u"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"i"'''
|}
|
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ proj<sub>''XZ''</sub>('''L'''<sub>B</sub>)
|- style="background:paleturquoise"
! style="width:50%" | Object
! style="width:50%" | Interpretant
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"u"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"i"'''
|}
|
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ proj<sub>''YZ''</sub>('''L'''<sub>B</sub>)
|- style="background:paleturquoise"
! style="width:50%" | Sign
! style="width:50%" | Interpretant
|-
| '''"A"''' || '''"A"'''
|-
| '''"A"''' || '''"u"'''
|-
| '''"u"''' || '''"A"'''
|-
| '''"u"''' || '''"u"'''
|-
| '''"B"''' || '''"B"'''
|-
| '''"B"''' || '''"i"'''
|-
| '''"i"''' || '''"B"'''
|-
| '''"i"''' || '''"i"'''
|}
|}
<br>

{| align="center" cellpadding="4" style="text-align:center; width:90%"
| proj<sub>''XY''</sub>('''L'''<sub>A</sub>) &#8800; proj<sub>''XY''</sub>('''L'''<sub>B</sub>)
| proj<sub>''XZ''</sub>('''L'''<sub>A</sub>) &#8800; proj<sub>''XZ''</sub>('''L'''<sub>B</sub>)
| proj<sub>''YZ''</sub>('''L'''<sub>A</sub>) &#8800; proj<sub>''YZ''</sub>('''L'''<sub>B</sub>)
|}
<br>

====Method 2 : Subtitles as Top Rows====

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
|+ '''L'''<sub>0</sub> = {(''x'', ''y'', ''z'') &#8712; '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 0}
|- style="background:paleturquoise"
! X !! Y !! Z
|-
| '''0''' || '''0''' || '''0'''
|-
| '''0''' || '''1''' || '''1'''
|-
| '''1''' || '''0''' || '''1'''
|-
| '''1''' || '''1''' || '''0'''
|}
<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
|+ '''L'''<sub>1</sub> = {(''x'', ''y'', ''z'') &#8712; '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 1}
|- style="background:paleturquoise"
! X !! Y !! Z
|-
| '''0''' || '''0''' || '''1'''
|-
| '''0''' || '''1''' || '''0'''
|-
| '''1''' || '''0''' || '''0'''
|-
| '''1''' || '''1''' || '''1'''
|}
<br>

{| align="center" style="width:90%"
| align="center" | proj<sub>''XY''</sub>('''L'''<sub>0</sub>)
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! X !! Y
|-
| '''0''' || '''0'''
|-
| '''0''' || '''1'''
|-
| '''1''' || '''0'''
|-
| '''1''' || '''1'''
|}
| align="center" | proj<sub>''XZ''</sub>('''L'''<sub>0</sub>)
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! X !! Z
|-
| '''0''' || '''0'''
|-
| '''0''' || '''1'''
|-
| '''1''' || '''1'''
|-
| '''1''' || '''0'''
|}
| align="center" | proj<sub>''YZ''</sub>('''L'''<sub>0</sub>)
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! Y !! Z
|-
| '''0''' || '''0'''
|-
| '''1''' || '''1'''
|-
| '''0''' || '''1'''
|-
| '''1''' || '''0'''
|}
|}
<br>

{| align="center" style="width:90%"
| align="center" | proj<sub>''XY''</sub>('''L'''<sub>1</sub>)
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! X !! Y
|-
| '''0''' || '''0'''
|-
| '''0''' || '''1'''
|-
| '''1''' || '''0'''
|-
| '''1''' || '''1'''
|}
| align="center" | proj<sub>''XZ''</sub>('''L'''<sub>1</sub>)
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! X !! Z
|-
| '''0''' || '''1'''
|-
| '''0''' || '''0'''
|-
| '''1''' || '''0'''
|-
| '''1''' || '''1'''
|}
| align="center" | proj<sub>''YZ''</sub>('''L'''<sub>1</sub>)
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! Y !! Z
|-
| '''0''' || '''1'''
|-
| '''1''' || '''0'''
|-
| '''0''' || '''0'''
|-
| '''1''' || '''1'''
|}
|}
<br>

{| align="center" cellpadding="4" style="text-align:center; width:90%"
| proj<sub>''XY''</sub>('''L'''<sub>0</sub>) = proj<sub>''XY''</sub>('''L'''<sub>1</sub>)
| proj<sub>''XZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''XZ''</sub>('''L'''<sub>1</sub>)
| proj<sub>''YZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''YZ''</sub>('''L'''<sub>1</sub>)
|}
<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
|+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A
|- style="background:paleturquoise"
! style="width:20%" | Object
! style="width:20%" | Sign
! style="width:20%" | Interpretant
|-
| '''A''' || '''"A"''' || '''"A"'''
|-
| '''A''' || '''"A"''' || '''"i"'''
|-
| '''A''' || '''"i"''' || '''"A"'''
|-
| '''A''' || '''"i"''' || '''"i"'''
|-
| '''B''' || '''"B"''' || '''"B"'''
|-
| '''B''' || '''"B"''' || '''"u"'''
|-
| '''B''' || '''"u"''' || '''"B"'''
|-
| '''B''' || '''"u"''' || '''"u"'''
|}
<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
|+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B
|- style="background:paleturquoise"
! style="width:20%" | Object
! style="width:20%" | Sign
! style="width:20%" | Interpretant
|-
| '''A''' || '''"A"''' || '''"A"'''
|-
| '''A''' || '''"A"''' || '''"u"'''
|-
| '''A''' || '''"u"''' || '''"A"'''
|-
| '''A''' || '''"u"''' || '''"u"'''
|-
| '''B''' || '''"B"''' || '''"B"'''
|-
| '''B''' || '''"B"''' || '''"i"'''
|-
| '''B''' || '''"i"''' || '''"B"'''
|-
| '''B''' || '''"i"''' || '''"i"'''
|}
<br>

{| align="center" style="width:90%"
| align="center" style="width:30%" | proj<sub>''XY''</sub>('''L'''<sub>A</sub>)
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! style="width:50%" | Object
! style="width:50%" | Sign
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"i"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"u"'''
|}
| align="center" style="width:30%" | proj<sub>''XZ''</sub>('''L'''<sub>A</sub>)
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! style="width:50%" | Object
! style="width:50%" | Interpretant
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"i"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"u"'''
|}
| align="center" style="width:30%" | proj<sub>''YZ''</sub>('''L'''<sub>A</sub>)
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! style="width:50%" | Sign
! style="width:50%" | Interpretant
|-
| '''"A"''' || '''"A"'''
|-
| '''"A"''' || '''"i"'''
|-
| '''"i"''' || '''"A"'''
|-
| '''"i"''' || '''"i"'''
|-
| '''"B"''' || '''"B"'''
|-
| '''"B"''' || '''"u"'''
|-
| '''"u"''' || '''"B"'''
|-
| '''"u"''' || '''"u"'''
|}
|}
<br>

{| align="center" style="width:90%"
| align="center" style="width:30%" | proj<sub>''XY''</sub>('''L'''<sub>B</sub>)
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! style="width:50%" | Object
! style="width:50%" | Sign
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"u"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"i"'''
|}
| align="center" style="width:30%" | proj<sub>''XZ''</sub>('''L'''<sub>B</sub>)
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! style="width:50%" | Object
! style="width:50%" | Interpretant
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"u"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"i"'''
|}
| align="center" style="width:30%" | proj<sub>''YZ''</sub>('''L'''<sub>B</sub>)
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! style="width:50%" | Sign
! style="width:50%" | Interpretant
|-
| '''"A"''' || '''"A"'''
|-
| '''"A"''' || '''"u"'''
|-
| '''"u"''' || '''"A"'''
|-
| '''"u"''' || '''"u"'''
|-
| '''"B"''' || '''"B"'''
|-
| '''"B"''' || '''"i"'''
|-
| '''"i"''' || '''"B"'''
|-
| '''"i"''' || '''"i"'''
|}
|}
<br>

{| align="center" cellpadding="4" style="text-align:center; width:90%"
| proj<sub>''XY''</sub>('''L'''<sub>A</sub>) &#8800; proj<sub>''XY''</sub>('''L'''<sub>B</sub>)
| proj<sub>''XZ''</sub>('''L'''<sub>A</sub>) &#8800; proj<sub>''XZ''</sub>('''L'''<sub>B</sub>)
| proj<sub>''YZ''</sub>('''L'''<sub>A</sub>) &#8800; proj<sub>''YZ''</sub>('''L'''<sub>B</sub>)
|}
<br>

===Formatted Text Display===

: So in a triadic fact, say, the example <br>
{| align="center" cellspacing="8" style="width:72%"
| align="center" | ''A'' gives ''B'' to ''C''
|}
: we make no distinction in the ordinary logic of relations between the ''[[subject (grammar)|subject]] [[nominative]]'', the ''[[direct object]]'', and the ''[[indirect object]]''. We say that the proposition has three ''logical subjects''. We regard it as a mere affair of English grammar that there are six ways of expressing this: <br>
{| align="center" cellspacing="8" style="width:72%"
| style="width:36%" | ''A'' gives ''B'' to ''C''
| style="width:36%" | ''A'' benefits ''C'' with ''B''
|-
| ''B'' enriches ''C'' at expense of ''A''
| ''C'' receives ''B'' from ''A''
|-
| ''C'' thanks ''A'' for ''B''
| ''B'' leaves ''A'' for ''C''
|}
: These six sentences express one and the same indivisible phenomenon. (C.S. Peirce, "The Categories Defended", MS 308 (1903), EP 2, 170-171).

==Work Area==

{| border="1" cellspacing="0" cellpadding="0" style="text-align:center"
|+ Binary Operations
|-
! style="width:2em" | x<sub>0</sub>
! style="width:2em" | x<sub>1</sub>
| style="width:2em" | <sup>2</sup>f<sub>0</sub>
| style="width:2em" | <sup>2</sup>f<sub>1</sub>
| style="width:2em" | <sup>2</sup>f<sub>2</sub>
| style="width:2em" | <sup>2</sup>f<sub>3</sub>
| style="width:2em" | <sup>2</sup>f<sub>4</sub>
| style="width:2em" | <sup>2</sup>f<sub>5</sub>
| style="width:2em" | <sup>2</sup>f<sub>6</sub>
| style="width:2em" | <sup>2</sup>f<sub>7</sub>
| style="width:2em" | <sup>2</sup>f<sub>8</sub>
| style="width:2em" | <sup>2</sup>f<sub>9</sub>
| style="width:2em" | <sup>2</sup>f<sub>10</sub>
| style="width:2em" | <sup>2</sup>f<sub>11</sub>
| style="width:2em" | <sup>2</sup>f<sub>12</sub>
| style="width:2em" | <sup>2</sup>f<sub>13</sub>
| style="width:2em" | <sup>2</sup>f<sub>14</sub>
| style="width:2em" | <sup>2</sup>f<sub>15</sub>
|-
| 0 || 0 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1
|-
| 1 || 0 || 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1
|-
| 0 || 1 || 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1
|-
| 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
|}
<br>

===Draft 1===

<center><table>
<caption>TRUTH TABLES FOR THE BOOLEAN OPERATIONS OF ARITY UP TO 2</caption>
<tr valign="top">
<td><table border=5 cellspacing=0>
<caption>Constants</caption>
<tr><td></td>
<td><sup>0</sup>f<sub>0</sub></td> <td><sup>0</sup>f<sub>1</sub></td>
</tr> <tr><td></td>
<td align=center>0</td> <td align=center>1</td>
</tr></table></td><td>&nbsp;&nbsp;&nbsp;&nbsp;</td>
<td><table border=5 cellspacing=0><caption>Unary Operations</caption><tr>
<td>x<sub>0</sub></td> <td></td>
<td><sup>1</sup>f<sub>0 </sub></td> <td><sup>1</sup>f<sub>1 </sub></td>
<td><sup>1</sup>f<sub>2 </sub></td> <td><sup>1</sup>f<sub>3 </sub></td>
</tr><tr> <td align=center>0</td> <td></td>
<td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
</tr> <tr> <td align=center>1</td> <td></td>
<td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
</tr></table></td><td>&nbsp;&nbsp;&nbsp;&nbsp;</td>
<td><table border=5 cellspacing=0><caption>Binary Operations</caption><tr>
<td>x<sub>0</sub></td> <td>x<sub>1</sub></td>
<td></td>
<td><sup>2</sup>f<sub>0</sub></td> <td><sup>2</sup>f<sub>1 </sub></td>
<td><sup> 2</sup>f<sub>2 </sub></td> <td><sup>2</sup>f<sub>3 </sub></td>
<td><sup>2</sup>f<sub>4 </sub></td> <td><sup>2</sup>f<sub>5 </sub></td>
<td><sup>2</sup>f<sub>6 </sub></td> <td><sup>2</sup>f<sub>7 </sub></td>
<td><sup>2</sup>f<sub>8 </sub></td> <td><sup>2</sup>f<sub>9 </sub></td>
<td><sup>2</sup>f<sub>10</sub></td> <td><sup>2</sup>f<sub>11</sub></td>
<td><sup>2</sup>f<sub>12</sub></td> <td><sup>2</sup>f<sub>13</sub></td>
<td><sup>2</sup>f<sub>14</sub></td> <td><sup>2</sup>f<sub>15</sub></td>
</tr><tr> <td align=center>0</td> <td align=center>0</td> <td></td>
<td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
<td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
<td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
<td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
</tr> <tr> <td align=center>1</td> <td align=center>0</td> <td></td>
<td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
<td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
<td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
<td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
</tr> <tr> <td align=center>0</td> <td align=center>1</td> <td></td>
<td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td>
<td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td>
<td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td>
<td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td>
</tr> <tr> <td align=center>1</td> <td align=center>1</td> <td></td>
<td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td>
<td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td>
<td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td>
<td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td>
</tr> </table></td>
</table></center>

===Draft 2===

<center><table>
<caption>TRUTH TABLES FOR THE BOOLEAN OPERATIONS OF ARITY UP TO 2</caption>
<tr valign="top">
<td><table border=5 cellspacing=0>
<caption>Constants</caption>
<tr><td></td>
<td><sup>0</sup>f<sub>0</sub></td> <td><sup>0</sup>f<sub>1</sub></td>
</tr> <tr><td></td>
<td align=center>0</td> <td align=center>1</td>
</tr></table></td><td>&nbsp;&nbsp;&nbsp;&nbsp;</td>
<td><table border=5 cellspacing=0><caption>Unary Operations</caption><tr>
<td>x<sub>0</sub></td> <td></td>
<td><sup>1</sup>f<sub>0 </sub></td> <td><sup>1</sup>f<sub>1 </sub></td>
<td><sup>1</sup>f<sub>2 </sub></td> <td><sup>1</sup>f<sub>3 </sub></td>
</tr><tr> <td align=center>0</td> <td></td>
<td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
</tr> <tr> <td align=center>1</td> <td></td>
<td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
</tr></table></td><td>&nbsp;&nbsp;&nbsp;&nbsp;</td>
<td><table border=5 cellspacing=0><caption>Binary Operations</caption><tr>
<td>x<sub>0</sub></td> <td>x<sub>1</sub></td>
<td></td>
<td><sup>2</sup>f<sub>0</sub></td> <td><sup>2</sup>f<sub>1 </sub></td>
<td><sup> 2</sup>f<sub>2 </sub></td> <td><sup>2</sup>f<sub>3 </sub></td>
<td><sup>2</sup>f<sub>4 </sub></td> <td><sup>2</sup>f<sub>5 </sub></td>
<td><sup>2</sup>f<sub>6 </sub></td> <td><sup>2</sup>f<sub>7 </sub></td>
<td><sup>2</sup>f<sub>8 </sub></td> <td><sup>2</sup>f<sub>9 </sub></td>
<td><sup>2</sup>f<sub>10</sub></td> <td><sup>2</sup>f<sub>11</sub></td>
<td><sup>2</sup>f<sub>12</sub></td> <td><sup>2</sup>f<sub>13</sub></td>
<td><sup>2</sup>f<sub>14</sub></td> <td><sup>2</sup>f<sub>15</sub></td>
</tr><tr> <td align=center>0</td> <td align=center>0</td> <td></td>
<td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
<td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
<td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
<td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
</tr> <tr> <td align=center>1</td> <td align=center>0</td> <td></td>
<td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
<td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
<td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
<td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
</tr> <tr> <td align=center>0</td> <td align=center>1</td> <td></td>
<td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td>
<td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td>
<td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td>
<td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td>
</tr> <tr> <td align=center>1</td> <td align=center>1</td> <td></td>
<td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td>
<td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td>
<td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td>
<td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td>
</tr> </table></td>
</table></center>
Jon Awbrey
12,080

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