Difference between revisions of "User:Jon Awbrey/TABLE"

MyWikiBiz, Author Your Legacy — Friday December 27, 2024
Jump to navigationJump to search
(→‎Wiki TeX Tables: add example)
 
(96 intermediate revisions by the same user not shown)
Line 1: Line 1:
−
==Differential Logic==
+
==Cactus Language==
  
 
===Ascii Tables===
 
===Ascii Tables===
  
 +
{| align="center" cellpadding="6" style="text-align:center; width:90%"
 +
|
 +
<pre>
 +
o-------------------o
 +
|                  |
 +
|        @        |
 +
|                  |
 +
o-------------------o
 +
|                  |
 +
|        o        |
 +
|        |        |
 +
|        @        |
 +
|                  |
 +
o-------------------o
 +
|                  |
 +
|        a        |
 +
|        @        |
 +
|                  |
 +
o-------------------o
 +
|                  |
 +
|        a        |
 +
|        o        |
 +
|        |        |
 +
|        @        |
 +
|                  |
 +
o-------------------o
 +
|                  |
 +
|      a b c      |
 +
|        @        |
 +
|                  |
 +
o-------------------o
 +
|                  |
 +
|      a b c      |
 +
|      o o o      |
 +
|        \|/        |
 +
|        o        |
 +
|        |        |
 +
|        @        |
 +
|                  |
 +
o-------------------o
 +
|                  |
 +
|        a  b    |
 +
|        o---o    |
 +
|        |        |
 +
|        @        |
 +
|                  |
 +
o-------------------o
 +
|                  |
 +
|      a  b      |
 +
|      o---o      |
 +
|        \ /        |
 +
|        @        |
 +
|                  |
 +
o-------------------o
 +
|                  |
 +
|      a  b      |
 +
|      o---o      |
 +
|        \ /        |
 +
|        o        |
 +
|        |        |
 +
|        @        |
 +
|                  |
 +
o-------------------o
 +
|                  |
 +
|      a  b  c      |
 +
|      o--o--o      |
 +
|      \  /      |
 +
|        \ /        |
 +
|        @        |
 +
|                  |
 +
o-------------------o
 +
|                  |
 +
|      a  b  c      |
 +
|      o  o  o      |
 +
|      |  |  |      |
 +
|      o--o--o      |
 +
|      \  /      |
 +
|        \ /        |
 +
|        @        |
 +
|                  |
 +
o-------------------o
 +
|                  |
 +
|        b  c      |
 +
|        o  o      |
 +
|      a  |  |      |
 +
|      o--o--o      |
 +
|      \  /      |
 +
|        \ /        |
 +
|        @        |
 +
|                  |
 +
o-------------------o
 +
</pre>
 +
|}
 +
 +
{| align="center" cellpadding="6" style="text-align:center; width:90%"
 +
|
 
<pre>
 
<pre>
−
Table 1.  Propositional Forms On Two Variables
+
Table 13.  The Existential Interpretation
−
o---------o---------o---------o----------o------------------o----------o
+
o----o-------------------o-------------------o-------------------o
−
| L_1    | L_2    | L_3    | L_4      | L_5              | L_6      |
+
| Ex |  Cactus Graph    | Cactus Expression |    Existential    |
−
|        |        |        |          |                  |          |
+
|    |                  |                  |  Interpretation  |
−
| Decimal | Binary  | Vector  | Cactus  | English          | Ordinary |
+
o----o-------------------o-------------------o-------------------o
−
o---------o---------o---------o----------o------------------o----------o
+
|    |                  |                  |                  |
−
|        |     
+
|  1 |        @        |        " "        |      true.      |
 +
|    |                  |                  |                  |
 +
o----o-------------------o-------------------o-------------------o
 +
|    |                  |                  |                  |
 +
|    |        o        |                  |                  |
 +
|    |        |        |                  |                  |
 +
|  2 |        @        |        ( )        |      untrue.      |
 +
|    |                  |                  |                  |
 +
o----o-------------------o-------------------o-------------------o
 +
|    |                  |                  |                  |
 +
|    |        a        |                  |                  |
 +
|  3 |        @        |        a        |        a.        |
 +
|    |                  |                  |                  |
 +
o----o-------------------o-------------------o-------------------o
 +
|    |                  |                  |                  |
 +
|    |        a        |                  |                  |
 +
|    |        o        |                  |                  |
 +
|    |        |        |                  |                  |
 +
|  4 |        @        |        (a)        |      not a.      |
 +
|    |                  |                  |                  |
 +
o----o-------------------o-------------------o-------------------o
 +
|    |                  |                  |                  |
 +
|    |      a b c      |                  |                  |
 +
|  5 |
 
<pre>
 
<pre>
−
Table 5.  Df Expanded Over Differential Features {dx, dy}
+
Table A2.  Propositional Forms On Two Variables
−
o------o------------o------------o------------o------------o------------o
+
o---------o---------o---------o----------o------------------o----------o
−
|     |           |           |           |           |           |
+
| L_1    | L_2    | L_3    | L_4      | L_5              | L_6      |
−
|     |     f      | Df| dx dy  | Df| dx(dy) | Df| (dx)dy | Df|(dx)(dy)|
+
|         |         |         |         |                 |         |
−
|     |           |           |           |           |           |
+
| Decimal | Binary  | Vector  | Cactus  | English          | Ordinary |
−
o------o------------o------------o------------o------------o------------o
+
o---------o---------o---------o----------o------------------o----------o
−
|     |           |           |           |           |           |
+
|        |      x : 1 1 0 0 |          |                  |          |
−
| f_0  |     ()    |     ()     |     ()    |     ()    |    ()     |
+
|        |      y : 1 0 1 0 |          |                  |          |
−
|     |           |           |           |           |           |
+
o---------o---------o---------o----------o------------------o----------o
−
o------o------------o------------o------------o------------o------------o
+
|         |         |         |         |                 |         |
−
|     |           |           |           |           |           |
+
| f_0     | f_0000 | 0 0 0 0 |   ()   | false            |   0     |
−
| f_1  |   (x)(y)   | ((x, y)) |   (y)    |    (x)    |    ()    |
+
|         |         |         |         |                 |         |
−
|     |           |           |           |           |           |
+
o---------o---------o---------o----------o------------------o----------o
−
| f_2  |   (x) y   |  (x, y)   |     y      |   (x)    |    ()    |
+
|         |         |         |         |                 |         |
−
|     |           |           |           |           |           |
+
| f_1     | f_0001 | 0 0 0 1 |  (x)(y) | neither x nor y  | ~x & ~y |
−
| f_4  |   x (y)  |  (x, y)  |    (y)     |     x      |     ()    |
+
|         |         |         |         |                 |         |
−
|     |           |           |           |           |           |
+
| f_2     | f_0010 | 0 0 1 0 |  (x) y  | y and not x     | ~x &  y  |
−
| f_8  |   x  y   |  ((x, y)) |    y      |    x      |    ()    |
+
|         |         |         |         |                 |         |
−
|     |           |           |           |           |           |
+
| f_4     | f_0100 | 0 1 0 0 |  x (y) | x and not y     | x & ~y  |
−
o------o------------o------------o------------o------------o------------o
+
|         |         |         |         |                 |         |
−
|     |           |           |           |           |           |
+
| f_8     | f_1000 | 1 0 0 0 |  x  y   | x and y          |  x &  y  |
−
| f_3  |   (x)     |   (())    |   (())    |    ()    |    ()    |
+
|         |         |         |         |                 |         |
−
|     |           |           |           |           |           |
+
o---------o---------o---------o----------o------------------o----------o
−
| f_12 |   x      |   (())    |   (())    |     ()    |     ()    |
+
|         |         |         |         |                 |         |
−
|     |           |           |           |           |           |
+
| f_3     | f_0011 | 0 0 1 1 |  (x)     | not x            | ~x      |
−
o------o------------o------------o------------o------------o------------o
+
|         |         |         |         |                 |         |
−
|     |           |           |           |           |           |
+
| f_12   | f_1100  | 1 1 0 0 |   x      | x                | x      |
−
| f_6  |   (x, y)   |     ()    |   (())    |    (())    |    ()    |
+
|         |         |         |         |                 |         |
−
|     |           |           |           |           |           |
+
o---------o---------o---------o----------o------------------o----------o
−
| f_9  | ((x, y))  |     ()     |   (())    |   (())    |    ()     |
+
|         |         |         |         |                 |         |
−
|      |           |           |           |           |           |
+
| f_6     | f_0110 | 0 1 1 0 |  (x, y) | x not equal to y | x +  y  |
−
o------o------------o------------o------------o------------o------------o
+
|         |         |         |         |                 |         |
−
|     |           |           |           |           |           |
+
| f_9     | f_1001 | 1 0 0 1 | ((x, y)) | x equal to y    |  x = y  |
−
| f_5 |     (y)   |    (())    |     ()    |   (())   |     ()    |
+
|        |        |        |          |                  |          |
−
|     |           |           |           |           |           |
+
o---------o---------o---------o----------o------------------o----------o
−
| f_10 |       y    |   (())    |     ()    |   (())    |    ()     |
+
|        |        |        |          |                  |          |
−
|     |           |           |           |           |           |
+
| f_5     | f_0101  | 0 1 0 1 |    (y) | not y            |      ~y  |
 +
|        |        |        |          |                  |          |
 +
| f_10    | f_1010  | 1 0 1 0 |      y  | y                |      y  |
 +
|        |        |         |         |                 |         |
 +
o---------o---------o---------o----------o------------------o----------o
 +
|         |         |         |         |                 |         |
 +
| f_7    | f_0111 | 0 1 1 1 |  (x  y) | not both x and y | ~x v ~y  |
 +
|        |        |        |          |                  |          |
 +
| f_11   | f_1011  | 1 0 1 1 |  (x (y)) | not x without y  |  x => y  |
 +
|        |        |        |          |                  |          |
 +
| 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  |
 +
|        |        |        |          |                  |          |
 +
o---------o---------o---------o----------o------------------o----------o
 +
|        |        |        |          |                  |          |
 +
| f_15   | f_1111  | 1 1 1 1 |   (())   | true            |   1     |
 +
|         |         |         |         |                 |         |
 +
o---------o---------o---------o----------o------------------o----------o
 +
</pre>
 +
 
 +
<pre>
 +
Table A3.  Ef Expanded Over Differential Features {dx, dy}
 
o------o------------o------------o------------o------------o------------o
 
o------o------------o------------o------------o------------o------------o
 
|      |            |            |            |            |            |
 
|      |            |            |            |            |            |
−
| f_7  |  (x  y)   | ((x, y))  |     y      |     x      |    ()    |
+
|     |    f      |  T_11 f   |   T_10 f  |   T_01 f  |   T_00 f  |
 
|      |            |            |            |            |            |
 
|      |            |            |            |            |            |
−
| f_11 |   (x (y)) |   (x, y)   |   (y)     |     x      |     ()     |
+
|     |            | Ef| dx dy | Ef| dx(dy) | Ef| (dx)dy | Ef|(dx)(dy)|
 
|      |            |            |            |            |            |
 
|      |            |            |            |            |            |
−
| f_13 |  ((x) y)  |  (x, y)  |    y      |    (x)    |    ()    |
+
o------o------------o------------o------------o------------o------------o
 
|      |            |            |            |            |            |
 
|      |            |            |            |            |            |
−
| f_14 | ((x)(y))  | ((x, y))  |   (y)    |   (x)    |    ()    |
+
| f_0  |     ()     |     ()     |     ()    |     ()    |    ()    |
 
|      |            |            |            |            |            |
 
|      |            |            |            |            |            |
 
o------o------------o------------o------------o------------o------------o
 
o------o------------o------------o------------o------------o------------o
 
|      |            |            |            |            |            |
 
|      |            |            |            |            |            |
−
| f_15 |    (())    |     ()     |     ()     |     ()     |     ()     |
+
| 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
 
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>
  
−
===Wiki Tables===
+
<pre>
−
 
+
Table A4.  Df Expanded Over Differential Features {dx, dy}
−
<br>
+
o------o------------o------------o------------o------------o------------o
−
 
+
|     |            |           |           |           |           |
−
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+
|     |     f      | Df| dx dy  | Df| dx(dy) | Df| (dx)dy | Df|(dx)(dy)|
−
|+ '''Table 1.  Propositional Forms on Two Variables'''
+
|     |           |           |           |           |           |
−
|- style="background:paleturquoise"
+
o------o------------o------------o------------o------------o------------o
−
! style="width:15%" | L<sub>1</sub>
+
|     |            |            |            |           |           |
−
! style="width:15%" | L<sub>2</sub>
+
| f_0  |     ()    |     ()     |     ()    |     ()    |     ()    |
−
! style="width:15%" | L<sub>3</sub>
+
|     |            |            |            |            |            |
−
! style="width:15%" | L<sub>4</sub>
+
o------o------------o------------o------------o------------o------------o
−
! style="width:15%" | L<sub>5</sub>
+
|     |            |            |           |           |           |
−
! style="width:15%" | L<sub>6</sub>
+
| f_1  |   (x)(y)   | ((x, y))  |   (y)    |   (x)    |    ()    |
−
|- style="background:paleturquoise"
+
|     |            |           |           |           |           |
−
| &nbsp;
+
| f_2  |   (x) y   |   (x, y)  |     y     |    (x)    |     ()    |
−
| align="right" | x :
+
|     |            |            |           |           |           |
−
| 1 1 0 0
+
| f_4  |   x (y)  |   (x, y)   |   (y)    |     x     |     ()    |
−
| &nbsp;
+
|     |            |           |           |           |           |
−
| &nbsp;
+
| f_8  |   x y    |  ((x, y))  |     y      |     x     |     ()    |
−
| &nbsp;
+
|     |            |            |            |            |            |
−
|- style="background:paleturquoise"
+
o------o------------o------------o------------o------------o------------o
−
| &nbsp;
+
|     |            |            |            |           |           |
−
| align="right" | y :
+
| f_3  |   (x)      |   (())   |   (())    |     ()    |     ()    |
−
| 1 0 1 0
+
|     |            |            |           |           |           |
−
| &nbsp;
+
| f_12 |   x      |   (())   |   (())    |     ()    |     ()    |
−
| &nbsp;
+
|     |            |            |            |            |            |
−
| &nbsp;
+
o------o------------o------------o------------o------------o------------o
−
|-
+
|     |            |            |           |           |           |
−
| f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || (&nbsp;) || false || 0
+
| f_6  |   (x, y)   |     ()    |   (())    |    (())    |     ()    |
−
|-
+
|     |            |           |           |           |           |
−
| f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || &not;x &and; &not;y
+
| f_9  | ((x, y))  |    ()    |   (())    |   (())    |     ()    |
−
|-
+
|     |            |            |            |            |            |
−
| f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || &not;x &and; y
+
o------o------------o------------o------------o------------o------------o
−
|-
+
|     |            |            |           |           |           |
−
| f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || &not;x
+
| f_5  |     (y)  |    (())   |     ()    |   (())    |     ()    |
−
|-
+
|     |            |           |           |           |           |
−
| f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x &and; &not;y
+
| f_10 |       y   |   (())    |    ()    |   (())    |     ()    |
−
|-
+
|     |            |            |            |            |            |
−
| f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || &not;y
+
o------o------------o------------o------------o------------o------------o
−
|-
+
|     |            |            |           |           |           |
−
| f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x &ne; y
+
| f_7  |   (x y)  |  ((x, y)) |     y      |     x     |     ()    |
−
|-
+
|     |            |            |           |           |           |
−
| 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_11 |   (x (y))  |   (x, y)  |   (y)    |     x     |     ()    |
−
|-
+
|     |            |           |           |           |           |
−
| f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x&nbsp;y || x and y || x &and; y
+
| f_13 | ((x) y)   |   (x, y)  |     y     |    (x)    |     ()    |
−
|-
+
|     |            |           |           |           |           |
−
| f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y
+
| f_14 | ((x)(y)) | ((x, y)) |   (y)    |   (x)    |    ()    |
−
|-
+
|     |            |            |            |            |            |
−
| f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y
+
o------o------------o------------o------------o------------o------------o
−
|-
+
|     |            |            |           |           |           |
−
| f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x &rarr; y
+
| f_15 |   (())   |     ()    |     ()    |     ()    |    ()    |
−
|-
+
|     |            |            |            |            |            |
−
| f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x
+
o------o------------o------------o------------o------------o------------o
−
|-
+
</pre>
−
| 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>
+
<pre>
−
 
+
Table A5. Ef Expanded Over Ordinary Features {x, y}
−
===Wiki TeX Tables===
+
o------o------------o------------o------------o------------o------------o
−
 
+
|      |            |            |            |            |            |
−
<br>
+
|     |    f      |  Ef | xy  | Ef | x(y)  | Ef | (x)y  | Ef | (x)(y)|
−
 
+
|     |            |            |            |            |            |
−
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
+
o------o------------o------------o------------o------------o------------o
−
|+ <math>\text{Table 0.}~~\text{Propositional Forms on Two Variables}</math>
+
|     |            |            |            |            |            |
−
|- style="background:#f0f0ff"
+
| f_0  |    ()    |    ()    |     ()    |    ()    |     ()    |
−
| style="width:15%" |
+
|     |           |           |           |           |           |
−
<p><math>\mathcal{L}_1</math></p>
+
o------o------------o------------o------------o------------o------------o
−
<p><math>\text{Decimal}</math></p>
+
|     |            |            |            |            |           |
−
| style="width:15%" |
+
| f_1  |   (x)(y)  |   dx  dy  |   dx (dy) |  (dx) dy  |  (dx)(dy)  |
−
<p><math>\mathcal{L}_2</math></p>
+
|      |            |            |            |            |            |
−
<p><math>\text{Binary}</math></p>
+
| f_2  |  (x) y   |   dx (dy)  |  dx  dy  |  (dx)(dy)  |  (dx) dy  |
−
| style="width:15%" |  
+
|     |            |            |            |            |            |
−
<p><math>\mathcal{L}_3</math></p>
+
| f_4  |    x (y)  |  (dx) dy  |  (dx)(dy)  |  dx  dy  |   dx (dy) |
−
<p><math>\text{Vector}</math></p>
+
|     |            |            |            |            |            |
−
| style="width:15%" |
+
| f_8  |    x y   |  (dx)(dy)  |  (dx) dy  |  dx (dy)  |  dx  dy  |
−
<p><math>\mathcal{L}_4</math></p>
+
|     |            |           |            |            |           |
−
<p><math>\text{Cactus}</math></p>
+
o------o------------o------------o------------o------------o------------o
−
| style="width:25%" |
+
|     |            |            |            |            |            |
−
<p><math>\mathcal{L}_5</math></p>
+
| f_3  |  (x)      |  dx      |  dx      |  (dx)      |  (dx)      |
−
<p><math>\text{English}</math></p>
+
|     |            |            |            |            |            |
−
| style="width:15%" |
+
| f_12 |   x      | (dx)     |  (dx)      |  dx      |   dx      |
−
<p><math>\mathcal{L}_6</math></p>
+
|     |           |           |            |            |           |
−
<p><math>\text{Ordinary}</math></p>
+
o------o------------o------------o------------o------------o------------o
−
|- style="background:#f0f0ff"
+
|     |            |            |            |            |            |
−
| &nbsp;
+
| f_6  |  (x, y)  |  (dx, dy)  | ((dx, dy)) | ((dx, dy)) |  (dx, dy)  |
−
| align="right" | <math>x\colon\!</math>
+
|     |            |            |            |            |            |
−
| <math>1~1~0~0\!</math>
+
| f_9  |  ((x, y))  | ((dx, dy)) | (dx, dy)  | (dx, dy)  | ((dx, dy)) |
−
| &nbsp;
+
|     |           |           |           |           |           |
−
| &nbsp;
+
o------o------------o------------o------------o------------o------------o
−
| &nbsp;
+
|     |            |            |            |            |            |
−
|-
+
| f_5  |      (y)  |      dy  |      (dy)  |      dy  |      (dy)  |
−
|- style="background:#f0f0ff"
+
|     |            |            |            |            |            |
−
| &nbsp;
+
| f_10 |      y    |      (dy)  |      dy  |      (dy)  |      dy  |
−
| align="right" | <math>y\colon\!</math>
+
|     |           |           |           |           |           |
−
| <math>1~0~1~0\!</math>
+
o------o------------o------------o------------o------------o------------o
−
| &nbsp;
+
|     |            |            |            |            |            |
−
| &nbsp;
+
| f_7  |  (x  y)  | ((dx)(dy)) | ((dx) dy) |  (dx (dy)) |  (dx  dy) |
−
| &nbsp;
+
|     |            |            |            |            |            |
−
|-
+
| f_11 |  (x (y))  | ((dx) dy)  | ((dx)(dy)) |  (dx  dy)  |  (dx (dy)) |
−
| <math>f_{0}\!</math>
+
|      |            |            |            |            |            |
−
| <math>f_{0000}\!</math>
+
| f_13 |  ((x) y)  |  (dx (dy)) |  (dx  dy)  | ((dx)(dy)) | ((dx) dy)  |
−
| <math>0~0~0~0\!</math>
+
|      |            |            |            |            |            |
−
| <math>(~)\!</math>
+
| f_14 |  ((x)(y))  |  (dx  dy)  |  (dx (dy)) | ((dx) dy)  | ((dx)(dy)) |
−
| <math>\text{false}\!</math>
+
|      |            |            |            |            |            |
−
| <math>0\!</math>
+
o------o------------o------------o------------o------------o------------o
−
|-
+
|      |            |            |            |            |            |
−
| <math>f_{1}\!</math>
+
| f_15 |    (())    |    (())    |    (())    |    (())    |    (())    |
−
| <math>f_{0001}\!</math>
+
|      |            |            |            |            |            |
−
| <math>0~0~0~1\!</math>
+
o------o------------o------------o------------o------------o------------o
−
| <math>(x)(y)\!</math>
+
</pre>
−
| <math>\text{neither}~ x ~\text{nor}~ y\!</math>
+
 
−
| <math>\lnot x \land \lnot y\!</math>
+
<pre>
−
|-
+
Table A6.  Df Expanded Over Ordinary Features {x, y}
−
| <math>f_{2}\!</math>
+
o------o------------o------------o------------o------------o------------o
−
| <math>f_{0010}\!</math>
+
|      |            |            |            |            |            |
−
| <math>0~0~1~0\!</math>
+
|      |    f      |  Df | xy  | Df | x(y)  | Df | (x)y  | Df | (x)(y)|
−
| <math>(x)~y\!</math>
+
|      |            |            |            |            |            |
−
| <math>y ~\text{without}~ x\!</math>
+
o------o------------o------------o------------o------------o------------o
−
| <math>\lnot x \land y\!</math>
+
|      |            |            |            |            |            |
−
|-
+
| f_0  |    ()    |    ()    |    ()    |    ()    |    ()    |
−
| <math>f_{3}\!</math>
+
|      |            |            |            |            |            |
−
| <math>f_{0011}\!</math>
+
o------o------------o------------o------------o------------o------------o
−
| <math>0~0~1~1\!</math>
+
|      |            |            |            |            |            |
−
| <math>(x)\!</math>
+
| f_1  |  (x)(y)  |  dx  dy  |  dx (dy)  |  (dx) dy  | ((dx)(dy)) |
−
| <math>\text{not}~ x\!</math>
+
|      |            |            |            |            |            |
−
| <math>\lnot x\!</math>
+
| f_2  |  (x) y    |  dx (dy)  |  dx  dy  | ((dx)(dy)) |  (dx) dy  |
−
|-
+
|      |            |            |            |            |            |
−
| <math>f_{4}\!</math>
+
| f_4  |    x (y)  |  (dx) dy  | ((dx)(dy)) |  dx  dy  |  dx (dy)  |
−
| <math>f_{0100}\!</math>
+
|      |            |            |            |            |            |
−
| <math>0~1~0~0\!</math>
+
| f_8  |    x  y    | ((dx)(dy)) |  (dx) dy  |  dx (dy)  |  dx  dy  |
−
| <math>x~(y)\!</math>
+
|      |            |            |            |            |            |
−
| <math>x ~\text{without}~ y\!</math>
+
o------o------------o------------o------------o------------o------------o
−
| <math>x \land \lnot y\!</math>
+
|      |            |            |            |            |            |
−
|-
+
| f_3  |  (x)      |  dx      |  dx      |  dx      |  dx      |
−
| <math>f_{5}\!</math>
+
|      |            |            |            |            |            |
−
| <math>f_{0101}\!</math>
+
| f_12 |    x      |  dx      |  dx      |  dx      |  dx      |
−
| <math>0~1~0~1\!</math>
+
|      |            |            |            |            |            |
−
| <math>(y)\!</math>
+
o------o------------o------------o------------o------------o------------o
−
| <math>\text{not}~ y\!</math>
+
|      |            |            |            |            |            |
−
| <math>\lnot y\!</math>
+
| f_6  |  (x, y)  |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |
−
|-
+
|      |            |            |            |            |            |
−
| <math>f_{6}\!</math>
+
| f_9  |  ((x, y))  |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |
−
| <math>f_{0110}\!</math>
+
|      |            |            |            |            |            |
−
| <math>0~1~1~0\!</math>
+
o------o------------o------------o------------o------------o------------o
−
| <math>(x,~y)\!</math>
+
|      |            |            |            |            |            |
−
| <math>x ~\text{not equal to}~ y\!</math>
+
| f_5  |      (y)  |      dy  |      dy  |      dy  |      dy  |
−
| <math>x \ne y\!</math>
+
|      |            |            |            |            |            |
−
|-
+
| f_10 |      y    |      dy  |      dy  |      dy  |      dy  |
−
| <math>f_{7}\!</math>
+
|      |            |            |            |            |            |
−
| <math>f_{0111}\!</math>
+
o------o------------o------------o------------o------------o------------o
−
| <math>0~1~1~1\!</math>
+
|      |            |            |            |            |            |
−
| <math>(x~y)\!</math>
+
| f_7  |  (x  y)  | ((dx)(dy)) |  (dx) dy  |  dx (dy)  |  dx  dy  |
−
| <math>\text{not both}~ x ~\text{and}~ y\!</math>
+
|      |            |            |            |            |            |
−
| <math>\lnot x \lor \lnot y\!</math>
+
| f_11 |  (x (y))  |  (dx) dy  | ((dx)(dy)) |  dx  dy  |  dx (dy)  |
−
|-
+
|      |            |            |            |            |            |
−
| <math>f_{8}\!</math>
+
| f_13 |  ((x) y)  |  dx (dy)  |  dx  dy  | ((dx)(dy)) |  (dx) dy  |
−
| <math>f_{1000}\!</math>
+
|      |            |            |            |            |            |
−
| <math>1~0~0~0\!</math>
+
| f_14 |  ((x)(y))  |  dx  dy  |  dx (dy)  |  (dx) dy  | ((dx)(dy)) |
−
| <math>x~y\!</math>
+
|      |            |            |            |            |            |
−
| <math>x ~\text{and}~ y\!</math>
+
o------o------------o------------o------------o------------o------------o
−
| <math>x \land y\!</math>
+
|      |            |            |            |            |            |
−
|-
+
| f_15 |    (())    |    ()    |    ()    |    ()    |    ()    |
−
| <math>f_{9}\!</math>
+
|      |            |            |            |            |            |
−
| <math>f_{1001}\!</math>
+
o------o------------o------------o------------o------------o------------o
−
| <math>1~0~0~1\!</math>
+
</pre>
−
| <math>((x,~y))\!</math>
+
 
−
| <math>x ~\text{equal to}~ y\!</math>
+
<pre>
−
| <math>x = y\!</math>
+
o----------o----------o----------o----------o----------o
−
|-
+
|          %          |          |          |          |
−
| <math>f_{10}\!</math>
+
|    ·    %  T_00  |  T_01  |  T_10  |  T_11  |
−
| <math>f_{1010}\!</math>
+
|          %          |          |          |          |
−
| <math>1~0~1~0\!</math>
+
o==========o==========o==========o==========o==========o
−
| <math>y\!</math>
+
|          %          |          |          |          |
−
| <math>y\!</math>
+
|  T_00  %  T_00  |  T_01  |  T_10  |  T_11  |
−
| <math>y\!</math>
+
|          %          |          |          |          |
−
|-
+
o----------o----------o----------o----------o----------o
−
| <math>f_{11}\!</math>
+
|          %          |          |          |          |
−
| <math>f_{1011}\!</math>
+
|  T_01  %  T_01  |  T_00  |  T_11  |  T_10  |
−
| <math>1~0~1~1\!</math>
+
|          %          |          |          |          |
−
| <math>(x~(y))\!</math>
+
o----------o----------o----------o----------o----------o
−
| <math>\text{not}~ x ~\text{without}~ y\!</math>
+
|          %          |          |          |          |
−
| <math>x \Rightarrow y\!</math>
+
|  T_10  %  T_10  |  T_11  |  T_00  |  T_01  |
−
|-
+
|          %          |          |          |          |
−
| <math>f_{12}\!</math>
+
o----------o----------o----------o----------o----------o
−
| <math>f_{1100}\!</math>
+
|          %          |          |          |          |
−
| <math>1~1~0~0\!</math>
+
|  T_11  %  T_11  |  T_10  |  T_01  |  T_00  |
−
| <math>x\!</math>
+
|          %          |          |          |          |
−
| <math>x\!</math>
+
o----------o----------o----------o----------o----------o
−
| <math>x\!</math>
+
</pre>
−
|-
+
 
−
| <math>f_{13}\!</math>
+
<pre>
−
| <math>f_{1101}\!</math>
+
o---------o---------o---------o---------o---------o
−
| <math>1~1~0~1\!</math>
+
|        %        |        |        |        |
−
| <math>((x)~y)\!</math>
+
|    ·    %    e    |    f    |    g    |    h    |
−
| <math>\text{not}~ y ~\text{without}~ x\!</math>
+
|        %        |        |        |        |
−
| <math>x \Leftarrow y\!</math>
+
o=========o=========o=========o=========o=========o
−
|-
+
|        %        |        |        |        |
−
| <math>f_{14}\!</math>
+
|    e    %    e    |    f    |    g    |    h    |
−
| <math>f_{1110}\!</math>
+
|        %        |        |        |        |
−
| <math>1~1~1~0\!</math>
+
o---------o---------o---------o---------o---------o
−
| <math>((x)(y))\!</math>
+
|        %        |        |        |        |
−
| <math>x ~\text{or}~ y\!</math>
+
|    f    %    f    |    e    |    h    |    g    |
−
| <math>x \lor y\!</math>
+
|        %        |        |        |        |
−
|-
+
o---------o---------o---------o---------o---------o
−
| <math>f_{15}\!</math>
+
|        %        |        |        |        |
−
| <math>f_{1111}\!</math>
+
|    g    %    g    |    h    |    e    |    f    |
−
| <math>1~1~1~1\!</math>
+
|        %        |        |        |        |
−
| <math>((~))\!</math>
+
o---------o---------o---------o---------o---------o
−
| <math>\text{true}\!</math>
+
|        %        |        |        |        |
−
| <math>1\!</math>
+
|    h    %    h    |    g    |    f    |    e    |
−
|}
+
|        %        |        |        |        |
 +
o---------o---------o---------o---------o---------o
 +
</pre>
 +
 
 +
<pre>
 +
Permutation Substitutions in Sym {A, B, C}
 +
o---------o---------o---------o---------o---------o---------o
 +
|        |        |        |        |        |        |
 +
|    e    |    f    |    g    |    h    |    i    |    j    |
 +
|        |        |        |        |        |        |
 +
o=========o=========o=========o=========o=========o=========o
 +
|        |        |        |        |        |        |
 +
|  A B C  |  A B C  |  A B C  |  A B C  |  A B C  |  A B C  |
 +
|        |        |        |        |        |        |
 +
|  | | |  |  | | |  |  | | |  |  | | |  |  | | |  |  | | |  |
 +
|  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |
 +
|        |        |        |        |        |        |
 +
|  A B C  |  C A B  |  B C A  |  A C B  |  C B A  |  B A C  |
 +
|        |        |        |        |        |        |
 +
o---------o---------o---------o---------o---------o---------o
 +
</pre>
 +
 
 +
<pre>
 +
Matrix Representations of Permutations in Sym(3)
 +
o---------o---------o---------o---------o---------o---------o
 +
|        |        |        |        |        |        |
 +
|    e    |    f    |    g    |    h    |    i    |    j    |
 +
|        |        |        |        |        |        |
 +
o=========o=========o=========o=========o=========o=========o
 +
|        |        |        |        |        |        |
 +
|  1 0 0  |  0 0 1  |  0 1 0  |  1 0 0  |  0 0 1  |  0 1 0  |
 +
|  0 1 0  |  1 0 0  |  0 0 1  |  0 0 1  |  0 1 0  |  1 0 0  |
 +
|  0 0 1  |  0 1 0  |  1 0 0  |  0 1 0  |  1 0 0  |  0 0 1  |
 +
|        |        |        |        |        |        |
 +
o---------o---------o---------o---------o---------o---------o
 +
</pre>
  
−
<br>
+
<pre>
 +
Symmetric Group S_3
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                        ^                        |
 +
|                    e / \ e                    |
 +
|                      /  \                      |
 +
|                    /  e  \                    |
 +
|                  f / \  / \ f                  |
 +
|                  /  \ /  \                  |
 +
|                  /  f  \  f  \                  |
 +
|              g / \  / \  / \ g              |
 +
|                /  \ /  \ /  \                |
 +
|              /  g  \  g  \  g  \              |
 +
|            h / \  / \  / \  / \ h            |
 +
|            /  \ /  \ /  \ /  \            |
 +
|            /  h  \  e  \  e  \  h  \            |
 +
|        i / \  / \  / \  / \  / \ i        |
 +
|          /  \ /  \ /  \ /  \ /  \          |
 +
|        /  i  \  i  \  f  \  j  \  i  \        |
 +
|      j / \  / \  / \  / \  / \  / \ j      |
 +
|      /  \ /  \ /  \ /  \ /  \ /  \      |
 +
|      (  j  \  j  \  j  \  i  \  h  \  j  )      |
 +
|      \  / \  / \  / \  / \  / \  /      |
 +
|        \ /  \ /  \ /  \ /  \ /  \ /        |
 +
|        \  h  \  h  \  e  \  j  \  i  /        |
 +
|          \  / \  / \  / \  / \  /          |
 +
|          \ /  \ /  \ /  \ /  \ /          |
 +
|            \  i  \  g  \  f  \  h  /            |
 +
|            \  / \  / \  / \  /            |
 +
|              \ /  \ /  \ /  \ /              |
 +
|              \  f  \  e  \  g  /              |
 +
|                \  / \  / \  /                |
 +
|                \ /  \ /  \ /                |
 +
|                  \  g  \  f  /                  |
 +
|                  \  / \  /                  |
 +
|                    \ /  \ /                    |
 +
|                    \  e  /                    |
 +
|                      \  /                      |
 +
|                      \ /                      |
 +
|                        v                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
</pre>
  
−
==Inquiry Driven Systems==
+
===Wiki Tables : New Versions===
  
−
===Table 1.  Sign Relation of Interpreter ''A''===
+
====Propositional Forms on Two Variables====
  
−
<pre>
+
<br>
−
Table 1.  Sign Relation of Interpreter A
 
−
o---------------o---------------o---------------o
 
−
| Object        | Sign          | Interpretant  |
 
−
o---------------o---------------o---------------o
 
−
| A            | "A"          | "A"          |
 
−
| A            | "A"          | "i"          |
 
−
| A            | "i"          | "A"          |
 
−
| A            | "i"          | "i"          |
 
−
| B            | "B"          | "B"          |
 
−
| B            | "B"          | "u"          |
 
−
| B            | "u"          | "B"          |
 
−
| B            | "u"          | "u"          |
 
−
o---------------o---------------o---------------o
 
−
</pre>
 
  
−
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
+
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%"
−
|+ Table 1. Sign Relation of Interpreter ''A''
+
|+ '''Table A1.&nbsp; Propositional Forms on Two Variables'''
−
|- style="background:paleturquoise"
+
|- style="background:#f0f0ff"
−
! style="width:20%" | Object
+
! width="15%" | L<sub>1</sub>
−
! style="width:20%" | Sign
+
! width="15%" | L<sub>2</sub>
−
! style="width:20%" | Interpretant
+
! width="15%" | L<sub>3</sub>
 +
! width="15%" | L<sub>4</sub>
 +
! width="25%" | L<sub>5</sub>
 +
! width="15%" | L<sub>6</sub>
 +
|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | x :
 +
| 1 1 0 0
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | y :
 +
| 1 0 1 0
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 
|-
 
|-
−
| ''A'' || "A" || "A"
+
| 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
 
|-
 
|-
−
| ''A'' || "A" || "i"
+
| f<sub>4</sub>
 +
| f<sub>0100</sub>
 +
| 0 1 0 0
 +
| x (y)
 +
| x and not y
 +
| x &and; &not;y
 
|-
 
|-
−
| ''A'' || "i" || "A"
+
| f<sub>5</sub>
 +
| f<sub>0101</sub>
 +
| 0 1 0 1
 +
| (y)
 +
| not y
 +
| &not;y
 
|-
 
|-
−
| ''A'' || "i" || "i"
+
| f<sub>6</sub>
 +
| f<sub>0110</sub>
 +
| 0 1 1 0
 +
| (x, y)
 +
| x not equal to y
 +
| x &ne; y
 
|-
 
|-
−
| ''B'' || "B" || "B"
+
| 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
 
|-
 
|-
−
| ''B'' || "B" || "u"
+
| f<sub>8</sub>
 +
| f<sub>1000</sub>
 +
| 1 0 0 0
 +
| x&nbsp;y
 +
| x and y
 +
| x &and; y
 
|-
 
|-
−
| ''B'' || "u" || "B"
+
| f<sub>9</sub>
 +
| f<sub>1001</sub>
 +
| 1 0 0 1
 +
| ((x, y))
 +
| x equal to y
 +
| x = y
 
|-
 
|-
−
| ''B'' || "u" || "u"
+
| f<sub>10</sub>
−
|}
+
| f<sub>1010</sub>
−
<br>
+
| 1 0 1 0
−
 
+
| y
−
===Table 2.  Sign Relation of Interpreter ''B''===
+
| y
−
 
+
| y
−
<pre>
 
−
Table 2.  Sign Relation of Interpreter B
 
−
o---------------o---------------o---------------o
 
−
| Object        | Sign          | Interpretant  |
 
−
o---------------o---------------o---------------o
 
−
| A            | "A"          | "A"          |
 
−
| A            | "A"          | "u"          |
 
−
| A            | "u"          | "A"          |
 
−
| A            | "u"          | "u"          |
 
−
| B            | "B"          | "B"          |
 
−
| B            | "B"          | "i"          |
 
−
| B            | "i"          | "B"          |
 
−
| B            | "i"          | "i"          |
 
−
o---------------o---------------o---------------o
 
−
</pre>
 
−
 
 
−
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
 
−
|+ Table 2.  Sign Relation of Interpreter ''B''
 
−
|- style="background:paleturquoise"
 
−
! style="width:20%" | Object
 
−
! style="width:20%" | Sign
 
−
! style="width:20%" | Interpretant
 
 
|-
 
|-
−
| ''A'' || "A" || "A"
+
| f<sub>11</sub>
 +
| f<sub>1011</sub>
 +
| 1 0 1 1
 +
| (x (y))
 +
| not x without y
 +
| x &rArr; y
 
|-
 
|-
−
| ''A'' || "A" || "u"
+
| f<sub>12</sub>
 +
| f<sub>1100</sub>
 +
| 1 1 0 0
 +
| x
 +
| x
 +
| x
 
|-
 
|-
−
| ''A'' || "u" || "A"
+
| f<sub>13</sub>
 +
| f<sub>1101</sub>
 +
| 1 1 0 1
 +
| ((x) y)
 +
| not y without x
 +
| x &lArr; y
 
|-
 
|-
−
| ''A'' || "u" || "u"
+
| f<sub>14</sub>
 +
| f<sub>1110</sub>
 +
| 1 1 1 0
 +
| ((x)(y))
 +
| x or y
 +
| x &or; y
 
|-
 
|-
−
| ''B'' || "B" || "B"
+
| f<sub>15</sub>
−
|-
+
| f<sub>1111</sub>
−
| ''B'' || "B" || "i"
+
| 1 1 1 1
 +
| ((&nbsp;))
 +
| true || 1
 +
|}
 +
 
 +
<br>
 +
 
 +
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%"
 +
|+ '''Table A2.&nbsp; Propositional Forms on Two Variables'''
 +
|- style="background:#f0f0ff"
 +
! width="15%" | L<sub>1</sub>
 +
! width="15%" | L<sub>2</sub>
 +
! width="15%" | L<sub>3</sub>
 +
! width="15%" | L<sub>4</sub>
 +
! width="25%" | L<sub>5</sub>
 +
! width="15%" | L<sub>6</sub>
 +
|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | x :
 +
| 1 1 0 0
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | y :
 +
| 1 0 1 0
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 
|-
 
|-
−
| ''B'' || "i" || "B"
+
| f<sub>0</sub>
 +
| f<sub>0000</sub>
 +
| 0 0 0 0
 +
| (&nbsp;)
 +
| false
 +
| 0
 
|-
 
|-
−
| ''B'' || "i" || "i"
+
|
 +
{| align="center"
 +
|
 +
<p>f<sub>1</sub></p>
 +
<p>f<sub>2</sub></p>
 +
<p>f<sub>4</sub></p>
 +
<p>f<sub>8</sub></p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p>f<sub>0001</sub></p>
 +
<p>f<sub>0010</sub></p>
 +
<p>f<sub>0100</sub></p>
 +
<p>f<sub>1000</sub></p>
 
|}
 
|}
−
<br>
 
−
 
−
===Table 3.  Semiotic Partition of Interpreter ''A''===
 
−
 
−
<pre>
 
−
Table 3.  A's Semiotic Partition
 
−
o-------------------------------o
 
−
|      "A"            "i"      |
 
−
o-------------------------------o
 
−
|      "u"            "B"      |
 
−
o-------------------------------o
 
−
</pre>
 
−
 
−
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
 
−
|+ Table 3.  Semiotic Partition of Interpreter ''A''
 
 
|
 
|
−
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+
{| align="center"
−
| width="50%" | "A"
+
|
−
| width="50%" | "i"
+
<p>0 0 0 1</p>
 +
<p>0 0 1 0</p>
 +
<p>0 1 0 0</p>
 +
<p>1 0 0 0</p>
 
|}
 
|}
−
|-
 
 
|
 
|
−
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
+
{| align="center"
−
| width="50%" | "u"
+
|
−
| width="50%" | "B"
+
<p>(x)(y)</p>
 +
<p>(x) y </p>
 +
<p> x (y)</p>
 +
<p> x  y </p>
 
|}
 
|}
 +
|
 +
{| align="center"
 +
|
 +
<p>neither x nor y</p>
 +
<p>not x but y</p>
 +
<p>x but not y</p>
 +
<p>x and y</p>
 
|}
 
|}
−
<br>
 
−
 
−
===Table 4.  Semiotic Partition of Interpreter ''B''===
 
−
 
−
<pre>
 
−
Table 4.  B's Semiotic Partition
 
−
o---------------o---------------o
 
−
|      "A"      |      "i"      |
 
−
|              |              |
 
−
|      "u"      |      "B"      |
 
−
o---------------o---------------o
 
−
</pre>
 
−
 
−
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
 
−
|+ Table 4.  Semiotic Partition of Interpreter ''B''
 
 
|
 
|
−
{| align="center" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:50%"
+
{| align="center"
−
| "A"
+
|
 +
<p>&not;x &and; &not;y</p>
 +
<p>&not;x &and; y</p>
 +
<p>x &and; &not;y</p>
 +
<p>x &and; y</p>
 +
|}
 
|-
 
|-
−
| "u"
+
|
 +
{| align="center"
 +
|
 +
<p>f<sub>3</sub></p>
 +
<p>f<sub>12</sub></p>
 
|}
 
|}
 
|
 
|
−
{| align="center" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:50%"
+
{| align="center"
−
| "i"
+
|
 +
<p>f<sub>0011</sub></p>
 +
<p>f<sub>1100</sub></p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p>0 0 1 1</p>
 +
<p>1 1 0 0</p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p>(x)</p>
 +
<p> x </p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p>not x</p>
 +
<p>x</p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p>&not;x</p>
 +
<p>x</p>
 +
|}
 
|-
 
|-
−
| "B"
+
|
 +
{| align="center"
 +
|
 +
<p>f<sub>6</sub></p>
 +
<p>f<sub>9</sub></p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p>f<sub>0110</sub></p>
 +
<p>f<sub>1001</sub></p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p>0 1 1 0</p>
 +
<p>1 0 0 1</p>
 
|}
 
|}
 +
|
 +
{| align="center"
 +
|
 +
<p> (x, y) </p>
 +
<p>((x, y))</p>
 
|}
 
|}
−
<br>
+
|
−
 
+
{| align="center"
−
===Table 5.  Alignments of Capacities===
+
|
−
 
+
<p>x not equal to y</p>
−
<pre>
+
<p>x equal to y</p>
−
Table 5.  Alignments of Capacities
+
|}
−
o-------------------o-----------------------------o
+
|
−
|     Formal      |         Formative          |
+
{| align="center"
−
o-------------------o-----------------------------o
+
|
−
|     Objective    |       Instrumental        |
+
<p>x &ne; y</p>
−
|     Passive      |          Active            |
+
<p>x = y</p>
−
o-------------------o--------------o--------------o
+
|}
−
|     Afforded      | Possessed  | Exercised  |
+
|-
−
o-------------------o--------------o--------------o
+
|
−
</pre>
+
{| align="center"
−
 
+
|
−
===Table 6.  Alignments of Capacities in Aristotle===
+
<p>f<sub>5</sub></p>
−
 
+
<p>f<sub>10</sub></p>
−
<pre>
+
|}
−
Table 6.  Alignments of Capacities in Aristotle
+
|
−
o-------------------o-----------------------------o
+
{| align="center"
−
|     Matter      |           Form            |
+
|
−
o-------------------o-----------------------------o
+
<p>f<sub>0101</sub></p>
−
|   Potentiality    |         Actuality          |
+
<p>f<sub>1010</sub></p>
−
|   Receptivity    | Possession  |   Exercise  |
+
|}
−
|       Life        |   Sleep    |   Waking    |
+
|
−
|       Wax        |         Impression          |
+
{| align="center"
−
|       Axe        |   Edge      |   Cutting    |
+
|
−
|       Eye        |   Vision    |   Seeing    |
+
<p>0 1 0 1</p>
−
|       Body        |           Soul            |
+
<p>1 0 1 0</p>
−
o-------------------o-----------------------------o
+
|}
−
|       Ship?      |           Sailor?          |
+
|
−
o-------------------o-----------------------------o
+
{| align="center"
−
</pre>
+
|
 +
<p>(y)</p>
 +
<p> y </p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p>not y</p>
 +
<p>y</p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p>&not;y</p>
 +
<p>y</p>
 +
|}
 +
|-
 +
|
 +
{| align="center"
 +
|
 +
<p>f<sub>7</sub></p>
 +
<p>f<sub>11</sub></p>
 +
<p>f<sub>13</sub></p>
 +
<p>f<sub>14</sub></p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p>f<sub>0111</sub></p>
 +
<p>f<sub>1011</sub></p>
 +
<p>f<sub>1101</sub></p>
 +
<p>f<sub>1110</sub></p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p>0 1 1 1</p>
 +
<p>1 0 1 1</p>
 +
<p>1 1 0 1</p>
 +
<p>1 1 1 0</p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p>(x y)</p>
 +
<p>(x (y))</p>
 +
<p>((x) y)</p>
 +
<p>((x)(y))</p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p>not both x and y</p>
 +
<p>not x without y</p>
 +
<p>not y without x</p>
 +
<p>x or y</p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p>&not;x &or; &not;y</p>
 +
<p>x &rArr; y</p>
 +
<p>x &lArr; y</p>
 +
<p>x &or; y</p>
 +
|}
 +
|-
 +
| f<sub>15</sub>
 +
| f<sub>1111</sub>
 +
| 1 1 1 1
 +
| ((&nbsp;))
 +
| true
 +
| 1
 +
|}
 +
 
 +
<br>
  
−
===Table 7.  Synthesis of Alignments===
+
====Differential Propositions====
  
−
<pre>
+
<br>
−
Table 7.  Synthesis of Alignments
 
−
o-------------------o-----------------------------o
 
−
|      Formal      |          Formative          |
 
−
o-------------------o-----------------------------o
 
−
|    Objective    |        Instrumental        |
 
−
|      Passive      |          Active            |
 
−
|    Afforded      |  Possessed  |  Exercised  |
 
−
|      To Hold      |  To Have    |    To Use    |
 
−
|    Receptivity    |  Possession  |  Exercise  |
 
−
|  Potentiality    |          Actuality          |
 
−
|      Matter      |            Form            |
 
−
o-------------------o-----------------------------o
 
−
</pre>
 
  
−
===Table 8.  Boolean Product===
+
{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%"
−
 
+
|+ '''Table 14.&nbsp; Differential Propositions'''
−
<pre>
+
|- style="background:#f0f0ff"
−
Table 8.  Boolean Product
+
| &nbsp;
−
o---------o---------o---------o
+
| align="right" | A :
−
|  %*%  %  %0%  |  %1%  |
+
| 1 1 0 0
−
o=========o=========o=========o
+
| &nbsp;
−
|  %0%  %  %0%  |  %0%  |
+
| &nbsp;
−
o---------o---------o---------o
+
| &nbsp;
−
|  %1%  %  %0%  |  %1%  |
+
|- style="background:#f0f0ff"
−
o---------o---------o---------o
+
| &nbsp;
−
</pre>
+
| align="right" | dA :
−
 
+
| 1 0 1 0
−
===Table 9.  Boolean Sum===
+
| &nbsp;
−
 
+
| &nbsp;
−
<pre>
+
| &nbsp;
−
Table 9.  Boolean Sum
+
|-
−
o---------o---------o---------o
+
| f<sub>0</sub>
−
|  %+%  %  %0%  |  %1%  |
+
| g<sub>0</sub>
−
o=========o=========o=========o
+
| 0 0 0 0
−
|  %0%  %  %0%  |  %1%  |
+
| (&nbsp;)
−
o---------o---------o---------o
+
| False
−
|  %1%  %  %1%  |  %0%  |
+
| 0
−
o---------o---------o---------o
+
|-
−
</pre>
+
|
−
 
+
{|
−
==Logical Tables==
 
−
 
 
−
===Table Templates===
 
−
 
 
−
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
 
−
|+ Table 1.  Two Variable Template
 
−
|- style="background:paleturquoise"
 
 
|
 
|
−
{| align="right" style="background:paleturquoise; text-align:right"
+
&nbsp;<br>
−
| u :
+
&nbsp;<br>
−
|-
+
&nbsp;<br>
−
| v :
+
&nbsp;
 
|}
 
|}
 
|
 
|
−
{| style="background:paleturquoise"
+
{|
−
| 1 1 0 0
 
−
|-
 
−
| 1 0 1 0
 
−
|}
 
 
|
 
|
−
{| style="background:paleturquoise"
+
g<sub>1</sub><br>
−
| f
+
g<sub>2</sub><br>
−
|-
+
g<sub>4</sub><br>
−
| &nbsp;
+
g<sub>8</sub>
 
|}
 
|}
 
|
 
|
−
{| style="background:paleturquoise"
+
{|
−
| f
+
|
−
|-
+
0 0 0 1<br>
−
| &nbsp;
+
0 0 1 0<br>
 +
0 1 0 0<br>
 +
1 0 0 0
 +
|}
 +
|
 +
{|
 +
|
 +
(A)(dA)<br>
 +
(A) dA <br>
 +
A (dA)<br>
 +
A dA
 +
|}
 +
|
 +
{|
 +
|
 +
Neither A nor dA<br>
 +
Not A but dA<br>
 +
A but not dA<br>
 +
A and dA
 
|}
 
|}
 
|
 
|
−
{| style="background:paleturquoise"
+
{|
−
| f
+
|
−
|-
+
&not;A &and; &not;dA<br>
−
| &nbsp;
+
&not;A &and; dA<br>
 +
A &and; &not;dA<br>
 +
A &and; dA
 
|}
 
|}
 
|-
 
|-
 
|
 
|
−
{| cellpadding="2" style="background:lightcyan"
+
{|
−
| f<sub>0</sub>
+
|
−
|-
+
f<sub>1</sub><br>
−
| f<sub>1</sub>
+
f<sub>2</sub>
−
|-
 
−
| f<sub>2</sub>
 
−
|-
 
−
| f<sub>3</sub>
 
−
|-
 
−
| f<sub>4</sub>
 
−
|-
 
−
| f<sub>5</sub>
 
−
|-
 
−
| f<sub>6</sub>
 
−
|-
 
−
| f<sub>7</sub>
 
 
|}
 
|}
 
|
 
|
−
{| cellpadding="2" style="background:lightcyan"
+
{|
−
| 0 0 0 0
+
|
−
|-
+
g<sub>3</sub><br>
−
| 0 0 0 1
+
g<sub>12</sub>
−
|-
+
|}
−
| 0 0 1 0
+
|
−
|-
+
{|
−
| 0 0 1 1
+
|
−
|-
+
0 0 1 1<br>
−
| 0 1 0 0
+
1 1 0 0
−
|-
 
−
| 0 1 0 1
 
−
|-
 
−
| 0 1 1 0
 
−
|-
 
−
| 0 1 1 1
 
 
|}
 
|}
 
|
 
|
−
{| cellpadding="2" style="background:lightcyan"
+
{|
−
| f<sub>0</sub>
+
|
−
|-
+
(A)<br>
−
| f<sub>1</sub>
+
A
−
|-
 
−
| f<sub>2</sub>
 
−
|-
 
−
| f<sub>3</sub>
 
−
|-
 
−
| f<sub>4</sub>
 
−
|-
 
−
| f<sub>5</sub>
 
−
|-
 
−
| f<sub>6</sub>
 
−
|-
 
−
| f<sub>7</sub>
 
 
|}
 
|}
 
|
 
|
−
{| cellpadding="2" style="background:lightcyan"
+
{|
−
| f<sub>0</sub>
+
|
−
|-
+
Not A<br>
−
| f<sub>1</sub>
+
A
 +
|}
 +
|
 +
{|
 +
|
 +
&not;A<br>
 +
A
 +
|}
 
|-
 
|-
−
| f<sub>2</sub>
+
|
−
|-
+
{|
−
| f<sub>3</sub>
+
|
−
|-
+
&nbsp;<br>
−
| f<sub>4</sub>
+
&nbsp;
−
|-
 
−
| f<sub>5</sub>
 
−
|-
 
−
| f<sub>6</sub>
 
−
|-
 
−
| f<sub>7</sub>
 
 
|}
 
|}
 
|
 
|
−
{| cellpadding="2" style="background:lightcyan"
+
{|
−
| f<sub>0</sub>
+
|
−
|-
+
g<sub>6</sub><br>
−
| f<sub>1</sub>
+
g<sub>9</sub>
−
|-
 
−
| f<sub>2</sub>
 
−
|-
 
−
| f<sub>3</sub>
 
−
|-
 
−
| f<sub>4</sub>
 
−
|-
 
−
| f<sub>5</sub>
 
−
|-
 
−
| f<sub>6</sub>
 
−
|-
 
−
| f<sub>7</sub>
 
 
|}
 
|}
−
|-
 
 
|
 
|
−
{| cellpadding="2" style="background:lightcyan"
+
{|
−
| f<sub>8</sub>
+
|
−
|-
+
0 1 1 0<br>
−
| f<sub>9</sub>
+
1 0 0 1
−
|-
+
|}
−
| f<sub>10</sub>
+
|
−
|-
+
{|
−
| f<sub>11</sub>
+
|
−
|-
+
(A, dA)<br>
−
| f<sub>12</sub>
+
((A, dA))
−
|-
 
−
| f<sub>13</sub>
 
−
|-
 
−
| f<sub>14</sub>
 
−
|-
 
−
| f<sub>15</sub>
 
 
|}
 
|}
 
|
 
|
−
{| cellpadding="2" style="background:lightcyan"
+
{|
−
| 1 0 0 0
+
|
−
|-
+
A not equal to dA<br>
−
| 1 0 0 1
+
A equal to dA
−
|-
 
−
| 1 0 1 0
 
−
|-
 
−
| 1 0 1 1
 
−
|-
 
−
| 1 1 0 0
 
−
|-
 
−
| 1 1 0 1
 
−
|-
 
−
| 1 1 1 0
 
−
|-
 
−
| 1 1 1 1
 
 
|}
 
|}
 
|
 
|
−
{| cellpadding="2" style="background:lightcyan"
+
{|
−
| f<sub>8</sub>
+
|
 +
A &ne; dA<br>
 +
A = dA
 +
|}
 
|-
 
|-
−
| f<sub>9</sub>
+
|
−
|-
+
{|
−
| f<sub>10</sub>
+
|
−
|-
+
&nbsp;<br>
−
| f<sub>11</sub>
+
&nbsp;
−
|-
+
|}
−
| f<sub>12</sub>
+
|
−
|-
+
{|
−
| f<sub>13</sub>
+
|
−
|-
+
g<sub>5</sub><br>
−
| f<sub>14</sub>
+
g<sub>10</sub>
−
|-
 
−
| f<sub>15</sub>
 
 
|}
 
|}
 
|
 
|
−
{| cellpadding="2" style="background:lightcyan"
+
{|
−
| f<sub>8</sub>
+
|
−
|-
+
0 1 0 1<br>
−
| f<sub>9</sub>
+
1 0 1 0
−
|-
 
−
| f<sub>10</sub>
 
−
|-
 
−
| f<sub>11</sub>
 
−
|-
 
−
| f<sub>12</sub>
 
−
|-
 
−
| f<sub>13</sub>
 
−
|-
 
−
| f<sub>14</sub>
 
−
|-
 
−
| f<sub>15</sub>
 
 
|}
 
|}
 
|
 
|
−
{| cellpadding="2" style="background:lightcyan"
+
{|
−
| f<sub>8</sub>
+
|
−
|-
+
(dA)<br>
−
| f<sub>9</sub>
+
dA
−
|-
 
−
| f<sub>10</sub>
 
−
|-
 
−
| f<sub>11</sub>
 
−
|-
 
−
| f<sub>12</sub>
 
−
|-
 
−
| f<sub>13</sub>
 
−
|-
 
−
| f<sub>14</sub>
 
−
|-
 
−
| f<sub>15</sub>
 
 
|}
 
|}
 +
|
 +
{|
 +
|
 +
Not dA<br>
 +
dA
 
|}
 
|}
−
<br>
 
−
 
−
<font face="courier new">
 
−
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
 
−
|+ Table 2.  Two Variable Template
 
−
|- style="background:paleturquoise"
 
 
|
 
|
−
{| align="right" style="background:paleturquoise; text-align:right"
+
{|
−
| u :
+
|
 +
&not;dA<br>
 +
dA
 +
|}
 
|-
 
|-
−
| v :
+
|
 +
{|
 +
|
 +
&nbsp;<br>
 +
&nbsp;<br>
 +
&nbsp;<br>
 +
&nbsp;
 
|}
 
|}
 
|
 
|
−
{| style="background:paleturquoise"
+
{|
−
| 1100
+
|
−
|-
+
g<sub>7</sub><br>
−
| 1010
+
g<sub>11</sub><br>
 +
g<sub>13</sub><br>
 +
g<sub>14</sub>
 
|}
 
|}
 
|
 
|
−
{| style="background:paleturquoise"
+
{|
−
| f
+
|
−
|-
+
0 1 1 1<br>
−
| &nbsp;
+
1 0 1 1<br>
 +
1 1 0 1<br>
 +
1 1 1 0
 
|}
 
|}
 
|
 
|
−
{| style="background:paleturquoise"
+
{|
−
| f
+
|
−
|-
+
(A dA)<br>
−
| &nbsp;
+
(A (dA))<br>
 +
((A) dA)<br>
 +
((A)(dA))
 
|}
 
|}
 
|
 
|
−
{| style="background:paleturquoise"
+
{|
−
| f
+
|
−
|-
+
Not both A and dA<br>
−
| &nbsp;
+
Not A without dA<br>
 +
Not dA without A<br>
 +
A or dA
 
|}
 
|}
−
|-
 
 
|
 
|
−
{| cellpadding="2" style="background:lightcyan"
+
{|
−
| f<sub>0</sub>
+
|
−
|-
+
&not;A &or; &not;dA<br>
−
| f<sub>1</sub>
+
A &rArr; dA<br>
−
|-
+
A &lArr; dA<br>
−
| f<sub>2</sub>
+
A &or; dA
 +
|}
 
|-
 
|-
 
| f<sub>3</sub>
 
| f<sub>3</sub>
−
|-
+
| g<sub>15</sub>
−
| f<sub>4</sub>
+
| 1 1 1 1
−
|-
+
| ((&nbsp;))
−
| f<sub>5</sub>
+
| True
−
|-
+
| 1
−
| f<sub>6</sub>
 
−
|-
 
−
| f<sub>7</sub>
 
 
|}
 
|}
−
|
+
 
−
{| cellpadding="2" style="background:lightcyan"
+
<br>
−
| 0000
+
 
−
|-
+
===Wiki Tables : Old Versions===
−
| 0001
+
 
 +
====Propositional Forms on Two Variables====
 +
 
 +
<br>
 +
 
 +
{| 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"
 +
! width="15%" | L<sub>1</sub>
 +
! width="15%" | L<sub>2</sub>
 +
! width="15%" | L<sub>3</sub>
 +
! width="15%" | L<sub>4</sub>
 +
! width="25%" | L<sub>5</sub>
 +
! 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;
 
|-
 
|-
−
| 0010
+
| f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || (&nbsp;) || false || 0
 
|-
 
|-
−
| 0011
+
| f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || &not;x &and; &not;y
 
|-
 
|-
−
| 0100
+
| f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || &not;x &and; y
 
|-
 
|-
−
| 0101
+
| f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || &not;x
 
|-
 
|-
−
| 0110
+
| f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x &and; &not;y
 
|-
 
|-
−
| 0111
+
| f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || &not;y
−
|}
 
−
|
 
−
{| cellpadding="2" style="background:lightcyan"
 
−
| ()
 
 
|-
 
|-
−
| &nbsp;(u)(v)&nbsp;
+
| f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x &ne; y
 
|-
 
|-
−
| &nbsp;(u)&nbsp;v&nbsp;&nbsp;
+
| 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
 
|-
 
|-
−
| &nbsp;(u)&nbsp;&nbsp;&nbsp;&nbsp;
+
| f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x&nbsp;y || x and y || x &and; y
 
|-
 
|-
−
| &nbsp;&nbsp;u&nbsp;(v)&nbsp;
+
| f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y
 
|-
 
|-
−
| &nbsp;&nbsp;&nbsp;&nbsp;(v)&nbsp;
+
| f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y
 
|-
 
|-
−
| &nbsp;(u,&nbsp;v)&nbsp;
+
| f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x &rarr; y
 
|-
 
|-
−
| &nbsp;(u&nbsp;&nbsp;v)&nbsp;
+
| f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x
−
|}
 
−
|
 
−
{| cellpadding="2" style="background:lightcyan"
 
−
| f<sub>0</sub>
 
 
|-
 
|-
−
| f<sub>1</sub>
+
| f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x &larr; y
 
|-
 
|-
−
| f<sub>2</sub>
+
| f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y  || x &or; y
 
|-
 
|-
−
| f<sub>3</sub>
+
| f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || ((&nbsp;)) || true || 1
−
|-
 
−
| f<sub>4</sub>
 
−
|-
 
−
| f<sub>5</sub>
 
−
|-
 
−
| f<sub>6</sub>
 
−
|-
 
−
| f<sub>7</sub>
 
 
|}
 
|}
−
|
+
 
−
{| cellpadding="2" style="background:lightcyan"
+
<br>
−
| f<sub>0</sub>
+
 
−
|-
+
====Differential Propositions====
−
| f<sub>1</sub>
+
 
 +
<br>
 +
 
 +
{| align="center" border="1" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"
 +
|+ '''Table 14.  Differential Propositions'''
 +
|- style="background:ghostwhite"
 +
| &nbsp;
 +
| align="right" | A :
 +
| 1 1 0 0  
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
|- style="background:ghostwhite"
 +
| &nbsp;
 +
| align="right" | dA :
 +
| 1 0 1 0
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 
|-
 
|-
−
| f<sub>2</sub>
+
| f<sub>0</sub>
 +
| g<sub>0</sub>
 +
| 0 0 0 0
 +
| (&nbsp;)
 +
| False
 +
| 0
 
|-
 
|-
−
| f<sub>3</sub>
+
|
−
|-
+
{|
−
| f<sub>4</sub>
+
|
−
|-
+
&nbsp;<br>
−
| f<sub>5</sub>
+
&nbsp;<br>
−
|-
+
&nbsp;<br>
−
| f<sub>6</sub>
+
&nbsp;
−
|-
+
|}
−
| f<sub>7</sub>
+
|
 +
{|
 +
|
 +
g<sub>1</sub><br>
 +
g<sub>2</sub><br>
 +
g<sub>4</sub><br>
 +
g<sub>8</sub>
 +
|}
 +
|
 +
{|
 +
|
 +
0 0 0 1<br>
 +
0 0 1 0<br>
 +
0 1 0 0<br>
 +
1 0 0 0
 
|}
 
|}
−
|-
 
 
|
 
|
−
{| cellpadding="2" style="background:lightcyan"
+
{|
−
| f<sub>8</sub>
+
|
−
|-
+
(A)(dA)<br>
−
| f<sub>9</sub>
+
(A) dA <br>
−
|-
+
A (dA)<br>
−
| f<sub>10</sub>
+
A dA
−
|-
+
|}
−
| f<sub>11</sub>
+
|
−
|-
+
{|
−
| f<sub>12</sub>
+
|
−
|-
+
Neither A nor dA<br>
−
| f<sub>13</sub>
+
Not A but dA<br>
−
|-
+
A but not dA<br>
−
| f<sub>14</sub>
+
A and dA
−
|-
 
−
| f<sub>15</sub>
 
 
|}
 
|}
 
|
 
|
−
{| cellpadding="2" style="background:lightcyan"
+
{|
−
| 1000
+
|
 +
&not;A &and; &not;dA<br>
 +
&not;A &and; dA<br>
 +
A &and; &not;dA<br>
 +
A &and; dA
 +
|}
 
|-
 
|-
−
| 1001
+
|
−
|-
+
{|
−
| 1010
+
|
−
|-
+
f<sub>1</sub><br>
−
| 1011
+
f<sub>2</sub>
−
|-
 
−
| 1100
 
−
|-
 
−
| 1101
 
−
|-
 
−
| 1110
 
−
|-
 
−
| 1111
 
 
|}
 
|}
 
|
 
|
−
{| cellpadding="2" style="background:lightcyan"
+
{|
−
| &nbsp;&nbsp;u&nbsp;&nbsp;v&nbsp;&nbsp;
+
|
−
|-
+
g<sub>3</sub><br>
−
| ((u,&nbsp;v))
+
g<sub>12</sub>
−
|-
+
|}
−
| &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;v&nbsp;&nbsp;
+
|
−
|-
+
{|
−
| &nbsp;(u&nbsp;(v))
+
|
−
|-
+
0 0 1 1<br>
−
| &nbsp;&nbsp;u&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+
1 1 0 0
−
|-
 
−
| ((u)&nbsp;v)&nbsp;
 
−
|-
 
−
| ((u)(v))
 
−
|-
 
−
| (())
 
 
|}
 
|}
 
|
 
|
−
{| cellpadding="2" style="background:lightcyan"
+
{|
−
| f<sub>8</sub>
+
|
−
|-
+
(A)<br>
−
| f<sub>9</sub>
+
A
−
|-
 
−
| f<sub>10</sub>
 
−
|-
 
−
| f<sub>11</sub>
 
−
|-
 
−
| f<sub>12</sub>
 
−
|-
 
−
| f<sub>13</sub>
 
−
|-
 
−
| f<sub>14</sub>
 
−
|-
 
−
| f<sub>15</sub>
 
 
|}
 
|}
 
|
 
|
−
{| cellpadding="2" style="background:lightcyan"
+
{|
−
| f<sub>8</sub>
+
|
−
|-
+
Not A<br>
−
| f<sub>9</sub>
+
A
 +
|}
 +
|
 +
{|
 +
|
 +
&not;A<br>
 +
A
 +
|}
 
|-
 
|-
−
| f<sub>10</sub>
+
|
 +
{|
 +
|
 +
&nbsp;<br>
 +
&nbsp;
 +
|}
 +
|
 +
{|
 +
|
 +
g<sub>6</sub><br>
 +
g<sub>9</sub>
 +
|}
 +
|
 +
{|
 +
|
 +
0 1 1 0<br>
 +
1 0 0 1
 +
|}
 +
|
 +
{|
 +
|
 +
(A, dA)<br>
 +
((A, dA))
 +
|}
 +
|
 +
{|
 +
|
 +
A not equal to dA<br>
 +
A equal to dA
 +
|}
 +
|
 +
{|
 +
|
 +
A &ne; dA<br>
 +
A = dA
 +
|}
 
|-
 
|-
−
| f<sub>11</sub>
+
|
−
|-
+
{|
−
| f<sub>12</sub>
+
|
−
|-
+
&nbsp;<br>
−
| f<sub>13</sub>
+
&nbsp;
−
|-
 
−
| f<sub>14</sub>
 
−
|-
 
−
| f<sub>15</sub>
 
 
|}
 
|}
 +
|
 +
{|
 +
|
 +
g<sub>5</sub><br>
 +
g<sub>10</sub>
 +
|}
 +
|
 +
{|
 +
|
 +
0 1 0 1<br>
 +
1 0 1 0
 +
|}
 +
|
 +
{|
 +
|
 +
(dA)<br>
 +
dA
 +
|}
 +
|
 +
{|
 +
|
 +
Not dA<br>
 +
dA
 +
|}
 +
|
 +
{|
 +
|
 +
&not;dA<br>
 +
dA
 
|}
 
|}
−
</font><br>
 
−
 
−
===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
+
|
−
|-
+
&nbsp;<br>
−
| ''F<sub>2</sub> || 1 0 ||  x  ||0||0||0||0||1||1||1||1||0||0||0||0||1||1||1||1
+
&nbsp;<br>
−
|-
+
&nbsp;<br>
−
| ''F<sub>3</sub> || 1 1 || 1  ||0||0||0||0||0||0||0||0||1||1||1||1||1||1||1||1
+
&nbsp;
 +
|}
 +
|
 +
{|
 +
|
 +
g<sub>7</sub><br>
 +
g<sub>11</sub><br>
 +
g<sub>13</sub><br>
 +
g<sub>14</sub>
 +
|}
 +
|
 +
{|
 +
|
 +
0 1 1 1<br>
 +
1 0 1 1<br>
 +
1 1 0 1<br>
 +
1 1 1 0
 +
|}
 +
|
 +
{|
 +
|
 +
(A dA)<br>
 +
(A (dA))<br>
 +
((A) dA)<br>
 +
((A)(dA))
 +
|}
 +
|
 +
{|
 +
|
 +
Not both A and dA<br>
 +
Not A without dA<br>
 +
Not dA without A<br>
 +
A or dA
 +
|}
 +
|
 +
{|
 +
|
 +
&not;A &or; &not;dA<br>
 +
A &rarr; dA<br>
 +
A &larr; dA<br>
 +
A &or; dA
 
|}
 
|}
−
<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;
+
| f<sub>3</sub>
−
|-
+
| g<sub>15</sub>
−
|''m''<sub>1</sub>||&nbsp;||just false||nothing exists||&nbsp;||&nbsp;||&nbsp;
+
| 1 1 1 1
−
|-
+
| ((&nbsp;))
−
|''m''<sub>2</sub>||&nbsp;||just not x||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| True
−
|-
+
| 1
−
|''m''<sub>3</sub>||&nbsp;||&nbsp;||nothing is x||&nbsp;||&nbsp;||&nbsp;
+
|}
−
|-
+
 
−
|''m''<sub>4</sub>||&nbsp;||just x||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
<br>
−
|-
+
 
−
|''m''<sub>5</sub>||&nbsp;||&nbsp;||everything is x||F is linear||&nbsp;||&nbsp;
+
===Wiki TeX Tables : PQ===
−
|-
+
 
−
|''m''<sub>6</sub>||&nbsp;||&nbsp;||&nbsp;||&nbsp;||F is not uniform||F is informed
+
<br>
−
|-
+
 
−
|''m''<sub>7</sub>||&nbsp;||not just true||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 +
|+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math>
 +
|- style="background:#f0f0ff"
 +
| width="15%" |
 +
<p><math>\mathcal{L}_1</math></p>
 +
<p><math>\text{Decimal}</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_2</math></p>
 +
<p><math>\text{Binary}</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_3</math></p>
 +
<p><math>\text{Vector}</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_4</math></p>
 +
<p><math>\text{Cactus}</math></p>
 +
| width="25%" |
 +
<p><math>\mathcal{L}_5</math></p>
 +
<p><math>\text{English}</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_6</math></p>
 +
<p><math>\text{Ordinary}</math></p>
 +
|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | <math>p\colon\!</math>
 +
| <math>1~1~0~0\!</math>
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | <math>q\colon\!</math>
 +
| <math>1~0~1~0\!</math>
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 
|-
 
|-
−
|''m''<sub>8</sub>||&nbsp;||just true||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
|
−
|-
+
<math>\begin{matrix}
−
|''m''<sub>9</sub>||&nbsp;||&nbsp;||&nbsp;||&nbsp;||F is uniform||F is not informed
+
f_0
−
|-
+
\\[4pt]
−
|''m''<sub>10</sub>||&nbsp;||&nbsp;||something is not x||F is not linear||&nbsp;||&nbsp;
+
f_1
−
|-
+
\\[4pt]
−
|''m''<sub>11</sub>||&nbsp;||not just x||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
f_2
 +
\\[4pt]
 +
f_3
 +
\\[4pt]
 +
f_4
 +
\\[4pt]
 +
f_5
 +
\\[4pt]
 +
f_6
 +
\\[4pt]
 +
f_7
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0000}
 +
\\[4pt]
 +
f_{0001}
 +
\\[4pt]
 +
f_{0010}
 +
\\[4pt]
 +
f_{0011}
 +
\\[4pt]
 +
f_{0100}
 +
\\[4pt]
 +
f_{0101}
 +
\\[4pt]
 +
f_{0110}
 +
\\[4pt]
 +
f_{0111}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0~0~0~0
 +
\\[4pt]
 +
0~0~0~1
 +
\\[4pt]
 +
0~0~1~0
 +
\\[4pt]
 +
0~0~1~1
 +
\\[4pt]
 +
0~1~0~0
 +
\\[4pt]
 +
0~1~0~1
 +
\\[4pt]
 +
0~1~1~0
 +
\\[4pt]
 +
0~1~1~1
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
(p)~~~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
~~~(q)
 +
\\[4pt]
 +
(p,~q)
 +
\\[4pt]
 +
(p~~q)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{false}
 +
\\[4pt]
 +
\text{neither}~ p ~\text{nor}~ q
 +
\\[4pt]
 +
q ~\text{without}~ p
 +
\\[4pt]
 +
\text{not}~ p
 +
\\[4pt]
 +
p ~\text{without}~ q
 +
\\[4pt]
 +
\text{not}~ q
 +
\\[4pt]
 +
p ~\text{not equal to}~ q
 +
\\[4pt]
 +
\text{not both}~ p ~\text{and}~ q
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0
 +
\\[4pt]
 +
\lnot p \land \lnot q
 +
\\[4pt]
 +
\lnot p \land q
 +
\\[4pt]
 +
\lnot p
 +
\\[4pt]
 +
p \land \lnot q
 +
\\[4pt]
 +
\lnot q
 +
\\[4pt]
 +
p \ne q
 +
\\[4pt]
 +
\lnot p \lor \lnot q
 +
\end{matrix}</math>
 
|-
 
|-
−
|''m''<sub>12</sub>||&nbsp;||&nbsp;||something is x||&nbsp;||&nbsp;||&nbsp;
+
|
−
|-
+
<math>\begin{matrix}
−
|''m''<sub>13</sub>||&nbsp;||not just not x||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
f_8
−
|-
+
\\[4pt]
−
|''m''<sub>14</sub>||&nbsp;||not just false||something exists||&nbsp;||&nbsp;||&nbsp;
+
f_9
−
|-
+
\\[4pt]
−
|''m''<sub>15</sub>||anything happens||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
f_{10}
 +
\\[4pt]
 +
f_{11}
 +
\\[4pt]
 +
f_{12}
 +
\\[4pt]
 +
f_{13}
 +
\\[4pt]
 +
f_{14}
 +
\\[4pt]
 +
f_{15}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{1000}
 +
\\[4pt]
 +
f_{1001}
 +
\\[4pt]
 +
f_{1010}
 +
\\[4pt]
 +
f_{1011}
 +
\\[4pt]
 +
f_{1100}
 +
\\[4pt]
 +
f_{1101}
 +
\\[4pt]
 +
f_{1110}
 +
\\[4pt]
 +
f_{1111}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
1~0~0~0
 +
\\[4pt]
 +
1~0~0~1
 +
\\[4pt]
 +
1~0~1~0
 +
\\[4pt]
 +
1~0~1~1
 +
\\[4pt]
 +
1~1~0~0
 +
\\[4pt]
 +
1~1~0~1
 +
\\[4pt]
 +
1~1~1~0
 +
\\[4pt]
 +
1~1~1~1
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~~p~~q~~
 +
\\[4pt]
 +
((p,~q))
 +
\\[4pt]
 +
~~~~~q~~
 +
\\[4pt]
 +
~(p~(q))
 +
\\[4pt]
 +
~~p~~~~~
 +
\\[4pt]
 +
((p)~q)~
 +
\\[4pt]
 +
((p)(q))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
p ~\text{and}~ q
 +
\\[4pt]
 +
p ~\text{equal to}~ q
 +
\\[4pt]
 +
q
 +
\\[4pt]
 +
\text{not}~ p ~\text{without}~ q
 +
\\[4pt]
 +
p
 +
\\[4pt]
 +
\text{not}~ q ~\text{without}~ p
 +
\\[4pt]
 +
p ~\text{or}~ q
 +
\\[4pt]
 +
\text{true}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
p \land q
 +
\\[4pt]
 +
p = q
 +
\\[4pt]
 +
q
 +
\\[4pt]
 +
p \Rightarrow q
 +
\\[4pt]
 +
p
 +
\\[4pt]
 +
p \Leftarrow q
 +
\\[4pt]
 +
p \lor q
 +
\\[4pt]
 +
1
 +
\end{matrix}</math>
 
|}
 
|}
 +
 
<br>
 
<br>
  
−
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
−
|+ '''Table 9. Higher Order Propositions (n = 2)'''
+
|+ <math>\text{Table A2.}~~\text{Propositional Forms on Two Variables}</math>
−
|- style="background:paleturquoise"
+
|- style="background:#f0f0ff"
−
| align=right | ''x'' : || 1100 || ''f''
+
| width="15%" |
−
|''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m''
+
<p><math>\mathcal{L}_1</math></p>
−
|''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m''
+
<p><math>\text{Decimal}</math></p>
−
|''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m''
+
| width="15%" |
−
|- style="background:paleturquoise"
+
<p><math>\mathcal{L}_2</math></p>
−
| align=right | ''y'' : || 1010 || &nbsp;
+
<p><math>\text{Binary}</math></p>
−
|0||1||2||3||4||5||6||7||8||9||10||11||12
+
| width="15%" |
−
|13||14||15||16||17||18||19||20||21||22||23
+
<p><math>\mathcal{L}_3</math></p>
 +
<p><math>\text{Vector}</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_4</math></p>
 +
<p><math>\text{Cactus}</math></p>
 +
| width="25%" |
 +
<p><math>\mathcal{L}_5</math></p>
 +
<p><math>\text{English}</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_6</math></p>
 +
<p><math>\text{Ordinary}</math></p>
 +
|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | <math>p\colon\!</math>
 +
| <math>1~1~0~0\!</math>
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | <math>q\colon\!</math>
 +
| <math>1~0~1~0\!</math>
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 
|-
 
|-
−
| ''f<sub>0</sub> || 0000 || ( )
+
| <math>f_0\!</math>
−
| 0   || 1    || 0   || 1    || 0   || 1    || 0   || 1
+
| <math>f_{0000}\!</math>
−
| 0    || 1    || 0    || 1    || 0    || 1    || 0    || 1
+
| <math>0~0~0~0</math>
−
| 0   || 1    || 0    || 1    || 0    || 1    || 0    || 1
+
| <math>(~)</math>
 +
| <math>\text{false}\!</math>
 +
| <math>0\!</math>
 
|-
 
|-
−
| ''f<sub>1</sub> || 0001 || (x)(y)
+
|
−
|&nbsp;||&nbsp;|| 1    || 1    || 0   || 0   || 1    || 1
+
<math>\begin{matrix}
−
| 0   || 0   || 1   || 1    || 0   || 0   || 1   || 1
+
f_1
−
| 0   || 0   || 1   || 1    || 0   || 0   || 1    || 1
+
\\[4pt]
 +
f_2
 +
\\[4pt]
 +
f_4
 +
\\[4pt]
 +
f_8
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0001}
 +
\\[4pt]
 +
f_{0010}
 +
\\[4pt]
 +
f_{0100}
 +
\\[4pt]
 +
f_{1000}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0~0~0~1
 +
\\[4pt]
 +
0~0~1~0
 +
\\[4pt]
 +
0~1~0~0
 +
\\[4pt]
 +
1~0~0~0
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
~p~~q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{neither}~ p ~\text{nor}~ q
 +
\\[4pt]
 +
q ~\text{without}~ p
 +
\\[4pt]
 +
p ~\text{without}~ q
 +
\\[4pt]
 +
p ~\text{and}~ q
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\lnot p \land \lnot q
 +
\\[4pt]
 +
\lnot p \land q
 +
\\[4pt]
 +
p \land \lnot q
 +
\\[4pt]
 +
p \land q
 +
\end{matrix}</math>
 
|-
 
|-
−
| ''f<sub>2</sub> || 0010 || (x) y
+
|
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1    || 1    || 1
+
<math>\begin{matrix}
−
| 0    || 0    || 0   || 0   || 1   || 1   || 1   || 1
+
f_3
−
| 0   || 0   || 0    || 0    || 1    || 1    || 1    || 1
+
\\[4pt]
 +
f_{12}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0011}
 +
\\[4pt]
 +
f_{1100}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0~0~1~1
 +
\\[4pt]
 +
1~1~0~0
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{not}~ p
 +
\\[4pt]
 +
p
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\lnot p
 +
\\[4pt]
 +
p
 +
\end{matrix}</math>
 
|-
 
|-
−
| ''f<sub>3</sub> || 0011 || (x)
+
|
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
<math>\begin{matrix}
−
| 1   || 1   || 1   || 1   || 1    || 1    || 1    || 1
+
f_6
−
| 0    || 0    || 0    || 0    || 0    || 0    || 0    || 0
+
\\[4pt]
−
|-
+
f_9
−
| ''f<sub>4</sub> || 0100 || x (y)
+
\end{matrix}</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
|
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
<math>\begin{matrix}
−
| 1    || 1    || 1    || 1    || 1    || 1    || 1    || 1
+
f_{0110}
 +
\\[4pt]
 +
f_{1001}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0~1~1~0
 +
\\[4pt]
 +
1~0~0~1
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
p ~\text{not equal to}~ q
 +
\\[4pt]
 +
p ~\text{equal to}~ q
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
p \ne q
 +
\\[4pt]
 +
p = q
 +
\end{matrix}</math>
 
|-
 
|-
−
| ''f<sub>5</sub> || 0101 || (y)
+
|
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
<math>\begin{matrix}
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
f_5
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
\\[4pt]
 +
f_{10}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0101}
 +
\\[4pt]
 +
f_{1010}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0~1~0~1
 +
\\[4pt]
 +
1~0~1~0
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{not}~ q
 +
\\[4pt]
 +
q
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\lnot q
 +
\\[4pt]
 +
q
 +
\end{matrix}</math>
 
|-
 
|-
−
| ''f<sub>6</sub> || 0110 || (x, y)
+
|
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
<math>\begin{matrix}
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
f_7
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
\\[4pt]
−
|-
+
f_{11}
−
| ''f<sub>7</sub> || 0111 || (x  y)
+
\\[4pt]
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
f_{13}
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
\\[4pt]
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
f_{14}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0111}
 +
\\[4pt]
 +
f_{1011}
 +
\\[4pt]
 +
f_{1101}
 +
\\[4pt]
 +
f_{1110}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0~1~1~1
 +
\\[4pt]
 +
1~0~1~1
 +
\\[4pt]
 +
1~1~0~1
 +
\\[4pt]
 +
1~1~1~0
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p~~q)~
 +
\\[4pt]
 +
~(p~(q))
 +
\\[4pt]
 +
((p)~q)~
 +
\\[4pt]
 +
((p)(q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{not both}~ p ~\text{and}~ q
 +
\\[4pt]
 +
\text{not}~ p ~\text{without}~ q
 +
\\[4pt]
 +
\text{not}~ q ~\text{without}~ p
 +
\\[4pt]
 +
p ~\text{or}~ q
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\lnot p \lor \lnot q
 +
\\[4pt]
 +
p \Rightarrow q
 +
\\[4pt]
 +
p \Leftarrow q
 +
\\[4pt]
 +
p \lor q
 +
\end{matrix}</math>
 
|-
 
|-
−
| ''f<sub>8</sub> || 1000 || x  y
+
| <math>f_{15}\!</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>f_{1111}\!</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>1~1~1~1</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>((~))</math>
 +
| <math>\text{true}\!</math>
 +
| <math>1\!</math>
 +
|}
 +
 
 +
<br>
 +
 
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 +
|+ <math>\text{Table A3.}~~\operatorname{E}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}p, \operatorname{d}q \}</math>
 +
|- style="background:#f0f0ff"
 +
| width="10%" | &nbsp;
 +
| width="18%" | <math>f\!</math>
 +
| width="18%" |  
 +
<p><math>\operatorname{T}_{11} f</math></p>
 +
<p><math>\operatorname{E}f|_{\operatorname{d}p~\operatorname{d}q}</math></p>
 +
| width="18%" |
 +
<p><math>\operatorname{T}_{10} f</math></p>
 +
<p><math>\operatorname{E}f|_{\operatorname{d}p(\operatorname{d}q)}</math></p>
 +
| width="18%" |
 +
<p><math>\operatorname{T}_{01} f</math></p>
 +
<p><math>\operatorname{E}f|_{(\operatorname{d}p)\operatorname{d}q}</math></p>
 +
| width="18%" |
 +
<p><math>\operatorname{T}_{00} f</math></p>
 +
<p><math>\operatorname{E}f|_{(\operatorname{d}p)(\operatorname{d}q)}</math></p>
 
|-
 
|-
−
| ''f<sub>9</sub> || 1001 || ((x, y))
+
| <math>f_0\!</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>(~)</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>(~)</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 
|-
 
|-
−
| ''f<sub>10</sub> || 1010 || y
+
|
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
<math>\begin{matrix}
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
f_1
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
\\[4pt]
−
|-
+
f_2
−
| ''f<sub>11</sub> || 1011 || (x (y))
+
\\[4pt]
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
f_4
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
\\[4pt]
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
f_8
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
~p~~q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~p~~q~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
(p)(q)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~p~(q)
 +
\\[4pt]
 +
~p~~q~
 +
\\[4pt]
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)~q~
 +
\\[4pt]
 +
(p)(q)
 +
\\[4pt]
 +
~p~~q~
 +
\\[4pt]
 +
~p~(q)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
~p~~q~
 +
\end{matrix}</math>
 
|-
 
|-
−
| ''f<sub>12</sub> || 1100 || x
+
|
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
<math>\begin{matrix}
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
f_3
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
\\[4pt]
 +
f_{12}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~p~
 +
\\[4pt]
 +
(p)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~p~
 +
\\[4pt]
 +
(p)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\end{matrix}</math>
 
|-
 
|-
−
| ''f<sub>13</sub> || 1101 || ((x) y)
+
|
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
<math>\begin{matrix}
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
f_6
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
\\[4pt]
−
|-
+
f_9
−
| ''f<sub>14</sub> || 1110 || ((x)(y))
+
\end{matrix}</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
|
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
<math>\begin{matrix}
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((p,~q))
 +
\\[4pt]
 +
~(p,~q)~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((p,~q))
 +
\\[4pt]
 +
~(p,~q)~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}</math>
 
|-
 
|-
−
| ''f<sub>15</sub> || 1111 || (( ))
+
|
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
<math>\begin{matrix}
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
f_5
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
\\[4pt]
−
|}
+
f_{10}
−
<br>
+
\end{matrix}</math>
−
 
+
|
−
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+
<math>\begin{matrix}
−
|+ '''Table 10.  Qualifiers of Implication Ordering:  &alpha;<sub>''i''&nbsp;</sub>''f'' = &Upsilon;(''f''<sub>''i''</sub> &rArr; ''f'')'''
+
(q)
−
|- style="background:paleturquoise"
+
\\[4pt]
−
| align=right | ''x'' : || 1100 || ''f''
+
~q~
−
|&alpha;||&alpha;||&alpha;||&alpha;||&alpha;||&alpha;||&alpha;||&alpha;
+
\end{matrix}</math>
−
|&alpha;||&alpha;||&alpha;||&alpha;||&alpha;||&alpha;||&alpha;||&alpha;
+
|
−
|- style="background:paleturquoise"
+
<math>\begin{matrix}
−
| align=right | ''y'' : || 1010 || &nbsp;
+
~q~
−
|15||14||13||12||11||10||9||8||7||6||5||4||3||2||1||0
+
\\[4pt]
 +
(q)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~q~
 +
\\[4pt]
 +
(q)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\end{matrix}</math>
 
|-
 
|-
−
| ''f<sub>0</sub> || 0000 || ( )
+
|
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
<math>\begin{matrix}
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
+
f_7
 +
\\[4pt]
 +
f_{11}
 +
\\[4pt]
 +
f_{13}
 +
\\[4pt]
 +
f_{14}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~p~~q~)
 +
\\[4pt]
 +
(~p~(q))
 +
\\[4pt]
 +
((p)~q~)
 +
\\[4pt]
 +
((p)(q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((p)(q))
 +
\\[4pt]
 +
((p)~q~)
 +
\\[4pt]
 +
(~p~(q))
 +
\\[4pt]
 +
(~p~~q~)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((p)~q~)
 +
\\[4pt]
 +
((p)(q))
 +
\\[4pt]
 +
(~p~~q~)
 +
\\[4pt]
 +
(~p~(q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~p~(q))
 +
\\[4pt]
 +
(~p~~q~)
 +
\\[4pt]
 +
((p)(q))
 +
\\[4pt]
 +
((p)~q~)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~p~~q~)
 +
\\[4pt]
 +
(~p~(q))
 +
\\[4pt]
 +
((p)~q~)
 +
\\[4pt]
 +
((p)(q))
 +
\end{matrix}</math>
 
|-
 
|-
−
| ''f<sub>1</sub> || 0001 || (x)(y)
+
| <math>f_{15}\!</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>((~))</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1
+
| <math>((~))</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
|- style="background:#f0f0ff"
 +
| colspan="2" | <math>\text{Fixed Point Total}\!</math>
 +
| <math>4\!</math>
 +
| <math>4\!</math>
 +
| <math>4\!</math>
 +
| <math>16\!</math>
 +
|}
 +
 
 +
<br>
 +
 
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 +
|+ <math>\text{Table A4.}~~\operatorname{D}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}p, \operatorname{d}q \}</math>
 +
|- style="background:#f0f0ff"
 +
| width="10%" | &nbsp;
 +
| width="18%" | <math>f\!</math>
 +
| width="18%" |
 +
<math>\operatorname{D}f|_{\operatorname{d}p~\operatorname{d}q}</math>
 +
| width="18%" |
 +
<math>\operatorname{D}f|_{\operatorname{d}p(\operatorname{d}q)}</math>
 +
| width="18%" |
 +
<math>\operatorname{D}f|_{(\operatorname{d}p)\operatorname{d}q}</math>
 +
| width="18%" |
 +
<math>\operatorname{D}f|_{(\operatorname{d}p)(\operatorname{d}q)}</math>
 
|-
 
|-
−
| ''f<sub>2</sub> || 0010 || (x) y
+
| <math>f_0\!</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>(~)</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;|| 1
+
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 
|-
 
|-
−
| ''f<sub>3</sub> || 0011 || (x)  
+
|
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
<math>\begin{matrix}
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1    || 1    || 1
+
f_1
−
|-
+
\\[4pt]
−
| ''f<sub>4</sub> || 0100 || x (y)
+
f_2
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
\\[4pt]
−
|&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;||&nbsp;||&nbsp;|| 1
+
f_4
 +
\\[4pt]
 +
f_8
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
~p~~q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((p,~q))
 +
\\[4pt]
 +
~(p,~q)~
 +
\\[4pt]
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\\[4pt]
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\\[4pt]
 +
~p~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}</math>
 
|-
 
|-
−
| ''f<sub>5</sub> || 0101 || (y)
+
|
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
<math>\begin{matrix}
−
|&nbsp;||&nbsp;|| 1    || 1    ||&nbsp;||&nbsp;|| 1    || 1
+
f_3
 +
\\[4pt]
 +
f_{12}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((~))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((~))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}</math>
 
|-
 
|-
−
| ''f<sub>6</sub> || 0110 || (x, y)
+
|
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
<math>\begin{matrix}
−
|&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1
+
f_6
 +
\\[4pt]
 +
f_9
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((~))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((~))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}</math>
 
|-
 
|-
−
| ''f<sub>7</sub> || 0111 || (x  y)
+
|
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
<math>\begin{matrix}
−
| 1    || 1    || 1    || 1    || 1    || 1    || 1    || 1
+
f_5
 +
\\[4pt]
 +
f_{10}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((~))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((~))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_7
 +
\\[4pt]
 +
f_{11}
 +
\\[4pt]
 +
f_{13}
 +
\\[4pt]
 +
f_{14}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p~~q)~
 +
\\[4pt]
 +
~(p~(q))
 +
\\[4pt]
 +
((p)~q)~
 +
\\[4pt]
 +
((p)(q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((p,~q))
 +
\\[4pt]
 +
~(p,~q)~
 +
\\[4pt]
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~q~
 +
\\[4pt]
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\\[4pt]
 +
(q)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~p~
 +
\\[4pt]
 +
~p~
 +
\\[4pt]
 +
(p)
 +
\\[4pt]
 +
(p)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}</math>
 
|-
 
|-
−
| ''f<sub>8</sub> || 1000 || x  y
+
| <math>f_{15}\!</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
+
| <math>((~))</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
+
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
|}
 +
 
 +
<br>
 +
 
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 +
|+ <math>\text{Table A5.}~~\operatorname{E}f ~\text{Expanded Over Ordinary Features}~ \{ p, q \}</math>
 +
|- style="background:#f0f0ff"
 +
| width="10%" | &nbsp;
 +
| width="18%" | <math>f\!</math>
 +
| width="18%" | <math>\operatorname{E}f|_{xy}</math>
 +
| width="18%" | <math>\operatorname{E}f|_{p(q)}</math>
 +
| width="18%" | <math>\operatorname{E}f|_{(p)q}</math>
 +
| width="18%" | <math>\operatorname{E}f|_{(p)(q)}</math>
 
|-
 
|-
−
| ''f<sub>9</sub> || 1001 || ((x, y))
+
| <math>f_0\!</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1
+
| <math>(~)</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1
+
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 
|-
 
|-
−
| ''f<sub>10</sub> || 1010 || y
+
|
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;|| 1
+
<math>\begin{matrix}
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;|| 1
+
f_1
 +
\\[4pt]
 +
f_2
 +
\\[4pt]
 +
f_4
 +
\\[4pt]
 +
f_8
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
~p~~q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~\operatorname{d}p~~\operatorname{d}q~
 +
\\[4pt]
 +
~\operatorname{d}p~(\operatorname{d}q)
 +
\\[4pt]
 +
(\operatorname{d}p)~\operatorname{d}q~
 +
\\[4pt]
 +
(\operatorname{d}p)(\operatorname{d}q)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~\operatorname{d}p~(\operatorname{d}q)
 +
\\[4pt]
 +
~\operatorname{d}p~~\operatorname{d}q~
 +
\\[4pt]
 +
(\operatorname{d}p)(\operatorname{d}q)
 +
\\[4pt]
 +
(\operatorname{d}p)~\operatorname{d}q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(\operatorname{d}p)~\operatorname{d}q~
 +
\\[4pt]
 +
(\operatorname{d}p)(\operatorname{d}q)
 +
\\[4pt]
 +
~\operatorname{d}p~~\operatorname{d}q~
 +
\\[4pt]
 +
~\operatorname{d}p~(\operatorname{d}q)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(\operatorname{d}p)(\operatorname{d}q)
 +
\\[4pt]
 +
(\operatorname{d}p)~\operatorname{d}q~
 +
\\[4pt]
 +
~\operatorname{d}p~(\operatorname{d}q)
 +
\\[4pt]
 +
~\operatorname{d}p~~\operatorname{d}q~
 +
\end{matrix}</math>
 
|-
 
|-
−
| ''f<sub>11</sub> || 1011 || (x (y))
+
|
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1    || 1    || 1
+
<math>\begin{matrix}
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1    || 1    || 1
+
f_3
 +
\\[4pt]
 +
f_{12}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~\operatorname{d}p~
 +
\\[4pt]
 +
(\operatorname{d}p)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~\operatorname{d}p~
 +
\\[4pt]
 +
(\operatorname{d}p)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(\operatorname{d}p)
 +
\\[4pt]
 +
~\operatorname{d}p~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(\operatorname{d}p)
 +
\\[4pt]
 +
~\operatorname{d}p~
 +
\end{matrix}</math>
 
|-
 
|-
−
| ''f<sub>12</sub> || 1100 || x
+
|
−
|&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;||&nbsp;||&nbsp;|| 1
+
<math>\begin{matrix}
−
|&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;||&nbsp;||&nbsp;|| 1
+
f_6
−
|-
+
\\[4pt]
−
| ''f<sub>13</sub> || 1101 || ((x) y)
+
f_9
−
|&nbsp;||&nbsp;|| 1    || 1    ||&nbsp;||&nbsp;|| 1    || 1
+
\end{matrix}</math>
−
|&nbsp;||&nbsp;|| 1    || 1    ||&nbsp;||&nbsp;|| 1    || 1
+
|
 +
<math>\begin{matrix}
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(\operatorname{d}p,~\operatorname{d}q)~
 +
\\[4pt]
 +
((\operatorname{d}p,~\operatorname{d}q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((\operatorname{d}p,~\operatorname{d}q))
 +
\\[4pt]
 +
~(\operatorname{d}p,~\operatorname{d}q)~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((\operatorname{d}p,~\operatorname{d}q))
 +
\\[4pt]
 +
~(\operatorname{d}p,~\operatorname{d}q)~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(\operatorname{d}p,~\operatorname{d}q)~
 +
\\[4pt]
 +
((\operatorname{d}p,~\operatorname{d}q))
 +
\end{matrix}</math>
 
|-
 
|-
−
| ''f<sub>14</sub> || 1110 || ((x)(y))
+
|
−
|&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1
+
<math>\begin{matrix}
−
|&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1
+
f_5
 +
\\[4pt]
 +
f_{10}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~\operatorname{d}q~
 +
\\[4pt]
 +
(\operatorname{d}q)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(\operatorname{d}q)
 +
\\[4pt]
 +
~\operatorname{d}q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~\operatorname{d}q~
 +
\\[4pt]
 +
(\operatorname{d}q)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(\operatorname{d}q)
 +
\\[4pt]
 +
~\operatorname{d}q~
 +
\end{matrix}</math>
 
|-
 
|-
−
| ''f<sub>15</sub> || 1111 || (( ))
+
|
−
| 1    || 1    || 1    || 1    || 1    || 1    || 1    || 1
+
<math>\begin{matrix}
−
| 1    || 1    || 1    || 1    || 1    || 1    || 1    || 1
+
f_7
−
|}
+
\\[4pt]
−
<br>
+
f_{11}
−
 
+
\\[4pt]
−
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+
f_{13}
−
|+ '''Table 11.  Qualifiers of Implication Ordering:  &beta;<sub>''i''&nbsp;</sub>''f'' = &Upsilon;(''f'' &rArr; ''f''<sub>''i''</sub>)'''
+
\\[4pt]
−
|- style="background:paleturquoise"
+
f_{14}
−
| align=right | ''x'' : || 1100 || ''f''
+
\end{matrix}</math>
−
|&beta;||&beta;||&beta;||&beta;||&beta;||&beta;||&beta;||&beta;
+
|
−
|&beta;||&beta;||&beta;||&beta;||&beta;||&beta;||&beta;||&beta;
+
<math>\begin{matrix}
−
|- style="background:paleturquoise"
+
(~p~~q~)
−
| align=right | ''y'' : || 1010 || &nbsp;
+
\\[4pt]
−
|0||1||2||3||4||5||6||7||8||9||10||11||12||13||14||15
+
(~p~(q))
 +
\\[4pt]
 +
((p)~q~)
 +
\\[4pt]
 +
((p)(q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((\operatorname{d}p)(\operatorname{d}q))
 +
\\[4pt]
 +
((\operatorname{d}p)~\operatorname{d}q~)
 +
\\[4pt]
 +
(~\operatorname{d}p~(\operatorname{d}q))
 +
\\[4pt]
 +
(~\operatorname{d}p~~\operatorname{d}q~)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((\operatorname{d}p)~\operatorname{d}q~)
 +
\\[4pt]
 +
((\operatorname{d}p)(\operatorname{d}q))
 +
\\[4pt]
 +
(~\operatorname{d}p~~\operatorname{d}q~)
 +
\\[4pt]
 +
(~\operatorname{d}p~(\operatorname{d}q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~\operatorname{d}p~(\operatorname{d}q))
 +
\\[4pt]
 +
(~\operatorname{d}p~~\operatorname{d}q~)
 +
\\[4pt]
 +
((\operatorname{d}p)(\operatorname{d}q))
 +
\\[4pt]
 +
((\operatorname{d}p)~\operatorname{d}q~)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~\operatorname{d}p~~\operatorname{d}q~)
 +
\\[4pt]
 +
(~\operatorname{d}p~(\operatorname{d}q))
 +
\\[4pt]
 +
((\operatorname{d}p)~\operatorname{d}q~)
 +
\\[4pt]
 +
((\operatorname{d}p)(\operatorname{d}q))
 +
\end{matrix}</math>
 
|-
 
|-
−
| ''f<sub>0</sub> || 0000 || ( )
+
| <math>f_{15}\!</math>
−
| 1    || 1   || 1    || 1    || 1    || 1    || 1    || 1
+
| <math>((~))</math>
−
| 1    || 1    || 1    || 1    || 1    || 1    || 1    || 1
+
| <math>((~))</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
|}
 +
 
 +
<br>
 +
 
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 +
|+ <math>\text{Table A6.}~~\operatorname{D}f ~\text{Expanded Over Ordinary Features}~ \{ p, q \}</math>
 +
|- style="background:#f0f0ff"
 +
| width="10%" | &nbsp;
 +
| width="18%" | <math>f\!</math>
 +
| width="18%" | <math>\operatorname{D}f|_{xy}</math>
 +
| width="18%" | <math>\operatorname{D}f|_{p(q)}</math>
 +
| width="18%" | <math>\operatorname{D}f|_{(p)q}</math>
 +
| width="18%" | <math>\operatorname{D}f|_{(p)(q)}</math>
 
|-
 
|-
−
| ''f<sub>1</sub> || 0001 || (x)(y)
+
| <math>f_0\!</math>
−
|&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1
+
| <math>(~)</math>
−
|&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1
+
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 
|-
 
|-
−
| ''f<sub>2</sub> || 0010 || (x) y
+
|
−
|&nbsp;||&nbsp;|| 1    || 1    ||&nbsp;||&nbsp;|| 1    || 1
+
<math>\begin{matrix}
−
|&nbsp;||&nbsp;|| 1    || 1    ||&nbsp;||&nbsp;|| 1    || 1
+
f_1
−
|-
+
\\[4pt]
−
| ''f<sub>3</sub> || 0011 || (x)  
+
f_2
−
|&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;||&nbsp;||&nbsp;|| 1
+
\\[4pt]
−
|&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;||&nbsp;||&nbsp;|| 1
+
f_4
 +
\\[4pt]
 +
f_8
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
~p~~q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~~\operatorname{d}p~~\operatorname{d}q~~
 +
\\[4pt]
 +
~~\operatorname{d}p~(\operatorname{d}q)~
 +
\\[4pt]
 +
~(\operatorname{d}p)~\operatorname{d}q~~
 +
\\[4pt]
 +
((\operatorname{d}p)(\operatorname{d}q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~~\operatorname{d}p~(\operatorname{d}q)~
 +
\\[4pt]
 +
~~\operatorname{d}p~~\operatorname{d}q~~
 +
\\[4pt]
 +
((\operatorname{d}p)(\operatorname{d}q))
 +
\\[4pt]
 +
~(\operatorname{d}p)~\operatorname{d}q~~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(\operatorname{d}p)~\operatorname{d}q~~
 +
\\[4pt]
 +
((\operatorname{d}p)(\operatorname{d}q))
 +
\\[4pt]
 +
~~\operatorname{d}p~~\operatorname{d}q~~
 +
\\[4pt]
 +
~~\operatorname{d}p~(\operatorname{d}q)~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((\operatorname{d}p)(\operatorname{d}q))
 +
\\[4pt]
 +
~(\operatorname{d}p)~\operatorname{d}q~~
 +
\\[4pt]
 +
~~\operatorname{d}p~(\operatorname{d}q)~
 +
\\[4pt]
 +
~~\operatorname{d}p~~\operatorname{d}q~~
 +
\end{matrix}</math>
 
|-
 
|-
−
| ''f<sub>4</sub> || 0100 || x (y)
+
|
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1    || 1    || 1
+
<math>\begin{matrix}
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1    || 1    || 1
+
f_3
 +
\\[4pt]
 +
f_{12}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{d}p
 +
\\[4pt]
 +
\operatorname{d}p
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{d}p
 +
\\[4pt]
 +
\operatorname{d}p
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{d}p
 +
\\[4pt]
 +
\operatorname{d}p
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{d}p
 +
\\[4pt]
 +
\operatorname{d}p
 +
\end{matrix}</math>
 
|-
 
|-
−
| ''f<sub>5</sub> || 0101 || (y)
+
|
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;|| 1
+
<math>\begin{matrix}
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;|| 1
+
f_6
 +
\\[4pt]
 +
f_9
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(\operatorname{d}p,~\operatorname{d}q)
 +
\\[4pt]
 +
(\operatorname{d}p,~\operatorname{d}q)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(\operatorname{d}p,~\operatorname{d}q)
 +
\\[4pt]
 +
(\operatorname{d}p,~\operatorname{d}q)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(\operatorname{d}p,~\operatorname{d}q)
 +
\\[4pt]
 +
(\operatorname{d}p,~\operatorname{d}q)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(\operatorname{d}p,~\operatorname{d}q)
 +
\\[4pt]
 +
(\operatorname{d}p,~\operatorname{d}q)
 +
\end{matrix}</math>
 
|-
 
|-
−
| ''f<sub>6</sub> || 0110 || (x, y)
+
|
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1
+
<math>\begin{matrix}
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1
+
f_5
 +
\\[4pt]
 +
f_{10}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{d}q
 +
\\[4pt]
 +
\operatorname{d}q
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{d}q
 +
\\[4pt]
 +
\operatorname{d}q
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{d}q
 +
\\[4pt]
 +
\operatorname{d}q
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{d}q
 +
\\[4pt]
 +
\operatorname{d}q
 +
\end{matrix}</math>
 
|-
 
|-
−
| ''f<sub>7</sub> || 0111 || (x  y)
+
|
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
+
<math>\begin{matrix}
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
+
f_7
 +
\\[4pt]
 +
f_{11}
 +
\\[4pt]
 +
f_{13}
 +
\\[4pt]
 +
f_{14}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~p~~q~)
 +
\\[4pt]
 +
(~p~(q))
 +
\\[4pt]
 +
((p)~q~)
 +
\\[4pt]
 +
((p)(q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((\operatorname{d}p)(\operatorname{d}q))
 +
\\[4pt]
 +
~(\operatorname{d}p)~\operatorname{d}q~~
 +
\\[4pt]
 +
~~\operatorname{d}p~(\operatorname{d}q)~
 +
\\[4pt]
 +
~~\operatorname{d}p~~\operatorname{d}q~~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(\operatorname{d}p)~\operatorname{d}q~~
 +
\\[4pt]
 +
((\operatorname{d}p)(\operatorname{d}q))
 +
\\[4pt]
 +
~~\operatorname{d}p~~\operatorname{d}q~~
 +
\\[4pt]
 +
~~\operatorname{d}p~(\operatorname{d}q)~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~~\operatorname{d}p~(\operatorname{d}q)~
 +
\\[4pt]
 +
~~\operatorname{d}p~~\operatorname{d}q~~
 +
\\[4pt]
 +
((\operatorname{d}p)(\operatorname{d}q))
 +
\\[4pt]
 +
~(\operatorname{d}p)~\operatorname{d}q~~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~~\operatorname{d}p~~\operatorname{d}q~~
 +
\\[4pt]
 +
~~\operatorname{d}p~(\operatorname{d}q)~
 +
\\[4pt]
 +
~(\operatorname{d}p)~\operatorname{d}q~~
 +
\\[4pt]
 +
((\operatorname{d}p)(\operatorname{d}q))
 +
\end{matrix}</math>
 
|-
 
|-
−
| ''f<sub>8</sub> || 1000 || x  y
+
| <math>f_{15}\!</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>((~))</math>
−
| 1    || 1    || 1    || 1    || 1    || 1    || 1    || 1
+
| <math>((~))</math>
−
|-
+
| <math>((~))</math>
−
| ''f<sub>9</sub> || 1001 || ((x, y))
+
| <math>((~))</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>((~))</math>
−
|&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1
+
|}
−
|-
+
 
−
| ''f<sub>10</sub> || 1010 || y
+
<br>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
 
−
|&nbsp;||&nbsp;|| 1    || 1    ||&nbsp;||&nbsp;|| 1    || 1
+
===Wiki TeX Tables : XY===
−
|-
+
 
−
| ''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>
 
<br>
  
−
{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
−
|+ '''Table 13. Syllogistic Premisses as Higher Order Indicator Functions'''
+
|+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math>
−
| A
+
|- style="background:#f0f0ff"
−
| align=left | Universal Affirmative
+
| width="15%" |
−
| align=left | All
+
<p><math>\mathcal{L}_1</math></p>
−
| x || is || y
+
<p><math>\text{Decimal}</math></p>
−
| align=left | Indicator of " x (y)" = 0
+
| width="15%" |
 +
<p><math>\mathcal{L}_2</math></p>
 +
<p><math>\text{Binary}</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_3</math></p>
 +
<p><math>\text{Vector}</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_4</math></p>
 +
<p><math>\text{Cactus}</math></p>
 +
| width="25%" |
 +
<p><math>\mathcal{L}_5</math></p>
 +
<p><math>\text{English}</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_6</math></p>
 +
<p><math>\text{Ordinary}</math></p>
 +
|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | <math>x\colon\!</math>
 +
| <math>1~1~0~0\!</math>
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | <math>y\colon\!</math>
 +
| <math>1~0~1~0\!</math>
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 
|-
 
|-
−
| E
+
| <math>f_{0}\!</math>
−
| align=left | Universal Negative
+
| <math>f_{0000}\!</math>
−
| align=left | All
+
| <math>0~0~0~0\!</math>
−
| x || is || (y)
+
| <math>(~)\!</math>
−
| align=left | Indicator of " x  y " = 0
+
| <math>\text{false}\!</math>
 +
| <math>0\!</math>
 
|-
 
|-
−
| I
+
| <math>f_{1}\!</math>
−
| align=left | Particular Affirmative
+
| <math>f_{0001}\!</math>
−
| align=left | Some
+
| <math>0~0~0~1\!</math>
−
| x || is || y
+
| <math>(x)(y)\!</math>
−
| align=left | Indicator of " x y " = 1
+
| <math>\text{neither}~ x ~\text{nor}~ y\!</math>
 +
| <math>\lnot x \land \lnot y\!</math>
 
|-
 
|-
−
| O
+
| <math>f_{2}\!</math>
−
| align=left | Particular Negative
+
| <math>f_{0010}\!</math>
−
| align=left | Some
+
| <math>0~0~1~0\!</math>
−
| x || is || (y)
+
| <math>(x)~y\!</math>
−
| align=left | Indicator of " x (y)" = 1
+
| <math>y ~\text{without}~ x\!</math>
−
|}
+
| <math>\lnot x \land y\!</math>
−
<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
+
| <math>f_{3}\!</math>
−
| Universal<br>Negative
+
| <math>f_{0011}\!</math>
−
| align=left | All x is (y)
+
| <math>0~0~1~1\!</math>
−
| align=left | &nbsp;
+
| <math>(x)\!</math>
−
| align=left | No x is y
+
| <math>\text{not}~ x\!</math>
−
| (''L''<sub>11</sub>)
+
| <math>\lnot x\!</math>
 
|-
 
|-
−
| A<br>Absolute
+
| <math>f_{4}\!</math>
−
| Universal<br>Affirmative
+
| <math>f_{0100}\!</math>
−
| align=left | All x is y
+
| <math>0~1~0~0\!</math>
−
| align=left | &nbsp;
+
| <math>x~(y)\!</math>
−
| align=left | No x is (y)
+
| <math>x ~\text{without}~ y\!</math>
−
| (''L''<sub>10</sub>)
+
| <math>x \land \lnot y\!</math>
 
|-
 
|-
−
| &nbsp;
+
| <math>f_{5}\!</math>
−
| &nbsp;
+
| <math>f_{0101}\!</math>
−
| align=left | All y is x
+
| <math>0~1~0~1\!</math>
−
| align=left | No y is (x)
+
| <math>(y)\!</math>
−
| align=left | No (x) is y
+
| <math>\text{not}~ y\!</math>
−
| (''L''<sub>01</sub>)
+
| <math>\lnot y\!</math>
 
|-
 
|-
−
| &nbsp;
+
| <math>f_{6}\!</math>
−
| &nbsp;
+
| <math>f_{0110}\!</math>
−
| align=left | All (y) is x
+
| <math>0~1~1~0\!</math>
−
| align=left | No (y) is (x)
+
| <math>(x,~y)\!</math>
−
| align=left | No (x) is (y)
+
| <math>x ~\text{not equal to}~ y\!</math>
−
| (''L''<sub>00</sub>)
+
| <math>x \ne y\!</math>
 
|-
 
|-
−
| &nbsp;
+
| <math>f_{7}\!</math>
−
| &nbsp;
+
| <math>f_{0111}\!</math>
−
| align=left | Some (x) is (y)
+
| <math>0~1~1~1\!</math>
−
| align=left | &nbsp;
+
| <math>(x~y)\!</math>
−
| align=left | Some (x) is (y)
+
| <math>\text{not both}~ x ~\text{and}~ y\!</math>
−
| ''L''<sub>00</sub>
+
| <math>\lnot x \lor \lnot y\!</math>
 
|-
 
|-
−
| &nbsp;
+
| <math>f_{8}\!</math>
−
| &nbsp;
+
| <math>f_{1000}\!</math>
−
| align=left | Some (x) is y
+
| <math>1~0~0~0\!</math>
−
| align=left | &nbsp;
+
| <math>x~y\!</math>
−
| align=left | Some (x) is y
+
| <math>x ~\text{and}~ y\!</math>
−
| ''L''<sub>01</sub>
+
| <math>x \land y\!</math>
 
|-
 
|-
−
| O<br>Obtrusive
+
| <math>f_{9}\!</math>
−
| Particular<br>Negative
+
| <math>f_{1001}\!</math>
−
| align=left | Some x is (y)
+
| <math>1~0~0~1\!</math>
−
| align=left | &nbsp;
+
| <math>((x,~y))\!</math>
−
| align=left | Some x is (y)
+
| <math>x ~\text{equal to}~ y\!</math>
−
| ''L''<sub>10</sub>
+
| <math>x = y\!</math>
 
|-
 
|-
−
| I<br>Indefinite
+
| <math>f_{10}\!</math>
−
| Particular<br>Affirmative
+
| <math>f_{1010}\!</math>
−
| align=left | Some x is y
+
| <math>1~0~1~0\!</math>
−
| align=left | &nbsp;
+
| <math>y\!</math>
−
| align=left | Some x is y
+
| <math>y\!</math>
−
| ''L''<sub>11</sub>
+
| <math>y\!</math>
−
|}
 
−
<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 || ( )
+
| <math>f_{11}\!</math>
−
| 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0
+
| <math>f_{1011}\!</math>
 +
| <math>1~0~1~1\!</math>
 +
| <math>(x~(y))\!</math>
 +
| <math>\text{not}~ x ~\text{without}~ y\!</math>
 +
| <math>x \Rightarrow y\!</math>
 
|-
 
|-
−
| ''f<sub>1</sub> || 0001 || (x)(y)
+
| <math>f_{12}\!</math>
−
| 1 || 1 || 1 || 0 || 1 || 0 || 0 || 0
+
| <math>f_{1100}\!</math>
−
 
+
| <math>1~1~0~0\!</math>
 +
| <math>x\!</math>
 +
| <math>x\!</math>
 +
| <math>x\!</math>
 
|-
 
|-
−
| ''f<sub>2</sub> || 0010 || (x) y  
+
| <math>f_{13}\!</math>
−
| 1 || 1 || 0 || 1 || 0 || 1 || 0 || 0
+
| <math>f_{1101}\!</math>
 +
| <math>1~1~0~1\!</math>
 +
| <math>((x)~y)\!</math>
 +
| <math>\text{not}~ y ~\text{without}~ x\!</math>
 +
| <math>x \Leftarrow y\!</math>
 
|-
 
|-
−
| ''f<sub>3</sub> || 0011 || (x)  
+
| <math>f_{14}\!</math>
−
| 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0
+
| <math>f_{1110}\!</math>
 +
| <math>1~1~1~0\!</math>
 +
| <math>((x)(y))\!</math>
 +
| <math>x ~\text{or}~ y\!</math>
 +
| <math>x \lor y\!</math>
 
|-
 
|-
−
| ''f<sub>4</sub> || 0100 || x (y)
+
| <math>f_{15}\!</math>
−
| 1 || 0 || 1 || 1 || 0 || 0 || 1 || 0
+
| <math>f_{1111}\!</math>
−
|-
+
| <math>1~1~1~1\!</math>
−
| ''f<sub>5</sub> || 0101 || (y)
+
| <math>((~))\!</math>
−
| 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0
+
| <math>\text{true}\!</math>
−
|-
+
| <math>1\!</math>
−
| ''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>
 
<br>
  
−
Table 7. Higher Order Propositions (n = 1)
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
−
o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o
+
|+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math>
−
| \ x | 1 0 | F  |m |m |m |m |m |m |m |m |m |m |m |m |m |m |m |m  |
+
|- style="background:#f0f0ff"
−
| F \ |     |     |00|01|02|03|04|05|06|07|08|09|10|11|12|13|14|15 |
+
| width="15%" |
−
o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o
+
<p><math>\mathcal{L}_1</math></p>
−
|     |     |     |                                                |
+
<p><math>\text{Decimal}</math></p>
−
| F_0  | 0 0 |  0  | 0  1  0  1 0  1 0 1  0 1  0  1  0  1  0  1 |
+
| width="15%" |
−
|     |     |     |                                                |
+
<p><math>\mathcal{L}_2</math></p>
−
| F_1  | 0 1 | (x) | 0  0  1  1 0 0  1 1  0 0  1  1  0  0  1  1 |
+
<p><math>\text{Binary}</math></p>
−
|     |     |     |                                                |
+
| width="15%" |
−
| F_2  | 1 0 |  x  | 0  0  0  0  1  1  1  1  0  0  0  0  1  1  1  1 |
+
<p><math>\mathcal{L}_3</math></p>
−
|      |    |    |                                                |
+
<p><math>\text{Vector}</math></p>
−
| F_3  | 1 1 |  1  | 0  0  0  0  0  0  0  0  1  1  1  1  1  1  1  1 |
+
| width="15%" |
−
|      |    |    |                                                |
+
<p><math>\mathcal{L}_4</math></p>
−
o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o
+
<p><math>\text{Cactus}</math></p>
−
<br>
+
| width="25%" |
−
 
+
<p><math>\mathcal{L}_5</math></p>
−
Table 8.  Interpretive Categories for Higher Order Propositions (n = 1)
+
<p><math>\text{English}</math></p>
−
o-------o----------o------------o------------o----------o----------o-----------o
+
| width="15%" |
−
|Measure| Happening| Exactness  | Existence  | Linearity|Uniformity|Information|
+
<p><math>\mathcal{L}_6</math></p>
−
o-------o----------o------------o------------o----------o----------o-----------o
+
<p><math>\text{Ordinary}</math></p>
−
| m_0  | nothing  |            |            |          |          |          |
+
|- style="background:#f0f0ff"
−
|      | happens  |            |            |          |          |          |
+
| &nbsp;
−
o-------o----------o------------o------------o----------o----------o-----------o
+
| align="right" | <math>x\colon\!</math>
−
| m_1  |          |            | nothing    |          |          |          |
+
| <math>1~1~0~0\!</math>
−
|      |          | just false | exists    |          |          |          |
+
| &nbsp;
−
o-------o----------o------------o------------o----------o----------o-----------o
+
| &nbsp;
−
| m_2  |          |            |            |          |          |          |
+
| &nbsp;
−
|      |          | just not x |            |          |          |          |
+
|- style="background:#f0f0ff"
−
o-------o----------o------------o------------o----------o----------o-----------o
+
| &nbsp;
−
| m_3  |          |            | nothing    |          |          |          |
+
| align="right" | <math>y\colon\!</math>
−
|      |          |            | is x      |          |          |          |
+
| <math>1~0~1~0\!</math>
−
o-------o----------o------------o------------o----------o----------o-----------o
+
| &nbsp;
−
| m_4  |          |            |            |          |          |          |
+
| &nbsp;
−
|      |          | just x    |            |          |          |          |
+
| &nbsp;
−
o-------o----------o------------o------------o----------o----------o-----------o
+
|-
−
| m_5  |          |            | everything | F is    |          |          |
+
|
−
|      |          |            | is x       | linear  |          |          |
+
<math>\begin{matrix}
−
o-------o----------o------------o------------o----------o----------o-----------o
+
f_0
−
| m_6  |          |            |            |          | F is not | F is      |
+
\\[4pt]
−
|      |          |            |            |          | uniform  | informed  |
+
f_1
−
o-------o----------o------------o------------o----------o----------o-----------o
+
\\[4pt]
−
| m_7  |          | not        |            |          |          |          |
+
f_2
−
|      |          | just true  |            |          |          |          |
+
\\[4pt]
−
o-------o----------o------------o------------o----------o----------o-----------o
+
f_3
−
| m_8  |         |            |           |         |         |          |
+
\\[4pt]
−
|      |          | just true  |           |          |          |          |
+
f_4
−
o-------o----------o------------o------------o----------o----------o-----------o
+
\\[4pt]
−
| m_9  |          |            |            |          | F is    | F is not  |
+
f_5
−
|       |          |            |            |          | uniform  | informed  |
+
\\[4pt]
−
o-------o----------o------------o------------o----------o----------o-----------o
+
f_6
−
| m_10  |          |           | something  | F is not |          |          |
+
\\[4pt]
−
|      |          |            | is not x  | linear  |          |          |
+
f_7
−
o-------o----------o------------o------------o----------o----------o-----------o
+
\end{matrix}</math>
−
| m_11  |          | not        |            |          |          |          |
+
|
−
|      |          | just x    |            |          |          |          |
+
<math>\begin{matrix}
−
o-------o----------o------------o------------o----------o----------o-----------o
+
f_{0000}
−
| m_12  |          |            | something  |          |          |          |
+
\\[4pt]
−
|      |          |            | is x       |          |          |          |
+
f_{0001}
−
o-------o----------o------------o------------o----------o----------o-----------o
+
\\[4pt]
−
| m_13  |          | not        |            |          |          |          |
+
f_{0010}
−
|      |          | just not x |            |          |          |          |
+
\\[4pt]
−
o-------o----------o------------o------------o----------o----------o-----------o
+
f_{0011}
−
| m_14  |          | not        | something  |          |          |          |
+
\\[4pt]
−
|       |          | just false | exists    |          |          |          |
+
f_{0100}
−
o-------o----------o------------o------------o----------o----------o-----------o
+
\\[4pt]
−
| m_15  | anything |            |            |          |          |          |
+
f_{0101}
−
|      | happens  |            |            |          |          |          |
+
\\[4pt]
−
o-------o----------o------------o------------o----------o----------o-----------o
+
f_{0110}
−
<br>
+
\\[4pt]
−
 
+
f_{0111}
−
Table 9.  Higher Order Propositions (n = 2)
+
\end{matrix}</math>
−
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|.|
+
<math>\begin{matrix}
−
|  | y | 1010 |          |0|0|0|0|0|0|0|0|0|0|1|1|1|1|1|1|.|
+
0~0~0~0
−
| f \ |      |         |0|1|2|3|4|5|6|7|8|9|0|1|2|3|4|5|.|
+
\\[4pt]
−
o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
+
0~0~0~1
−
|      |      |          |                                |
+
\\[4pt]
−
| f_0  | 0000 |    ()    |0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1  |
+
0~0~1~0
−
|      |      |          |                                |
+
\\[4pt]
−
| f_1  | 0001 |  (x)(y)  |    1 1 0 0 1 1 0 0 1 1 0 0 1 1  |
+
0~0~1~1
−
|      |      |          |                                |
+
\\[4pt]
−
| f_2  | 0010 |  (x) y   |        1 1 1 1 0 0 0 0 1 1 1 1  |
+
0~1~0~0
−
|      |      |          |                                |
+
\\[4pt]
−
| f_3  | 0011 |  (x)     |                1 1 1 1 1 1 1 1  |
+
0~1~0~1
−
|      |      |          |                                |
+
\\[4pt]
−
| f_4  | 0100 |  x (y)  |                                |
+
0~1~1~0
−
|      |      |          |                                |
+
\\[4pt]
−
| f_5  | 0101 |    (y)  |                                |
+
0~1~1~1
−
|      |      |          |                                |
+
\end{matrix}</math>
−
| f_6  | 0110 |  (x, y)  |                                |
+
|
−
|      |      |          |                                |
+
<math>\begin{matrix}
−
| f_7  | 0111 |  (x y)  |                                |
+
(~)
−
|      |      |          |                                |
+
\\[4pt]
−
o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
+
(x)(y)
−
|      |      |          |                                |
+
\\[4pt]
−
| f_8  | 1000 |  x  y  |                                |
+
(x)~y~
−
|      |      |          |                                |
+
\\[4pt]
−
| f_9  | 1001 | ((x, y)) |                                |
+
(x)~~~
−
|      |      |          |                                |
+
\\[4pt]
−
| f_10 | 1010 |      y  |                                |
+
~x~(y)
−
|      |      |          |                                |
+
\\[4pt]
−
| f_11 | 1011 |  (x (y)) |                                |
+
~~~(y)
−
|      |      |          |                                |
+
\\[4pt]
−
| f_12 | 1100 |  x      |                                |
+
(x,~y)
−
|      |      |          |                                |
+
\\[4pt]
−
| f_13 | 1101 | ((x) y)  |                                |
+
(x~~y)
−
|      |      |          |                                |
+
\end{matrix}</math>
−
| f_14 | 1110 | ((x)(y)) |                                |
+
|
−
|      |      |          |                                |
+
<math>\begin{matrix}
−
| f_15 | 1111 |  (())  |                                |
+
\text{false}
−
|      |      |          |                                |
+
\\[4pt]
−
o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
+
\text{neither}~ x ~\text{nor}~ y
−
<br>
+
\\[4pt]
−
 
+
y ~\text{without}~ x
−
Table 10.  Qualifiers of Implication Ordering:  !a!_i f  =  !Y!(f_i => f)
+
\\[4pt]
−
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
+
\text{not}~ x
−
|  | x | 1100 |    f    |a |a |a |a |a |a |a |a |a |a |a |a |a |a |a |a |
+
\\[4pt]
−
|  | y | 1010 |          |1 |1 |1 |1 |1 |1 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |
+
x ~\text{without}~ y
−
| f \  |      |          |5 |4 |3 |2 |1 |0 |9 |8 |7 |6 |5 |4 |3 |2 |1 |0 |
+
\\[4pt]
−
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
+
\text{not}~ y
−
|      |      |          |                                              |
+
\\[4pt]
−
| f_0  | 0000 |    ()    |                                            1 |
+
x ~\text{not equal to}~ y
−
|      |      |          |                                              |
+
\\[4pt]
−
| f_1  | 0001 |  (x)(y)  |                                          1  1 |
+
\text{not both}~ x ~\text{and}~ y
−
|      |      |          |                                              |
+
\end{matrix}</math>
−
| f_2  | 0010 |  (x) y  |                                      1    1 |
+
|
−
|      |      |          |                                              |
+
<math>\begin{matrix}
−
| f_3  | 0011 |  (x)    |                                    1  1  1  1 |
+
0
−
|      |      |          |                                              |
+
\\[4pt]
−
| f_4  | 0100 |  x (y)  |                                1          1 |
+
\lnot x \land \lnot y
−
|      |      |          |                                              |
+
\\[4pt]
−
| f_5  | 0101 |    (y)  |                              1  1        1  1 |
+
\lnot x \land y
−
|      |      |          |                                              |
+
\\[4pt]
−
| f_6  | 0110 |  (x, y)  |                          1    1    1    1 |
+
\lnot x
−
|      |      |          |                                              |
+
\\[4pt]
−
| f_7  | 0111 |  (x  y)  |                        1  1  1  1  1  1  1  1 |
+
x \land \lnot y
−
|      |      |          |                                              |
+
\\[4pt]
−
| f_8  | 1000 |  x  y  |                    1                      1 |
+
\lnot y
−
|      |      |          |                                              |
+
\\[4pt]
−
| f_9  | 1001 | ((x, y)) |                  1  1                    1  1 |
+
x \ne y
−
|      |      |          |                                              |
+
\\[4pt]
−
| f_10 | 1010 |      y  |              1    1                1    1 |
+
\lnot x \lor \lnot y
−
|      |      |          |                                              |
+
\end{matrix}</math>
−
| f_11 | 1011 |  (x (y)) |            1  1  1  1              1  1  1  1 |
+
|-
−
|      |      |          |                                              |
+
|
−
| f_12 | 1100 |  x      |        1          1          1          1 |
+
<math>\begin{matrix}
−
|      |      |          |                                              |
+
f_8
−
| f_13 | 1101 | ((x) y)  |      1  1        1  1        1  1        1  1 |
+
\\[4pt]
−
|      |      |          |                                              |
+
f_9
−
| f_14 | 1110 | ((x)(y)) |  1    1    1    1    1    1    1    1 |
+
\\[4pt]
−
|      |      |          |                                              |
+
f_{10}
−
| f_15 | 1111 |  (())  |1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 |
+
\\[4pt]
−
|      |      |          |                                              |
+
f_{11}
−
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
+
\\[4pt]
 +
f_{12}
 +
\\[4pt]
 +
f_{13}
 +
\\[4pt]
 +
f_{14}
 +
\\[4pt]
 +
f_{15}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{1000}
 +
\\[4pt]
 +
f_{1001}
 +
\\[4pt]
 +
f_{1010}
 +
\\[4pt]
 +
f_{1011}
 +
\\[4pt]
 +
f_{1100}
 +
\\[4pt]
 +
f_{1101}
 +
\\[4pt]
 +
f_{1110}
 +
\\[4pt]
 +
f_{1111}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
1~0~0~0
 +
\\[4pt]
 +
1~0~0~1
 +
\\[4pt]
 +
1~0~1~0
 +
\\[4pt]
 +
1~0~1~1
 +
\\[4pt]
 +
1~1~0~0
 +
\\[4pt]
 +
1~1~0~1
 +
\\[4pt]
 +
1~1~1~0
 +
\\[4pt]
 +
1~1~1~1
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~~x~~y~~
 +
\\[4pt]
 +
((x,~y))
 +
\\[4pt]
 +
~~~~~y~~
 +
\\[4pt]
 +
~(x~(y))
 +
\\[4pt]
 +
~~x~~~~~
 +
\\[4pt]
 +
((x)~y)~
 +
\\[4pt]
 +
((x)(y))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
x ~\text{and}~ y
 +
\\[4pt]
 +
x ~\text{equal to}~ y
 +
\\[4pt]
 +
y
 +
\\[4pt]
 +
\text{not}~ x ~\text{without}~ y
 +
\\[4pt]
 +
x
 +
\\[4pt]
 +
\text{not}~ y ~\text{without}~ x
 +
\\[4pt]
 +
x ~\text{or}~ y
 +
\\[4pt]
 +
\text{true}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
x \land y
 +
\\[4pt]
 +
x = y
 +
\\[4pt]
 +
y
 +
\\[4pt]
 +
x \Rightarrow y
 +
\\[4pt]
 +
x
 +
\\[4pt]
 +
x \Leftarrow y
 +
\\[4pt]
 +
x \lor y
 +
\\[4pt]
 +
1
 +
\end{matrix}</math>
 +
|}
 +
 
 +
<br>
 +
 
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 +
|+ <math>\text{Table A2.}~~\text{Propositional Forms on Two Variables}</math>
 +
|- style="background:#f0f0ff"
 +
| width="15%" |
 +
<p><math>\mathcal{L}_1</math></p>
 +
<p><math>\text{Decimal}</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_2</math></p>
 +
<p><math>\text{Binary}</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_3</math></p>
 +
<p><math>\text{Vector}</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_4</math></p>
 +
<p><math>\text{Cactus}</math></p>
 +
| width="25%" |
 +
<p><math>\mathcal{L}_5</math></p>
 +
<p><math>\text{English}</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_6</math></p>
 +
<p><math>\text{Ordinary}</math></p>
 +
|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | <math>x\colon\!</math>
 +
| <math>1~1~0~0\!</math>
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | <math>y\colon\!</math>
 +
| <math>1~0~1~0\!</math>
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
|-
 +
| <math>f_0\!</math>
 +
| <math>f_{0000}\!</math>
 +
| <math>0~0~0~0</math>
 +
| <math>(~)</math>
 +
| <math>\text{false}\!</math>
 +
| <math>0\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_1
 +
\\[4pt]
 +
f_2
 +
\\[4pt]
 +
f_4
 +
\\[4pt]
 +
f_8
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0001}
 +
\\[4pt]
 +
f_{0010}
 +
\\[4pt]
 +
f_{0100}
 +
\\[4pt]
 +
f_{1000}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0~0~0~1
 +
\\[4pt]
 +
0~0~1~0
 +
\\[4pt]
 +
0~1~0~0
 +
\\[4pt]
 +
1~0~0~0
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(x)(y)
 +
\\[4pt]
 +
(x)~y~
 +
\\[4pt]
 +
~x~(y)
 +
\\[4pt]
 +
~x~~y~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{neither}~ x ~\text{nor}~ y
 +
\\[4pt]
 +
y ~\text{without}~ x
 +
\\[4pt]
 +
x ~\text{without}~ y
 +
\\[4pt]
 +
x ~\text{and}~ y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\lnot x \land \lnot y
 +
\\[4pt]
 +
\lnot x \land y
 +
\\[4pt]
 +
x \land \lnot y
 +
\\[4pt]
 +
x \land y
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_3
 +
\\[4pt]
 +
f_{12}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0011}
 +
\\[4pt]
 +
f_{1100}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0~0~1~1
 +
\\[4pt]
 +
1~1~0~0
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(x)
 +
\\[4pt]
 +
~x~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{not}~ x
 +
\\[4pt]
 +
x
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\lnot x
 +
\\[4pt]
 +
x
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_6
 +
\\[4pt]
 +
f_9
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0110}
 +
\\[4pt]
 +
f_{1001}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0~1~1~0
 +
\\[4pt]
 +
1~0~0~1
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(x,~y)~
 +
\\[4pt]
 +
((x,~y))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
x ~\text{not equal to}~ y
 +
\\[4pt]
 +
x ~\text{equal to}~ y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
x \ne y
 +
\\[4pt]
 +
x = y
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_5
 +
\\[4pt]
 +
f_{10}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0101}
 +
\\[4pt]
 +
f_{1010}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0~1~0~1
 +
\\[4pt]
 +
1~0~1~0
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(y)
 +
\\[4pt]
 +
~y~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{not}~ y
 +
\\[4pt]
 +
y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\lnot y
 +
\\[4pt]
 +
y
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_7
 +
\\[4pt]
 +
f_{11}
 +
\\[4pt]
 +
f_{13}
 +
\\[4pt]
 +
f_{14}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0111}
 +
\\[4pt]
 +
f_{1011}
 +
\\[4pt]
 +
f_{1101}
 +
\\[4pt]
 +
f_{1110}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0~1~1~1
 +
\\[4pt]
 +
1~0~1~1
 +
\\[4pt]
 +
1~1~0~1
 +
\\[4pt]
 +
1~1~1~0
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(x~~y)~
 +
\\[4pt]
 +
~(x~(y))
 +
\\[4pt]
 +
((x)~y)~
 +
\\[4pt]
 +
((x)(y))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{not both}~ x ~\text{and}~ y
 +
\\[4pt]
 +
\text{not}~ x ~\text{without}~ y
 +
\\[4pt]
 +
\text{not}~ y ~\text{without}~ x
 +
\\[4pt]
 +
x ~\text{or}~ y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\lnot x \lor \lnot y
 +
\\[4pt]
 +
x \Rightarrow y
 +
\\[4pt]
 +
x \Leftarrow y
 +
\\[4pt]
 +
x \lor y
 +
\end{matrix}</math>
 +
|-
 +
| <math>f_{15}\!</math>
 +
| <math>f_{1111}\!</math>
 +
| <math>1~1~1~1</math>
 +
| <math>((~))</math>
 +
| <math>\text{true}\!</math>
 +
| <math>1\!</math>
 +
|}
 +
 
 
<br>
 
<br>
  
−
Table 11. Qualifiers of Implication Ordering:  !b!_i f =  !Y!(f => f_i)
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
−
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
+
|+ <math>\text{Table A3.}~~\operatorname{E}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}x, \operatorname{d}y \}</math>
−
| | x | 1100 |   f     |b |b |b |b |b |b |b |b |b |b |b |b |b |b |b |b |
+
|- style="background:#f0f0ff"
−
|  | y | 1010 |          |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |1 |1 |1 |1 |1 |1 |
+
| width="10%" | &nbsp;
−
| f \ |     |         |0 |1 |2 |3 |4 |5 |6 |7 |8 |9 |0 |1 |2 |3 |4 |5 |
+
| width="18%" | <math>f\!</math>
−
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
+
| width="18%" |  
−
|     |      |          |                                              |
+
<p><math>\operatorname{T}_{11} f</math></p>
−
| f_0 | 0000 |   ()   |1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 |
+
<p><math>\operatorname{E}f|_{\operatorname{d}x~\operatorname{d}y}</math></p>
−
|     |     |          |                                              |
+
| width="18%" |
−
| f_1 | 0001 | (x)(y) |   1    1    1    1    1    1    1    1 |
+
<p><math>\operatorname{T}_{10} f</math></p>
−
|      |      |          |                                              |
+
<p><math>\operatorname{E}f|_{\operatorname{d}x(\operatorname{d}y)}</math></p>
−
| f_2  | 0010 |  (x) y   |     1  1        1  1        1  1        1  1 |
+
| width="18%" |
−
|      |      |          |                                              |
+
<p><math>\operatorname{T}_{01} f</math></p>
−
| f_3 | 0011 | (x)     |         1          1          1          1 |
+
<p><math>\operatorname{E}f|_{(\operatorname{d}x)\operatorname{d}y}</math></p>
−
|     |     |         |                                               |
+
| width="18%" |
−
| f_4  | 0100 |  x (y) |            1  1  1  1              1  1  1  1 |
+
<p><math>\operatorname{T}_{00} f</math></p>
−
|      |      |          |                                              |
+
<p><math>\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}</math></p>
−
| f_5  | 0101 |    (y) |              1    1                1    1 |
+
|-
−
|      |      |          |                                              |
+
| <math>f_0\!</math>
−
| f_6  | 0110 |  (x, y) |                  1  1                    1  1 |
+
| <math>(~)</math>
−
|      |      |          |                                              |
+
| <math>(~)</math>
−
| f_7  | 0111 |  (x y) |                    1                      1 |
+
| <math>(~)</math>
−
|      |      |          |                                              |
+
| <math>(~)</math>
−
| f_8  | 1000 |  x y   |                        1  1  1  1  1  1  1  1 |
+
| <math>(~)</math>
−
|      |      |          |                                              |
+
|-
−
| f_9  | 1001 | ((x, y)) |                           1    1    1    1 |
+
|
−
|     |     |         |                                               |
+
<math>\begin{matrix}
−
| f_10 | 1010 |     y   |                              1  1        1  1 |
+
f_1
−
|      |      |          |                                              |
+
\\[4pt]
−
| f_11 | 1011 |  (x (y)) |                                1          1 |
+
f_2
−
|      |      |          |                                              |
+
\\[4pt]
−
| f_12 | 1100 |  x     |                                    1  1  1  1 |
+
f_4
−
|      |      |          |                                              |
+
\\[4pt]
−
| f_13 | 1101 | ((x) y) |                                      1    1 |
+
f_8
−
|      |      |          |                                              |
+
\end{matrix}</math>
−
| f_14 | 1110 | ((x)(y)) |                                          1  1 |
+
|
−
|      |      |          |                                              |
+
<math>\begin{matrix}
−
| f_15 | 1111 |  (())   |                                            1 |
+
(x)(y)
−
|      |      |         |                                              |
+
\\[4pt]
−
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
+
(x)~y~
−
<br>
+
\\[4pt]
−
 
+
~x~(y)
−
Table 13.  Syllogistic Premisses as Higher Order Indicator Functions
+
\\[4pt]
−
o---o------------------------o-----------------o---------------------------o
+
~x~~y~
−
|  |                        |                |                          |
+
\end{matrix}</math>
−
| A | Universal Affirmative  | All  x is  y | Indicator of " x (y)" = 0 |
+
|
−
|  |                        |                |                          |
+
<math>\begin{matrix}
−
| E | Universal Negative    | All  x is (y) | Indicator of " x y " = 0 |
+
~x~~y~
−
|  |                        |                |                          |
+
\\[4pt]
−
| I | Particular Affirmative | Some  x is  y | Indicator of " x y " = 1 |
+
~x~(y)
−
|   |                       |                 |                           |
+
\\[4pt]
−
| O | Particular Negative    | Some  x  is (y) | Indicator of " x (y)" = 1 |
+
(x)~y~
−
|   |                       |                 |                           |
+
\\[4pt]
−
o---o------------------------o-----------------o---------------------------o
+
(x)(y)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~x~(y)
 +
\\[4pt]
 +
~x~~y~
 +
\\[4pt]
 +
(x)(y)
 +
\\[4pt]
 +
(x)~y~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(x)~y~
 +
\\[4pt]
 +
(x)(y)
 +
\\[4pt]
 +
~x~~y~
 +
\\[4pt]
 +
~x~(y)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(x)(y)
 +
\\[4pt]
 +
(x)~y~
 +
\\[4pt]
 +
~x~(y)
 +
\\[4pt]
 +
~x~~y~
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_3
 +
\\[4pt]
 +
f_{12}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(x)
 +
\\[4pt]
 +
~x~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~x~
 +
\\[4pt]
 +
(x)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~x~
 +
\\[4pt]
 +
(x)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(x)
 +
\\[4pt]
 +
~x~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(x)
 +
\\[4pt]
 +
~x~
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_6
 +
\\[4pt]
 +
f_9
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(x,~y)~
 +
\\[4pt]
 +
((x,~y))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(x,~y)~
 +
\\[4pt]
 +
((x,~y))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((x,~y))
 +
\\[4pt]
 +
~(x,~y)~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((x,~y))
 +
\\[4pt]
 +
~(x,~y)~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(x,~y)~
 +
\\[4pt]
 +
((x,~y))
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_5
 +
\\[4pt]
 +
f_{10}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(y)
 +
\\[4pt]
 +
~y~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~y~
 +
\\[4pt]
 +
(y)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(y)
 +
\\[4pt]
 +
~y~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~y~
 +
\\[4pt]
 +
(y)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(y)
 +
\\[4pt]
 +
~y~
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_7
 +
\\[4pt]
 +
f_{11}
 +
\\[4pt]
 +
f_{13}
 +
\\[4pt]
 +
f_{14}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~x~~y~)
 +
\\[4pt]
 +
(~x~(y))
 +
\\[4pt]
 +
((x)~y~)
 +
\\[4pt]
 +
((x)(y))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((x)(y))
 +
\\[4pt]
 +
((x)~y~)
 +
\\[4pt]
 +
(~x~(y))
 +
\\[4pt]
 +
(~x~~y~)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((x)~y~)
 +
\\[4pt]
 +
((x)(y))
 +
\\[4pt]
 +
(~x~~y~)
 +
\\[4pt]
 +
(~x~(y))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~x~(y))
 +
\\[4pt]
 +
(~x~~y~)
 +
\\[4pt]
 +
((x)(y))
 +
\\[4pt]
 +
((x)~y~)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~x~~y~)
 +
\\[4pt]
 +
(~x~(y))
 +
\\[4pt]
 +
((x)~y~)
 +
\\[4pt]
 +
((x)(y))
 +
\end{matrix}</math>
 +
|-
 +
| <math>f_{15}\!</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
|- style="background:#f0f0ff"
 +
| colspan="2" | <math>\text{Fixed Point Total}\!</math>
 +
| <math>4\!</math>
 +
| <math>4\!</math>
 +
| <math>4\!</math>
 +
| <math>16\!</math>
 +
|}
 +
 
 
<br>
 
<br>
  
−
Table 14. Relation of Quantifiers to Higher Order Propositions
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
−
o------------o------------o-----------o-----------o-----------o-----------o
+
|+ <math>\text{Table A4.}~~\operatorname{D}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}x, \operatorname{d}y \}</math>
−
| Mnemonic  | Category  | Classical | Alternate | Symmetric | Operator  |
+
|- style="background:#f0f0ff"
−
|           |            |  Form    |  Form    |  Form    |          |
+
| width="10%" | &nbsp;
−
o============o============o===========o===========o===========o===========o
+
| width="18%" | <math>f\!</math>
−
|     E      | Universal  |  All  x |           |   No  x  | (L_11)   |
+
| width="18%" |
−
| Exclusive  | Negative  |   is  (y) |           |  is  y  |          |
+
<math>\operatorname{D}f|_{\operatorname{d}x~\operatorname{d}y}</math>
−
o------------o------------o-----------o-----------o-----------o-----------o
+
| width="18%" |
−
|     A      | Universal  |  All  x  |          |  No  x  |  (L_10)   |
+
<math>\operatorname{D}f|_{\operatorname{d}x(\operatorname{d}y)}</math>
−
| Absolute  |  Affrmtve  |  is  y  |          |  is  (y) |          |
+
| width="18%" |
−
o------------o------------o-----------o-----------o-----------o-----------o
+
<math>\operatorname{D}f|_{(\operatorname{d}x)\operatorname{d}y}</math>
−
|           |            |  All  y  |  No  y  |  No  (x) | (L_01)   |
+
| width="18%" |
−
|           |            |  is  x  |  is  (x) |   is  y  |           |
+
<math>\operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)}</math>
−
o------------o------------o-----------o-----------o-----------o-----------o
+
|-
−
|            |            |  All  (y) |  No  (y) |  No  (x) |  (L_00)   |
+
| <math>f_0\!</math>
−
|            |            |  is  x |   is  (x) |  is  (y) |          |
+
| <math>(~)</math>
−
o------------o------------o-----------o-----------o-----------o-----------o
+
| <math>(~)</math>
−
|            |            | Some  (x) |          | Some  (x) |   L_00    |
+
| <math>(~)</math>
−
|            |            |  is  (y) |          |  is  (y) |          |
+
| <math>(~)</math>
−
o------------o------------o-----------o-----------o-----------o-----------o
+
| <math>(~)</math>
−
|            |            | Some  (x) |          | Some  (x) |  L_01    |
+
|-
−
|            |            |  is  y  |          |  is  y  |          |
+
|
−
o------------o------------o-----------o-----------o-----------o-----------o
+
<math>\begin{matrix}
−
|    O      | Particular | Some  x |          | Some  x  |  L_10    |
+
f_1
−
| Obtrusive  |  Negative  |  is  (y) |          |  is  (y) |           |
+
\\[4pt]
−
o------------o------------o-----------o-----------o-----------o-----------o
+
f_2
−
|     I      | Particular | Some  x |          | Some  x |  L_11    |
+
\\[4pt]
−
| Indefinite |  Affrmtve  |  is  y  |          |  is  y  |          |
+
f_4
−
o------------o------------o-----------o-----------o-----------o-----------o
+
\\[4pt]
−
<br>
+
f_8
−
 
+
\end{matrix}</math>
−
Table 15.  Simple Qualifiers of Propositions (n = 2)
+
|
−
o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o
+
<math>\begin{matrix}
−
|  | x | 1100 |    f    |(L11)|(L10)|(L01)|(L00)| L00 | L01 | L10 | L11 |
+
(x)(y)
−
|  | y | 1010 |          |no  x|no  x|no ~x|no ~x|sm ~x|sm ~x|sm  x|sm  x|
+
\\[4pt]
−
| f \ |      |          |is  y|is ~y|is  y|is ~y|is ~y|is  y|is ~y|is  y|
+
(x)~y~
−
o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o
+
\\[4pt]
−
|      |      |          |                                              |
+
~x~(y)
−
| f_0  | 0000 |    ()   |  1    1    1    1    0    0    0    0  |
+
\\[4pt]
−
|      |      |          |                                              |
+
~x~~y~
−
| f_1  | 0001 |  (x)(y) | 1    1    1    0    1    0    0    0  |
+
\end{matrix}</math>
−
|      |      |          |                                              |
+
|
−
| f_2  | 0010 |  (x) y  |  1    1    0    1    0    1    0    0  |
+
<math>\begin{matrix}
−
|      |      |          |                                              |
+
((x,~y))
−
| f_3  | 0011 |  (x)     |  1    1    0    0    1    1    0    0  |
+
\\[4pt]
−
|      |      |          |                                              |
+
~(x,~y)~
−
| f_4  | 0100 |  x (y) |  1    0    1    1    0    0    1    0  |
+
\\[4pt]
−
|      |      |          |                                              |
+
~(x,~y)~
−
| f_5  | 0101 |    (y) |  1    0    1    0    1    0    1    0  |
+
\\[4pt]
−
|      |      |          |                                              |
+
((x,~y))
−
| f_6  | 0110 |  (x, y) |  1    0    0    1    0    1    1    0  |
+
\end{matrix}</math>
−
|      |      |          |                                              |
+
|
−
| f_7  | 0111 |  (x y) |  1    0    0    0    1    1    1    0  |
+
<math>\begin{matrix}
−
|      |      |          |                                              |
+
(y)
−
| f_8  | 1000 |  x y   |  0    1    1    1    0    0    0    1  |
+
\\[4pt]
−
|      |      |          |                                              |
+
~y~
−
| f_9  | 1001 | ((x, y)) |  0    1    1    0    1    0    0    1  |
+
\\[4pt]
−
|      |      |          |                                              |
+
(y)
−
| f_10 | 1010 |      y   |  0    1    0    1    0    1    0    1  |
+
\\[4pt]
−
|      |      |          |                                              |
+
~y~
−
| f_11 | 1011 |  (x (y)) |  0    1    0    0    1    1    0    1  |
+
\end{matrix}</math>
−
|      |      |          |                                              |
+
|
−
| f_12 | 1100 |  x     |  0    0    1    1    0    0    1    1  |
+
<math>\begin{matrix}
−
|      |      |          |                                              |
+
(x)
−
| f_13 | 1101 | ((x) y)  |  0    0    1    0    1    0    1    1  |
+
\\[4pt]
−
|      |      |          |                                              |
+
(x)
−
| f_14 | 1110 | ((x)(y)) | 0    0    0    1    0    1    1    1  |
+
\\[4pt]
−
|      |      |          |                                              |
+
~x~
−
| f_15 | 1111 |  (())   | 0    0    0    0    1    1    1    1  |
+
\\[4pt]
−
|     |     |          |                                              |
+
~x~
−
o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o
+
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_3
 +
\\[4pt]
 +
f_{12}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(x)
 +
\\[4pt]
 +
~x~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((~))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((~))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_6
 +
\\[4pt]
 +
f_9
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(x,~y)~
 +
\\[4pt]
 +
((x,~y))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((~))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((~))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_5
 +
\\[4pt]
 +
f_{10}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(y)
 +
\\[4pt]
 +
~y~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((~))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((~))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_7
 +
\\[4pt]
 +
f_{11}
 +
\\[4pt]
 +
f_{13}
 +
\\[4pt]
 +
f_{14}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(x~~y)~
 +
\\[4pt]
 +
~(x~(y))
 +
\\[4pt]
 +
((x)~y)~
 +
\\[4pt]
 +
((x)(y))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((x,~y))
 +
\\[4pt]
 +
~(x,~y)~
 +
\\[4pt]
 +
~(x,~y)~
 +
\\[4pt]
 +
((x,~y))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~y~
 +
\\[4pt]
 +
(y)
 +
\\[4pt]
 +
~y~
 +
\\[4pt]
 +
(y)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~x~
 +
\\[4pt]
 +
~x~
 +
\\[4pt]
 +
(x)
 +
\\[4pt]
 +
(x)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}</math>
 +
|-
 +
| <math>f_{15}\!</math>
 +
| <math>((~))</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
|}
 +
 
 
<br>
 
<br>
  
−
===[[Zeroth Order Logic]]===
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
−
 
+
|+ <math>\text{Table A5.}~~\operatorname{E}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math>
−
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+
|- style="background:#f0f0ff"
−
|+ '''Table 1. Propositional Forms on Two Variables'''
+
| width="10%" | &nbsp;
−
|- style="background:paleturquoise"
+
| width="18%" | <math>f\!</math>
−
! style="width:15%" | L<sub>1</sub>
+
| width="18%" | <math>\operatorname{E}f|_{xy}</math>
−
! style="width:15%" | L<sub>2</sub>
+
| width="18%" | <math>\operatorname{E}f|_{x(y)}</math>
−
! style="width:15%" | L<sub>3</sub>
+
| width="18%" | <math>\operatorname{E}f|_{(x)y}</math>
−
! style="width:15%" | L<sub>4</sub>
+
| width="18%" | <math>\operatorname{E}f|_{(x)(y)}</math>
−
! 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
+
| <math>f_0\!</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 
|-
 
|-
−
| f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || &not;x &and; &not;y
+
|
 +
<math>\begin{matrix}
 +
f_1
 +
\\[4pt]
 +
f_2
 +
\\[4pt]
 +
f_4
 +
\\[4pt]
 +
f_8
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(x)(y)
 +
\\[4pt]
 +
(x)~y~
 +
\\[4pt]
 +
~x~(y)
 +
\\[4pt]
 +
~x~~y~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~\operatorname{d}x~~\operatorname{d}y~
 +
\\[4pt]
 +
~\operatorname{d}x~(\operatorname{d}y)
 +
\\[4pt]
 +
(\operatorname{d}x)~\operatorname{d}y~
 +
\\[4pt]
 +
(\operatorname{d}x)(\operatorname{d}y)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~\operatorname{d}x~(\operatorname{d}y)
 +
\\[4pt]
 +
~\operatorname{d}x~~\operatorname{d}y~
 +
\\[4pt]
 +
(\operatorname{d}x)(\operatorname{d}y)
 +
\\[4pt]
 +
(\operatorname{d}x)~\operatorname{d}y~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(\operatorname{d}x)~\operatorname{d}y~
 +
\\[4pt]
 +
(\operatorname{d}x)(\operatorname{d}y)
 +
\\[4pt]
 +
~\operatorname{d}x~~\operatorname{d}y~
 +
\\[4pt]
 +
~\operatorname{d}x~(\operatorname{d}y)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(\operatorname{d}x)(\operatorname{d}y)
 +
\\[4pt]
 +
(\operatorname{d}x)~\operatorname{d}y~
 +
\\[4pt]
 +
~\operatorname{d}x~(\operatorname{d}y)
 +
\\[4pt]
 +
~\operatorname{d}x~~\operatorname{d}y~
 +
\end{matrix}</math>
 
|-
 
|-
−
| f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || &not;x &and; y
+
|
 +
<math>\begin{matrix}
 +
f_3
 +
\\[4pt]
 +
f_{12}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(x)
 +
\\[4pt]
 +
~x~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~\operatorname{d}x~
 +
\\[4pt]
 +
(\operatorname{d}x)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~\operatorname{d}x~
 +
\\[4pt]
 +
(\operatorname{d}x)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(\operatorname{d}x)
 +
\\[4pt]
 +
~\operatorname{d}x~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(\operatorname{d}x)
 +
\\[4pt]
 +
~\operatorname{d}x~
 +
\end{matrix}</math>
 
|-
 
|-
−
| f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || &not;x
+
|
 +
<math>\begin{matrix}
 +
f_6
 +
\\[4pt]
 +
f_9
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(x,~y)~
 +
\\[4pt]
 +
((x,~y))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(\operatorname{d}x,~\operatorname{d}y)~
 +
\\[4pt]
 +
((\operatorname{d}x,~\operatorname{d}y))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((\operatorname{d}x,~\operatorname{d}y))
 +
\\[4pt]
 +
~(\operatorname{d}x,~\operatorname{d}y)~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((\operatorname{d}x,~\operatorname{d}y))
 +
\\[4pt]
 +
~(\operatorname{d}x,~\operatorname{d}y)~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(\operatorname{d}x,~\operatorname{d}y)~
 +
\\[4pt]
 +
((\operatorname{d}x,~\operatorname{d}y))
 +
\end{matrix}</math>
 
|-
 
|-
−
| f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x &and; &not;y
+
|
 +
<math>\begin{matrix}
 +
f_5
 +
\\[4pt]
 +
f_{10}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(y)
 +
\\[4pt]
 +
~y~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~\operatorname{d}y~
 +
\\[4pt]
 +
(\operatorname{d}y)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(\operatorname{d}y)
 +
\\[4pt]
 +
~\operatorname{d}y~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~\operatorname{d}y~
 +
\\[4pt]
 +
(\operatorname{d}y)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(\operatorname{d}y)
 +
\\[4pt]
 +
~\operatorname{d}y~
 +
\end{matrix}</math>
 
|-
 
|-
−
| f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || &not;y
+
|
 +
<math>\begin{matrix}
 +
f_7
 +
\\[4pt]
 +
f_{11}
 +
\\[4pt]
 +
f_{13}
 +
\\[4pt]
 +
f_{14}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~x~~y~)
 +
\\[4pt]
 +
(~x~(y))
 +
\\[4pt]
 +
((x)~y~)
 +
\\[4pt]
 +
((x)(y))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((\operatorname{d}x)(\operatorname{d}y))
 +
\\[4pt]
 +
((\operatorname{d}x)~\operatorname{d}y~)
 +
\\[4pt]
 +
(~\operatorname{d}x~(\operatorname{d}y))
 +
\\[4pt]
 +
(~\operatorname{d}x~~\operatorname{d}y~)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((\operatorname{d}x)~\operatorname{d}y~)
 +
\\[4pt]
 +
((\operatorname{d}x)(\operatorname{d}y))
 +
\\[4pt]
 +
(~\operatorname{d}x~~\operatorname{d}y~)
 +
\\[4pt]
 +
(~\operatorname{d}x~(\operatorname{d}y))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~\operatorname{d}x~(\operatorname{d}y))
 +
\\[4pt]
 +
(~\operatorname{d}x~~\operatorname{d}y~)
 +
\\[4pt]
 +
((\operatorname{d}x)(\operatorname{d}y))
 +
\\[4pt]
 +
((\operatorname{d}x)~\operatorname{d}y~)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~\operatorname{d}x~~\operatorname{d}y~)
 +
\\[4pt]
 +
(~\operatorname{d}x~(\operatorname{d}y))
 +
\\[4pt]
 +
((\operatorname{d}x)~\operatorname{d}y~)
 +
\\[4pt]
 +
((\operatorname{d}x)(\operatorname{d}y))
 +
\end{matrix}</math>
 
|-
 
|-
−
| f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x &ne; y
+
| <math>f_{15}\!</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
|}
 +
 
 +
<br>
 +
 
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 +
|+ <math>\text{Table A6.}~~\operatorname{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math>
 +
|- style="background:#f0f0ff"
 +
| width="10%" | &nbsp;
 +
| width="18%" | <math>f\!</math>
 +
| width="18%" | <math>\operatorname{D}f|_{xy}</math>
 +
| width="18%" | <math>\operatorname{D}f|_{x(y)}</math>
 +
| width="18%" | <math>\operatorname{D}f|_{(x)y}</math>
 +
| width="18%" | <math>\operatorname{D}f|_{(x)(y)}</math>
 
|-
 
|-
−
| 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
+
| <math>f_0\!</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 
|-
 
|-
−
| f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x&nbsp;y || x and y || x &and; y
+
|
−
|-
+
<math>\begin{matrix}
−
| f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y
+
f_1
 +
\\[4pt]
 +
f_2
 +
\\[4pt]
 +
f_4
 +
\\[4pt]
 +
f_8
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(x)(y)
 +
\\[4pt]
 +
(x)~y~
 +
\\[4pt]
 +
~x~(y)
 +
\\[4pt]
 +
~x~~y~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~~\operatorname{d}x~~\operatorname{d}y~~
 +
\\[4pt]
 +
~~\operatorname{d}x~(\operatorname{d}y)~
 +
\\[4pt]
 +
~(\operatorname{d}x)~\operatorname{d}y~~
 +
\\[4pt]
 +
((\operatorname{d}x)(\operatorname{d}y))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~~\operatorname{d}x~(\operatorname{d}y)~
 +
\\[4pt]
 +
~~\operatorname{d}x~~\operatorname{d}y~~
 +
\\[4pt]
 +
((\operatorname{d}x)(\operatorname{d}y))
 +
\\[4pt]
 +
~(\operatorname{d}x)~\operatorname{d}y~~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(\operatorname{d}x)~\operatorname{d}y~~
 +
\\[4pt]
 +
((\operatorname{d}x)(\operatorname{d}y))
 +
\\[4pt]
 +
~~\operatorname{d}x~~\operatorname{d}y~~
 +
\\[4pt]
 +
~~\operatorname{d}x~(\operatorname{d}y)~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((\operatorname{d}x)(\operatorname{d}y))
 +
\\[4pt]
 +
~(\operatorname{d}x)~\operatorname{d}y~~
 +
\\[4pt]
 +
~~\operatorname{d}x~(\operatorname{d}y)~
 +
\\[4pt]
 +
~~\operatorname{d}x~~\operatorname{d}y~~
 +
\end{matrix}</math>
 
|-
 
|-
−
| f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y
+
|
 +
<math>\begin{matrix}
 +
f_3
 +
\\[4pt]
 +
f_{12}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(x)
 +
\\[4pt]
 +
~x~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{d}x
 +
\\[4pt]
 +
\operatorname{d}x
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{d}x
 +
\\[4pt]
 +
\operatorname{d}x
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{d}x
 +
\\[4pt]
 +
\operatorname{d}x
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{d}x
 +
\\[4pt]
 +
\operatorname{d}x
 +
\end{matrix}</math>
 
|-
 
|-
−
| f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x &rarr; y
+
|
 +
<math>\begin{matrix}
 +
f_6
 +
\\[4pt]
 +
f_9
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(x,~y)~
 +
\\[4pt]
 +
((x,~y))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(\operatorname{d}x,~\operatorname{d}y)
 +
\\[4pt]
 +
(\operatorname{d}x,~\operatorname{d}y)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(\operatorname{d}x,~\operatorname{d}y)
 +
\\[4pt]
 +
(\operatorname{d}x,~\operatorname{d}y)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(\operatorname{d}x,~\operatorname{d}y)
 +
\\[4pt]
 +
(\operatorname{d}x,~\operatorname{d}y)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(\operatorname{d}x,~\operatorname{d}y)
 +
\\[4pt]
 +
(\operatorname{d}x,~\operatorname{d}y)
 +
\end{matrix}</math>
 
|-
 
|-
−
| f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x
+
|
 +
<math>\begin{matrix}
 +
f_5
 +
\\[4pt]
 +
f_{10}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(y)
 +
\\[4pt]
 +
~y~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{d}y
 +
\\[4pt]
 +
\operatorname{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{d}y
 +
\\[4pt]
 +
\operatorname{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{d}y
 +
\\[4pt]
 +
\operatorname{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{d}y
 +
\\[4pt]
 +
\operatorname{d}y
 +
\end{matrix}</math>
 
|-
 
|-
−
| f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x &larr; y
+
|
 +
<math>\begin{matrix}
 +
f_7
 +
\\[4pt]
 +
f_{11}
 +
\\[4pt]
 +
f_{13}
 +
\\[4pt]
 +
f_{14}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~x~~y~)
 +
\\[4pt]
 +
(~x~(y))
 +
\\[4pt]
 +
((x)~y~)
 +
\\[4pt]
 +
((x)(y))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((\operatorname{d}x)(\operatorname{d}y))
 +
\\[4pt]
 +
~(\operatorname{d}x)~\operatorname{d}y~~
 +
\\[4pt]
 +
~~\operatorname{d}x~(\operatorname{d}y)~
 +
\\[4pt]
 +
~~\operatorname{d}x~~\operatorname{d}y~~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(\operatorname{d}x)~\operatorname{d}y~~
 +
\\[4pt]
 +
((\operatorname{d}x)(\operatorname{d}y))
 +
\\[4pt]
 +
~~\operatorname{d}x~~\operatorname{d}y~~
 +
\\[4pt]
 +
~~\operatorname{d}x~(\operatorname{d}y)~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~~\operatorname{d}x~(\operatorname{d}y)~
 +
\\[4pt]
 +
~~\operatorname{d}x~~\operatorname{d}y~~
 +
\\[4pt]
 +
((\operatorname{d}x)(\operatorname{d}y))
 +
\\[4pt]
 +
~(\operatorname{d}x)~\operatorname{d}y~~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~~\operatorname{d}x~~\operatorname{d}y~~
 +
\\[4pt]
 +
~~\operatorname{d}x~(\operatorname{d}y)~
 +
\\[4pt]
 +
~(\operatorname{d}x)~\operatorname{d}y~~
 +
\\[4pt]
 +
((\operatorname{d}x)(\operatorname{d}y))
 +
\end{matrix}</math>
 
|-
 
|-
−
| f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y  || x &or; y
+
| <math>f_{15}\!</math>
−
|-
+
| <math>((~))</math>
−
| f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || ((&nbsp;)) || true || 1
+
| <math>((~))</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 
|}
 
|}
 +
 
<br>
 
<br>
  
−
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:90%"
+
===Klein Four-Group V<sub>4</sub>===
−
|+ '''Table 1.  Propositional Forms on Two Variables'''
+
 
−
|- style="background:aliceblue"
+
<br>
−
! style="width:15%" | L<sub>1</sub>
+
 
−
! style="width:15%" | L<sub>2</sub>
+
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
−
! style="width:15%" | L<sub>3</sub>
+
|- style="height:50px"
−
! style="width:15%" | L<sub>4</sub>
+
| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
−
! style="width:15%" | L<sub>5</sub>
+
| width="22%" style="border-bottom:1px solid black" |
−
! style="width:15%" | L<sub>6</sub>
+
<math>\operatorname{T}_{00}</math>
−
|- style="background:aliceblue"
+
| width="22%" style="border-bottom:1px solid black" |
−
| &nbsp;
+
<math>\operatorname{T}_{01}</math>
−
| align="right" | x :
+
| width="22%" style="border-bottom:1px solid black" |
−
| 1 1 0 0
+
<math>\operatorname{T}_{10}</math>
−
| &nbsp;
+
| width="22%" style="border-bottom:1px solid black" |
−
| &nbsp;
+
<math>\operatorname{T}_{11}</math>
−
| &nbsp;
+
|- style="height:50px"
−
|- style="background:aliceblue"
+
| style="border-right:1px solid black" | <math>\operatorname{T}_{00}</math>
−
| &nbsp;
+
| <math>\operatorname{T}_{00}</math>
−
| align="right" | y :
+
| <math>\operatorname{T}_{01}</math>
−
| 1 0 1 0
+
| <math>\operatorname{T}_{10}</math>
−
| &nbsp;
+
| <math>\operatorname{T}_{11}</math>
−
| &nbsp;
+
|- style="height:50px"
−
| &nbsp;
+
| style="border-right:1px solid black" | <math>\operatorname{T}_{01}</math>
 +
| <math>\operatorname{T}_{01}</math>
 +
| <math>\operatorname{T}_{00}</math>
 +
| <math>\operatorname{T}_{11}</math>
 +
| <math>\operatorname{T}_{10}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{T}_{10}</math>
 +
| <math>\operatorname{T}_{10}</math>
 +
| <math>\operatorname{T}_{11}</math>
 +
| <math>\operatorname{T}_{00}</math>
 +
| <math>\operatorname{T}_{01}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{T}_{11}</math>
 +
| <math>\operatorname{T}_{11}</math>
 +
| <math>\operatorname{T}_{10}</math>
 +
| <math>\operatorname{T}_{01}</math>
 +
| <math>\operatorname{T}_{00}</math>
 +
|}
 +
 
 +
<br>
 +
 
 +
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
 +
|- style="height:50px"
 +
| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
 +
| width="22%" style="border-bottom:1px solid black" |
 +
<math>\operatorname{e}</math>
 +
| width="22%" style="border-bottom:1px solid black" |
 +
<math>\operatorname{f}</math>
 +
| width="22%" style="border-bottom:1px solid black" |
 +
<math>\operatorname{g}</math>
 +
| width="22%" style="border-bottom:1px solid black" |
 +
<math>\operatorname{h}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{e}</math>
 +
| <math>\operatorname{e}</math>
 +
| <math>\operatorname{f}</math>
 +
| <math>\operatorname{g}</math>
 +
| <math>\operatorname{h}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{f}</math>
 +
| <math>\operatorname{f}</math>
 +
| <math>\operatorname{e}</math>
 +
| <math>\operatorname{h}</math>
 +
| <math>\operatorname{g}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{g}</math>
 +
| <math>\operatorname{g}</math>
 +
| <math>\operatorname{h}</math>
 +
| <math>\operatorname{e}</math>
 +
| <math>\operatorname{f}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{h}</math>
 +
| <math>\operatorname{h}</math>
 +
| <math>\operatorname{g}</math>
 +
| <math>\operatorname{f}</math>
 +
| <math>\operatorname{e}</math>
 +
|}
 +
 
 +
<br>
 +
 
 +
===Symmetric Group S<sub>3</sub>===
 +
 
 +
<br>
 +
 
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 +
|+ <math>\text{Permutation Substitutions in}~ \operatorname{Sym} \{ \mathrm{A}, \mathrm{B}, \mathrm{C} \}</math>
 +
|- style="background:#f0f0ff"
 +
| width="16%" | <math>\operatorname{e}</math>
 +
| width="16%" | <math>\operatorname{f}</math>
 +
| width="16%" | <math>\operatorname{g}</math>
 +
| width="16%" | <math>\operatorname{h}</math>
 +
| width="16%" | <math>\operatorname{i}</math>
 +
| width="16%" | <math>\operatorname{j}</math>
 
|-
 
|-
−
| f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || (&nbsp;) || false || 0
+
|
−
|-
+
<math>\begin{matrix}
−
| f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || &not;x &and; &not;y
+
\mathrm{A} & \mathrm{B} & \mathrm{C}
−
|-
+
\\[3pt]
−
| f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || &not;x &and; y
+
\downarrow & \downarrow & \downarrow
−
|-
+
\\[6pt]
−
| f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || &not;x
+
\mathrm{A} & \mathrm{B} & \mathrm{C}
−
|-
+
\end{matrix}</math>
−
| f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x &and; &not;y
+
|
−
|-
+
<math>\begin{matrix}
−
| f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || &not;y
+
\mathrm{A} & \mathrm{B} & \mathrm{C}
−
|-
+
\\[3pt]
−
| f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x &ne; y
+
\downarrow & \downarrow & \downarrow
−
|-
+
\\[6pt]
−
| 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
+
\mathrm{C} & \mathrm{A} & \mathrm{B}
−
|-
+
\end{matrix}</math>
−
| f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x&nbsp;y || x and y || x &and; y
+
|
−
|-
+
<math>\begin{matrix}
−
| f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y
+
\mathrm{A} & \mathrm{B} & \mathrm{C}
−
|-
+
\\[3pt]
−
| f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y
+
\downarrow & \downarrow & \downarrow
−
|-
+
\\[6pt]
−
| f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x &rarr; y
+
\mathrm{B} & \mathrm{C} & \mathrm{A}
−
|-
+
\end{matrix}</math>
−
| f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x
+
|
−
|-
+
<math>\begin{matrix}
−
| f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x &larr; y
+
\mathrm{A} & \mathrm{B} & \mathrm{C}
−
|-
+
\\[3pt]
−
| f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y  || x &or; y
+
\downarrow & \downarrow & \downarrow
−
|-
+
\\[6pt]
−
| f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || ((&nbsp;)) || true || 1
+
\mathrm{A} & \mathrm{C} & \mathrm{B}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{A} & \mathrm{B} & \mathrm{C}
 +
\\[3pt]
 +
\downarrow & \downarrow & \downarrow
 +
\\[6pt]
 +
\mathrm{C} & \mathrm{B} & \mathrm{A}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{A} & \mathrm{B} & \mathrm{C}
 +
\\[3pt]
 +
\downarrow & \downarrow & \downarrow
 +
\\[6pt]
 +
\mathrm{B} & \mathrm{A} & \mathrm{C}
 +
\end{matrix}</math>
 
|}
 
|}
 +
 
<br>
 
<br>
  
−
===Template Draft===
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 +
|+ <math>\text{Matrix Representations of Permutations in}~ \operatorname{Sym}(3)</math>
 +
|- style="background:#f0f0ff"
 +
| width="16%" | <math>\operatorname{e}</math>
 +
| width="16%" | <math>\operatorname{f}</math>
 +
| width="16%" | <math>\operatorname{g}</math>
 +
| width="16%" | <math>\operatorname{h}</math>
 +
| width="16%" | <math>\operatorname{i}</math>
 +
| width="16%" | <math>\operatorname{j}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
1 & 0 & 0
 +
\\
 +
0 & 1 & 0
 +
\\
 +
0 & 0 & 1
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0 & 0 & 1
 +
\\
 +
1 & 0 & 0
 +
\\
 +
0 & 1 & 0
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0 & 1 & 0
 +
\\
 +
0 & 0 & 1
 +
\\
 +
1 & 0 & 0
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
1 & 0 & 0
 +
\\
 +
0 & 0 & 1
 +
\\
 +
0 & 1 & 0
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0 & 0 & 1
 +
\\
 +
0 & 1 & 0
 +
\\
 +
1 & 0 & 0
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0 & 1 & 0
 +
\\
 +
1 & 0 & 0
 +
\\
 +
0 & 0 & 1
 +
\end{matrix}</math>
 +
|}
  
−
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:98%"
+
<br>
−
|+ '''Propositional Forms on Two Variables'''
+
 
−
|- style="background:aliceblue"
+
<pre>
−
! style="width:14%" | L<sub>1</sub>
+
Symmetric Group S_3
−
! style="width:14%" | L<sub>2</sub>
+
o-------------------------------------------------o
−
! style="width:14%" | L<sub>3</sub>
+
|                                                |
−
! style="width:14%" | L<sub>4</sub>
+
|                        ^                        |
−
! style="width:14%" | L<sub>5</sub>
+
|                    e / \ e                    |
−
! style="width:14%" | L<sub>6</sub>
+
|                     /   \                      |
−
! style="width:14%" | Name
+
|                     /  e  \                    |
−
|- style="background:aliceblue"
+
|                  f / \  / \ f                  |
−
| &nbsp;
+
|                   /  \ /   \                  |
−
| align="right" | x :
+
|                  /  f  \  f  \                  |
−
| 1 1 0 0
+
|               g / \  / \  / \ g              |
−
| &nbsp;
+
|               /  \ /  \ /  \                |
−
| &nbsp;
+
|               /  g  \  g  \  g  \              |
−
| &nbsp;
+
|           h / \  / \  / \  / \ h            |
−
| &nbsp;
+
|             /  \ /  \ /  \ /  \            |
−
|- style="background:aliceblue"
+
|           /  h  \  e  \  e  \  h  \            |
−
| &nbsp;
+
|         i / \  / \  / \  / \  / \ i        |
−
| align="right" | y :
+
|         /  \ /  \ /  \ /  \ /  \          |
−
| 1 0 1 0
+
|         /  i  \  i  \  f  \  j  \  i  \        |
−
| &nbsp;
+
|     j / \  / \  / \  / \  / \  / \ j      |
−
| &nbsp;
+
|       /  \ /  \ /  \ /  \ /  \ /  \      |
−
| &nbsp;
+
|     (  j  \  j  \  j  \  i  \  h  \  j  )      |
−
| &nbsp;
+
|       \  / \  / \  / \  / \  / \  /      |
−
|-
+
|       \ /  \ /  \ /  \ /  \ /  \ /       |
−
| f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || (&nbsp;) || false || 0 || Falsity
+
|         \  h  \  h  \  e  \  j  \  i  /         |
−
|-
+
|         \  / \  / \  / \  / \  /          |
−
| f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || &not;x &and; &not;y || [[NNOR]]
+
|           \ /  \ /  \ /  \ /  \ /          |
−
|-
+
|           \  i  \  g  \  f  \  h  /            |
−
| 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 \  e  \  g  /               |
−
|-
+
|               \  / \  / \  /                |
−
| f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x &and; &not;y || Imprecede
+
|                 \ /  \ /  \ /                |
−
|-
+
|                 \  g  \  f  /                  |
−
| 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
+
|                     \  e  /                     |
−
|-
+
|                     \  /                     |
−
| 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
+
|                       \ /                      |
−
|-
+
|                       v                        |
−
| f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x&nbsp;y || x and y || x &and; y || [[Conjunction]]
+
|                                                 |
−
|-
+
o-------------------------------------------------o
−
| f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y || Equality
+
</pre>
−
|-
+
 
−
| f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y || Two
+
<br>
−
|-
+
 
−
| f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x &rarr; y || [[Logical implcation|Implication]]
+
===TeX Tables===
−
|-
+
 
−
| f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x || One
+
<pre>
−
|-
+
\tableofcontents
−
| f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x &larr; y || [[Logical involution|Involution]]
+
 
−
|-
+
\subsection{Table A1.  Propositional Forms on Two Variables}
−
| f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y || x &or; y || [[Disjunction]]
+
 
−
|-
+
Table A1 lists equivalent expressions for the Boolean functions of two variables in a number of different notational systems.
−
| f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || ((&nbsp;)) || true || 1 || Tautology
+
 
 +
\begin{quote}\begin{tabular}{|c|c|c|c|c|c|c|}
 +
\multicolumn{7}{c}{\textbf{Table A1.  Propositional Forms on Two Variables}} \\
 +
\hline
 +
$\mathcal{L}_1$ &
 +
$\mathcal{L}_2$ &&
 +
$\mathcal{L}_3$ &
 +
$\mathcal{L}_4$ &
 +
$\mathcal{L}_5$ &
 +
$\mathcal{L}_6$ \\
 +
\hline
 +
& & $x =$ & 1 1 0 0 & & & \\
 +
& & $y =$ & 1 0 1 0 & & & \\
 +
\hline
 +
$f_{0}$    &
 +
$f_{0000}$  &&
 +
0 0 0 0     &
 +
$(~)$      &
 +
$\operatorname{false}$ &
 +
$0$        \\
 +
$f_{1}$    &
 +
$f_{0001}$  &&
 +
0 0 0 1    &
 +
$(x)(y)$    &
 +
$\operatorname{neither}\ x\ \operatorname{nor}\ y$ &
 +
$\lnot x \land \lnot y$ \\
 +
$f_{2}$    &
 +
$f_{0010}$  &&
 +
0 0 1 0    &
 +
$(x)\ y$    &
 +
$y\ \operatorname{without}\ x$ &
 +
$\lnot x \land y$ \\
 +
$f_{3}$    &
 +
$f_{0011}$  &&
 +
0 0 1 1     &
 +
$(x)$      &
 +
$\operatorname{not}\ x$ &
 +
$\lnot x$  \\
 +
$f_{4}$    &
 +
$f_{0100}$  &&
 +
0 1 0 0    &
 +
$x\ (y)$    &
 +
$x\ \operatorname{without}\ y$ &
 +
$x \land \lnot y$ \\
 +
$f_{5}$    &
 +
$f_{0101}$  &&
 +
0 1 0 1     &
 +
$(y)$      &
 +
$\operatorname{not}\ y$ &
 +
$\lnot y$  \\
 +
$f_{6}$    &
 +
$f_{0110}$  &&
 +
0 1 1 0     &
 +
$(x,\ y)$  &
 +
$x\ \operatorname{not~equal~to}\ y$ &
 +
$x \ne y$  \\
 +
$f_{7}$    &
 +
$f_{0111}$  &&
 +
0 1 1 1     &
 +
$(x\ y)$    &
 +
$\operatorname{not~both}\ x\ \operatorname{and}\ y$ &
 +
$\lnot x \lor \lnot y$ \\
 +
\hline
 +
$f_{8}$    &
 +
$f_{1000}$  &&
 +
1 0 0 0     &
 +
$x\ y$      &
 +
$x\ \operatorname{and}\ y$ &
 +
$x \land y$ \\
 +
$f_{9}$    &
 +
$f_{1001}$  &&
 +
1 0 0 1    &
 +
$((x,\ y))$ &
 +
$x\ \operatorname{equal~to}\ y$ &
 +
$x = y$    \\
 +
$f_{10}$    &
 +
$f_{1010}$  &&
 +
1 0 1 0    &
 +
$y$        &
 +
$y$        &
 +
$y$        \\
 +
$f_{11}$    &
 +
$f_{1011}$  &&
 +
1 0 1 1    &
 +
$(x\ (y))$  &
 +
$\operatorname{not}\ x\ \operatorname{without}\ y$ &
 +
$x \Rightarrow y$ \\
 +
$f_{12}$    &
 +
$f_{1100}$  &&
 +
1 1 0 0    &
 +
$x$        &
 +
$x$        &
 +
$x$        \\
 +
$f_{13}$    &
 +
$f_{1101}$  &&
 +
1 1 0 1    &
 +
$((x)\ y)$  &
 +
$\operatorname{not}\ y\ \operatorname{without}\ x$ &
 +
$x \Leftarrow y$ \\
 +
$f_{14}$    &
 +
$f_{1110}$  &&
 +
1 1 1 0    &
 +
$((x)(y))$  &
 +
$x\ \operatorname{or}\ y$ &
 +
$x \lor y$  \\
 +
$f_{15}$    &
 +
$f_{1111}$  &&
 +
1 1 1 1    &
 +
$((~))$    &
 +
$\operatorname{true}$ &
 +
$1$        \\
 +
\hline
 +
\end{tabular}\end{quote}
 +
 
 +
\subsection{Table A2.  Propositional Forms on Two Variables}
 +
 
 +
Table A2 lists the sixteen Boolean functions of two variables in a different order, grouping them by structural similarity into seven natural classes.
 +
 
 +
\begin{quote}\begin{tabular}{|c|c|c|c|c|c|c|}
 +
\multicolumn{7}{c}{\textbf{Table A2.  Propositional Forms on Two Variables}} \\
 +
\hline
 +
$\mathcal{L}_1$ &
 +
$\mathcal{L}_2$ &&
 +
$\mathcal{L}_3$ &
 +
$\mathcal{L}_4$ &
 +
$\mathcal{L}_5$ &
 +
$\mathcal{L}_6$ \\
 +
\hline
 +
& & $x =$ & 1 1 0 0 & & & \\
 +
& & $y =$ & 1 0 1 0 & & & \\
 +
\hline
 +
$f_{0}$    &
 +
$f_{0000}$  &&
 +
0 0 0 0    &
 +
$(~)$      &
 +
$\operatorname{false}$ &
 +
$0$        \\
 +
\hline
 +
$f_{1}$    &
 +
$f_{0001}$  &&
 +
0 0 0 1    &
 +
$(x)(y)$    &
 +
$\operatorname{neither}\ x\ \operatorname{nor}\ y$ &
 +
$\lnot x \land \lnot y$ \\
 +
$f_{2}$    &
 +
$f_{0010}$  &&
 +
0 0 1 0    &
 +
$(x)\ y$    &
 +
$y\ \operatorname{without}\ x$ &
 +
$\lnot x \land y$ \\
 +
$f_{4}$    &
 +
$f_{0100}$  &&
 +
0 1 0 0    &
 +
$x\ (y)$    &
 +
$x\ \operatorname{without}\ y$ &
 +
$x \land \lnot y$ \\
 +
$f_{8}$    &
 +
$f_{1000}$  &&
 +
1 0 0 0    &
 +
$x\ y$      &
 +
$x\ \operatorname{and}\ y$ &
 +
$x \land y$ \\
 +
\hline
 +
$f_{3}$    &
 +
$f_{0011}$  &&
 +
0 0 1 1    &
 +
$(x)$      &
 +
$\operatorname{not}\ x$ &
 +
$\lnot x$  \\
 +
$f_{12}$    &
 +
$f_{1100}$  &&
 +
1 1 0 0    &
 +
$x$        &
 +
$x$        &
 +
$x$        \\
 +
\hline
 +
$f_{6}$    &
 +
$f_{0110}$  &&
 +
0 1 1 0    &
 +
$(x,\ y)$  &
 +
$x\ \operatorname{not~equal~to}\ y$ &
 +
$x \ne y$  \\
 +
$f_{9}$    &
 +
$f_{1001}$  &&
 +
1 0 0 1    &
 +
$((x,\ y))$ &
 +
$x\ \operatorname{equal~to}\ y$ &
 +
$x = y$    \\
 +
\hline
 +
$f_{5}$    &
 +
$f_{0101}$  &&
 +
0 1 0 1    &
 +
$(y)$      &
 +
$\operatorname{not}\ y$ &
 +
$\lnot y$  \\
 +
$f_{10}$    &
 +
$f_{1010}$  &&
 +
1 0 1 0    &
 +
$y$        &
 +
$y$        &
 +
$y$        \\
 +
\hline
 +
$f_{7}$    &
 +
$f_{0111}$  &&
 +
0 1 1 1    &
 +
$(x\ y)$    &
 +
$\operatorname{not~both}\ x\ \operatorname{and}\ y$ &
 +
$\lnot x \lor \lnot y$ \\
 +
$f_{11}$    &
 +
$f_{1011}$  &&
 +
1 0 1 1    &
 +
$(x\ (y))$  &
 +
$\operatorname{not}\ x\ \operatorname{without}\ y$ &
 +
$x \Rightarrow y$ \\
 +
$f_{13}$    &
 +
$f_{1101}$  &&
 +
1 1 0 1    &
 +
$((x)\ y)$  &
 +
$\operatorname{not}\ y\ \operatorname{without}\ x$ &
 +
$x \Leftarrow y$ \\
 +
$f_{14}$    &
 +
$f_{1110}$  &&
 +
1 1 1 0    &
 +
$((x)(y))$  &
 +
$x\ \operatorname{or}\ y$ &
 +
$x \lor y$  \\
 +
\hline
 +
$f_{15}$    &
 +
$f_{1111}$  &&
 +
1 1 1 1    &
 +
$((~))$    &
 +
$\operatorname{true}$ &
 +
$1$        \\
 +
\hline
 +
\end{tabular}\end{quote}
 +
 
 +
\subsection{Table A3.  $\operatorname{E}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}
 +
 
 +
\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
 +
\multicolumn{6}{c}{\textbf{Table A3.  $\operatorname{E}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}} \\
 +
\hline
 +
& &
 +
$\operatorname{T}_{11}$ &
 +
$\operatorname{T}_{10}$ &
 +
$\operatorname{T}_{01}$ &
 +
$\operatorname{T}_{00}$ \\
 +
& $f$ &
 +
$\operatorname{E}f|_{\operatorname{d}x\ \operatorname{d}y}$  &
 +
$\operatorname{E}f|_{\operatorname{d}x (\operatorname{d}y)}$  &
 +
$\operatorname{E}f|_{(\operatorname{d}x) \operatorname{d}y}$  &
 +
$\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}$ \\
 +
\hline
 +
$f_{0}$  & $(~)$      & $(~)$      & $(~)$      & $(~)$      & $(~)$      \\
 +
\hline
 +
$f_{1}$  & $(x)(y)$    & $x\ y$      & $x\ (y)$    & $(x)\ y$    & $(x)(y)$    \\
 +
$f_{2}$  & $(x)\ y$    & $x\ (y)$    & $x\ y$      & $(x)(y)$    & $(x)\ y$    \\
 +
$f_{4}$  & $x\ (y)$    & $(x)\ y$    & $(x)(y)$    & $x\ y$      & $x\ (y)$    \\
 +
$f_{8}$  & $x\ y$      & $(x)(y)$    & $(x)\ y$    & $x\ (y)$    & $x\ y$      \\
 +
\hline
 +
$f_{3}$  & $(x)$      & $x$        & $x$        & $(x)$      & $(x)$      \\
 +
$f_{12}$ & $x$        & $(x)$      & $(x)$      & $x$        & $x$        \\
 +
\hline
 +
$f_{6}$  & $(x,\ y)$  & $(x,\ y)$  & $((x,\ y))$ & $((x,\ y))$ & $(x,\ y)$  \\
 +
$f_{9}$  & $((x,\ y))$ & $((x,\ y))$ & $(x,\ y)$  & $(x,\ y)$  & $((x,\ y))$ \\
 +
\hline
 +
$f_{5}$  & $(y)$      & $y$        & $(y)$      & $y$        & $(y)$      \\
 +
$f_{10}$ & $y$        & $(y)$      & $y$        & $(y)$      & $y$        \\
 +
\hline
 +
$f_{7}$  & $(x\ y)$    & $((x)(y))$  & $((x)\ y)$  & $(x\ (y))$  & $(x\ y)$    \\
 +
$f_{11}$ & $(x\ (y))$  & $((x)\ y)$  & $((x)(y))$  & $(x\ y)$    & $(x\ (y))$  \\
 +
$f_{13}$ & $((x)\ y)$  & $(x\ (y))$  & $(x\ y)$    & $((x)(y))$  & $((x)\ y)$  \\
 +
$f_{14}$ & $((x)(y))$  & $(x\ y)$    & $(x\ (y))$  & $((x)\ y)$  & $((x)(y))$  \\
 +
\hline
 +
$f_{15}$ & $((~))$    & $((~))$    & $((~))$    & $((~))$    & $((~))$    \\
 +
\hline
 +
\multicolumn{2}{|c||}{\PMlinkname{Fixed Point}{FixedPoint} Total:} & 4 & 4 & 4 & 16 \\
 +
\hline
 +
\end{tabular}\end{quote}
 +
 
 +
\subsection{Table A4.  $\operatorname{D}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}
 +
 
 +
\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
 +
\multicolumn{6}{c}{\textbf{Table A4.  $\operatorname{D}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}} \\
 +
\hline
 +
& $f$ &
 +
$\operatorname{D}f|_{\operatorname{d}x\ \operatorname{d}y}$  &
 +
$\operatorname{D}f|_{\operatorname{d}x (\operatorname{d}y)}$  &
 +
$\operatorname{D}f|_{(\operatorname{d}x) \operatorname{d}y}$  &
 +
$\operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)}$ \\
 +
\hline
 +
$f_{0}$  & $(~)$      & $(~)$      & $(~)$  & $(~)$  & $(~)$ \\
 +
\hline
 +
$f_{1}$  & $(x)(y)$    & $((x,\ y))$ & $(y)$  & $(x)$  & $(~)$ \\
 +
$f_{2}$  & $(x)\ y$    & $(x,\ y)$  & $y$    & $(x)$  & $(~)$ \\
 +
$f_{4}$  & $x\ (y)$    & $(x,\ y)$  & $(y)$  & $x$    & $(~)$ \\
 +
$f_{8}$  & $x\ y$      & $((x,\ y))$ & $y$    & $x$    & $(~)$ \\
 +
\hline
 +
$f_{3}$  & $(x)$      & $((~))$    & $((~))$ & $(~)$  & $(~)$ \\
 +
$f_{12}$ & $x$        & $((~))$    & $((~))$ & $(~)$  & $(~)$ \\
 +
\hline
 +
$f_{6}$  & $(x,\ y)$  & $(~)$      & $((~))$ & $((~))$ & $(~)$ \\
 +
$f_{9}$  & $((x,\ y))$ & $(~)$      & $((~))$ & $((~))$ & $(~)$ \\
 +
\hline
 +
$f_{5}$  & $(y)$      & $((~))$    & $(~)$  & $((~))$ & $(~)$ \\
 +
$f_{10}$ & $y$        & $((~))$    & $(~)$  & $((~))$ & $(~)$ \\
 +
\hline
 +
$f_{7}$  & $(x\ y)$    & $((x,\ y))$ & $y$    & $x$    & $(~)$ \\
 +
$f_{11}$ & $(x\ (y))$  & $(x,\ y)$  & $(y)$  & $x$    & $(~)$ \\
 +
$f_{13}$ & $((x)\ y)$  & $(x,\ y)$  & $y$    & $(x)$  & $(~)$ \\
 +
$f_{14}$ & $((x)(y))$  & $((x,\ y))$ & $(y)$  & $(x)$  & $(~)$ \\
 +
\hline
 +
$f_{15}$ & $((~))$    & $(~)$      & $(~)$  & $(~)$  & $(~)$ \\
 +
\hline
 +
\end{tabular}\end{quote}
 +
 
 +
\subsection{Table A5.  $\operatorname{E}f$ Expanded Over Ordinary Features $\{ x, y \}$}
 +
 
 +
\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
 +
\multicolumn{6}{c}{\textbf{Table A5.  $\operatorname{E}f$ Expanded Over Ordinary Features $\{ x, y \}$}} \\
 +
\hline
 +
& $f$ &
 +
$\operatorname{E}f|_{x\ y}$  &
 +
$\operatorname{E}f|_{x (y)}$  &
 +
$\operatorname{E}f|_{(x) y}$  &
 +
$\operatorname{E}f|_{(x)(y)}$ \\
 +
\hline
 +
$f_{0}$ &
 +
$(~)$  &
 +
$(~)$  &
 +
$(~)$  &
 +
$(~)$  &
 +
$(~)$  \\
 +
\hline
 +
$f_{1}$  &
 +
$(x)(y)$ &
 +
$\operatorname{d}x\ \operatorname{d}y$  &
 +
$\operatorname{d}x\ (\operatorname{d}y)$ &
 +
$(\operatorname{d}x)\ \operatorname{d}y$ &
 +
$(\operatorname{d}x)(\operatorname{d}y)$ \\
 +
$f_{2}$  &
 +
$(x)\ y$ &
 +
$\operatorname{d}x\ (\operatorname{d}y)$ &
 +
$\operatorname{d}x\ \operatorname{d}y$  &
 +
$(\operatorname{d}x)(\operatorname{d}y)$ &
 +
$(\operatorname{d}x)\ \operatorname{d}y$ \\
 +
$f_{4}$  &
 +
$x\ (y)$ &
 +
$(\operatorname{d}x)\ \operatorname{d}y$ &
 +
$(\operatorname{d}x)(\operatorname{d}y)$ &
 +
$\operatorname{d}x\ \operatorname{d}y$  &
 +
$\operatorname{d}x\ (\operatorname{d}y)$ \\
 +
$f_{8}$ &
 +
$x\ y$  &
 +
$(\operatorname{d}x)(\operatorname{d}y)$ &
 +
$(\operatorname{d}x)\ \operatorname{d}y$ &
 +
$\operatorname{d}x\ (\operatorname{d}y)$ &
 +
$\operatorname{d}x\ \operatorname{d}y$  \\
 +
\hline
 +
$f_{3}$ &
 +
$(x)$  &
 +
$\operatorname{d}x$  &
 +
$\operatorname{d}x$  &
 +
$(\operatorname{d}x)$ &
 +
$(\operatorname{d}x)$ \\
 +
$f_{12}$ &
 +
$x$      &
 +
$(\operatorname{d}x)$ &
 +
$(\operatorname{d}x)$ &
 +
$\operatorname{d}x$  &
 +
$\operatorname{d}x$  \\
 +
\hline
 +
$f_{6}$  &
 +
$(x,\ y)$ &
 +
$(\operatorname{d}x,\ \operatorname{d}y)$  &
 +
$((\operatorname{d}x,\ \operatorname{d}y))$ &
 +
$((\operatorname{d}x,\ \operatorname{d}y))$ &
 +
$(\operatorname{d}x,\ \operatorname{d}y)$  \\
 +
$f_{9}$    &
 +
$((x,\ y))$ &
 +
$((\operatorname{d}x,\ \operatorname{d}y))$ &
 +
$(\operatorname{d}x,\ \operatorname{d}y)$  &
 +
$(\operatorname{d}x,\ \operatorname{d}y)$  &
 +
$((\operatorname{d}x,\ \operatorname{d}y))$ \\
 +
\hline
 +
$f_{5}$ &
 +
$(y)$  &
 +
$\operatorname{d}y$  &
 +
$(\operatorname{d}y)$ &
 +
$\operatorname{d}y$  &
 +
$(\operatorname{d}y)$ \\
 +
$f_{10}$ &
 +
$y$      &
 +
$(\operatorname{d}y)$ &
 +
$\operatorname{d}y$  &
 +
$(\operatorname{d}y)$ &
 +
$\operatorname{d}y$  \\
 +
\hline
 +
$f_{7}$  &
 +
$(x\ y)$ &
 +
$((\operatorname{d}x)(\operatorname{d}y))$ &
 +
$((\operatorname{d}x)\ \operatorname{d}y)$ &
 +
$(\operatorname{d}x\ (\operatorname{d}y))$ &
 +
$(\operatorname{d}x\ \operatorname{d}y)$  \\
 +
$f_{11}$  &
 +
$(x\ (y))$ &
 +
$((\operatorname{d}x)\ \operatorname{d}y)$ &
 +
$((\operatorname{d}x)(\operatorname{d}y))$ &
 +
$(\operatorname{d}x\ \operatorname{d}y)$  &
 +
$(\operatorname{d}x\ (\operatorname{d}y))$ \\
 +
$f_{13}$  &
 +
$((x)\ y)$ &
 +
$(\operatorname{d}x\ (\operatorname{d}y))$ &
 +
$(\operatorname{d}x\ \operatorname{d}y)$  &
 +
$((\operatorname{d}x)(\operatorname{d}y))$ &
 +
$((\operatorname{d}x)\ \operatorname{d}y)$ \\
 +
$f_{14}$  &
 +
$((x)(y))$ &
 +
$(\operatorname{d}x\ \operatorname{d}y)$  &
 +
$(\operatorname{d}x\ (\operatorname{d}y))$ &
 +
$((\operatorname{d}x)\ \operatorname{d}y)$ &
 +
$((\operatorname{d}x)(\operatorname{d}y))$ \\
 +
\hline
 +
$f_{15}$ &
 +
$((~))$  &
 +
$((~))$  &
 +
$((~))$  &
 +
$((~))$  &
 +
$((~))$  \\
 +
\hline
 +
\end{tabular}\end{quote}
 +
 
 +
\subsection{Table A6.  $\operatorname{D}f$ Expanded Over Ordinary Features $\{ x, y \}$}
 +
 
 +
\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
 +
\multicolumn{6}{c}{\textbf{Table A6.  $\operatorname{D}f$ Expanded Over Ordinary Features $\{ x, y \}$}} \\
 +
\hline
 +
& $f$ &
 +
$\operatorname{D}f|_{x\ y}$  &
 +
$\operatorname{D}f|_{x (y)}$  &
 +
$\operatorname{D}f|_{(x) y}$  &
 +
$\operatorname{D}f|_{(x)(y)}$ \\
 +
\hline
 +
$f_{0}$ &
 +
$(~)$  &
 +
$(~)$  &
 +
$(~)$  &
 +
$(~)$  &
 +
$(~)$  \\
 +
\hline
 +
$f_{1}$  &
 +
$(x)(y)$ &
 +
$\operatorname{d}x\ \operatorname{d}y$    &
 +
$\operatorname{d}x\ (\operatorname{d}y)$  &
 +
$(\operatorname{d}x)\ \operatorname{d}y$  &
 +
$((\operatorname{d}x)(\operatorname{d}y))$ \\
 +
$f_{2}$  &
 +
$(x)\ y$ &
 +
$\operatorname{d}x\ (\operatorname{d}y)$  &
 +
$\operatorname{d}x\ \operatorname{d}y$    &
 +
$((\operatorname{d}x)(\operatorname{d}y))$ &
 +
$(\operatorname{d}x)\ \operatorname{d}y$  \\
 +
$f_{4}$  &
 +
$x\ (y)$ &
 +
$(\operatorname{d}x)\ \operatorname{d}y$  &
 +
$((\operatorname{d}x)(\operatorname{d}y))$ &
 +
$\operatorname{d}x\ \operatorname{d}y$    &
 +
$\operatorname{d}x\ (\operatorname{d}y)$  \\
 +
$f_{8}$ &
 +
$x\ y$  &
 +
$((\operatorname{d}x)(\operatorname{d}y))$ &
 +
$(\operatorname{d}x)\ \operatorname{d}y$  &
 +
$\operatorname{d}x\ (\operatorname{d}y)$  &
 +
$\operatorname{d}x\ \operatorname{d}y$    \\
 +
\hline
 +
$f_{3}$ &
 +
$(x)$  &
 +
$\operatorname{d}x$ &
 +
$\operatorname{d}x$ &
 +
$\operatorname{d}x$ &
 +
$\operatorname{d}x$ \\
 +
$f_{12}$ &
 +
$x$      &
 +
$\operatorname{d}x$ &
 +
$\operatorname{d}x$ &
 +
$\operatorname{d}x$ &
 +
$\operatorname{d}x$ \\
 +
\hline
 +
$f_{6}$  &
 +
$(x,\ y)$ &
 +
$(\operatorname{d}x,\ \operatorname{d}y)$ &
 +
$(\operatorname{d}x,\ \operatorname{d}y)$ &
 +
$(\operatorname{d}x,\ \operatorname{d}y)$ &
 +
$(\operatorname{d}x,\ \operatorname{d}y)$ \\
 +
$f_{9}$    &
 +
$((x,\ y))$ &
 +
$(\operatorname{d}x,\ \operatorname{d}y)$ &
 +
$(\operatorname{d}x,\ \operatorname{d}y)$ &
 +
$(\operatorname{d}x,\ \operatorname{d}y)$ &
 +
$(\operatorname{d}x,\ \operatorname{d}y)$ \\
 +
\hline
 +
$f_{5}$ &
 +
$(y)$  &
 +
$\operatorname{d}y$ &
 +
$\operatorname{d}y$ &
 +
$\operatorname{d}y$ &
 +
$\operatorname{d}y$ \\
 +
$f_{10}$ &
 +
$y$      &
 +
$\operatorname{d}y$ &
 +
$\operatorname{d}y$ &
 +
$\operatorname{d}y$ &
 +
$\operatorname{d}y$ \\
 +
\hline
 +
$f_{7}$  &
 +
$(x\ y)$ &
 +
$((\operatorname{d}x)(\operatorname{d}y))$ &
 +
$(\operatorname{d}x)\ \operatorname{d}y$  &
 +
$\operatorname{d}x\ (\operatorname{d}y)$  &
 +
$\operatorname{d}x\ \operatorname{d}y$    \\
 +
$f_{11}$  &
 +
$(x\ (y))$ &
 +
$(\operatorname{d}x)\ \operatorname{d}y$  &
 +
$((\operatorname{d}x)(\operatorname{d}y))$ &
 +
$\operatorname{d}x\ \operatorname{d}y$    &
 +
$\operatorname{d}x\ (\operatorname{d}y)$  \\
 +
$f_{13}$  &
 +
$((x)\ y)$ &
 +
$\operatorname{d}x\ (\operatorname{d}y)$  &
 +
$\operatorname{d}x\ \operatorname{d}y$    &
 +
$((\operatorname{d}x)(\operatorname{d}y))$ &
 +
$(\operatorname{d}x)\ \operatorname{d}y$  \\
 +
$f_{14}$  &
 +
$((x)(y))$ &
 +
$\operatorname{d}x\ \operatorname{d}y$    &
 +
$\operatorname{d}x\ (\operatorname{d}y)$  &
 +
$(\operatorname{d}x)\ \operatorname{d}y$  &
 +
$((\operatorname{d}x)(\operatorname{d}y))$ \\
 +
\hline
 +
$f_{15}$ &
 +
$((~))$  &
 +
$(~)$    &
 +
$(~)$    &
 +
$(~)$    &
 +
$(~)$    \\
 +
\hline
 +
\end{tabular}\end{quote}
 +
</pre>
 +
 
 +
==Group Operation Tables==
 +
 
 +
<br>
 +
 
 +
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:80%"
 +
|+ <math>\text{Table 32.1}~~\text{Scheme of a Group Operation Table}</math>
 +
|- style="height:50px"
 +
| style="border-bottom:1px solid black; border-right:1px solid black" | <math>*\!</math>
 +
| style="border-bottom:1px solid black" | <math>x_0\!</math>
 +
| style="border-bottom:1px solid black" | <math>\cdots\!</math>
 +
| style="border-bottom:1px solid black" | <math>x_j\!</math>
 +
| style="border-bottom:1px solid black" | <math>\cdots\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>x_0\!</math>
 +
| <math>x_0 * x_0\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>x_0 * x_j\!</math>
 +
| <math>\cdots\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\cdots\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>\cdots\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>x_i\!</math>
 +
| <math>x_i * x_0\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>x_i * x_j\!</math>
 +
| <math>\cdots\!</math>
 +
|- style="height:50px"
 +
| width="12%" style="border-right:1px solid black" | <math>\cdots\!</math>
 +
| width="22%" | <math>\cdots\!</math>
 +
| width="22%" | <math>\cdots\!</math>
 +
| width="22%" | <math>\cdots\!</math>
 +
| width="22%" | <math>\cdots\!</math>
 
|}
 
|}
 +
 
<br>
 
<br>
  
−
===[[Truth Tables]]===
+
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:80%"
 +
|+ <math>\text{Table 32.2}~~\text{Scheme of the Regular Ante-Representation}</math>
 +
|- style="height:50px"
 +
| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
 +
| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>x_0\!</math>
 +
| <math>\{\!</math>
 +
| <math>(x_0 ~,~ x_0 * x_0),\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>(x_j ~,~ x_0 * x_j),\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\cdots\!</math>
 +
| <math>\{\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>x_i\!</math>
 +
| <math>\{\!</math>
 +
| <math>(x_0 ~,~ x_i * x_0),\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>(x_j ~,~ x_i * x_j),\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| width="12%" style="border-right:1px solid black" | <math>\cdots\!</math>
 +
| width="4%"  | <math>\{\!</math>
 +
| width="18%" | <math>\cdots\!</math>
 +
| width="22%" | <math>\cdots\!</math>
 +
| width="22%" | <math>\cdots\!</math>
 +
| width="18%" | <math>\cdots\!</math>
 +
| width="4%"  | <math>\}\!</math>
 +
|}
  
−
====[[Logical negation]]====
+
<br>
  
−
'''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.
+
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:80%"
 +
|+ <math>\text{Table 32.3}~~\text{Scheme of the Regular Post-Representation}</math>
 +
|- style="height:50px"
 +
| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
 +
| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>x_0\!</math>
 +
| <math>\{\!</math>
 +
| <math>(x_0 ~,~ x_0 * x_0),\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>(x_j ~,~ x_j * x_0),\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\cdots\!</math>
 +
| <math>\{\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>x_i\!</math>
 +
| <math>\{\!</math>
 +
| <math>(x_0 ~,~ x_0 * x_i),\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>(x_j ~,~ x_j * x_i),\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| width="12%" style="border-right:1px solid black" | <math>\cdots\!</math>
 +
| width="4%"  | <math>\{\!</math>
 +
| width="18%" | <math>\cdots\!</math>
 +
| width="22%" | <math>\cdots\!</math>
 +
| width="22%" | <math>\cdots\!</math>
 +
| width="18%" | <math>\cdots\!</math>
 +
| width="4%"  | <math>\}\!</math>
 +
|}
  
−
The [[truth table]] of '''NOT p''' (also written as '''~p''' or '''&not;p''') is as follows:
+
<br>
  
−
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:40%"
+
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
−
|+ '''Logical Negation'''
+
|+ <math>\text{Table 33.1}~~\text{Multiplication Operation of the Group}~V_4</math>
−
|- style="background:aliceblue"
+
|- style="height:50px"
−
! style="width:20%" | p
+
| width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
−
! style="width:20%" | &not;p
+
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{e}</math>
−
|-
+
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{f}</math>
−
| F || T
+
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{g}</math>
−
|-
+
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{h}</math>
−
| T || F
+
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{e}</math>
 +
| <math>\operatorname{e}</math>
 +
| <math>\operatorname{f}</math>
 +
| <math>\operatorname{g}</math>
 +
| <math>\operatorname{h}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{f}</math>
 +
| <math>\operatorname{f}</math>
 +
| <math>\operatorname{e}</math>
 +
| <math>\operatorname{h}</math>
 +
| <math>\operatorname{g}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{g}</math>
 +
| <math>\operatorname{g}</math>
 +
| <math>\operatorname{h}</math>
 +
| <math>\operatorname{e}</math>
 +
| <math>\operatorname{f}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{h}</math>
 +
| <math>\operatorname{h}</math>
 +
| <math>\operatorname{g}</math>
 +
| <math>\operatorname{f}</math>
 +
| <math>\operatorname{e}</math>
 
|}
 
|}
 +
 
<br>
 
<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" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
 +
|+ <math>\text{Table 33.2}~~\text{Regular Representation of the Group}~V_4</math>
 +
|- style="height:50px"
 +
| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
 +
| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
 +
|- style="height:50px"
 +
| width="20%" style="border-right:1px solid black" | <math>\operatorname{e}</math>
 +
| width="4%"  | <math>\{\!</math>
 +
| width="16%" | <math>(\operatorname{e}, \operatorname{e}),</math>
 +
| width="20%" | <math>(\operatorname{f}, \operatorname{f}),</math>
 +
| width="20%" | <math>(\operatorname{g}, \operatorname{g}),</math>
 +
| width="16%" | <math>(\operatorname{h}, \operatorname{h})</math>
 +
| width="4%" | <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{f}</math>
 +
| <math>\{\!</math>
 +
| <math>(\operatorname{e}, \operatorname{f}),</math>
 +
| <math>(\operatorname{f}, \operatorname{e}),</math>
 +
| <math>(\operatorname{g}, \operatorname{h}),</math>
 +
| <math>(\operatorname{h}, \operatorname{g})</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{g}</math>
 +
| <math>\{\!</math>
 +
| <math>(\operatorname{e}, \operatorname{g}),</math>
 +
| <math>(\operatorname{f}, \operatorname{h}),</math>
 +
| <math>(\operatorname{g}, \operatorname{e}),</math>
 +
| <math>(\operatorname{h}, \operatorname{f})</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{h}</math>
 +
| <math>\{\!</math>
 +
| <math>(\operatorname{e}, \operatorname{h}),</math>
 +
| <math>(\operatorname{f}, \operatorname{g}),</math>
 +
| <math>(\operatorname{g}, \operatorname{f}),</math>
 +
| <math>(\operatorname{h}, \operatorname{e})</math>
 +
| <math>\}\!</math>
 +
|}
 +
 
 +
<br>
  
−
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; width:40%"
+
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
−
|+ '''Variant Notations'''
+
|+ <math>\text{Table 33.3}~~\text{Regular Representation of the Group}~V_4</math>
−
|- style="background:aliceblue"
+
|- style="height:50px"
−
! style="text-align:center" | Notation
+
| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
−
! Vocalization
+
| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Symbols}\!</math>
−
|-
+
|- style="height:50px"
−
| style="text-align:center" | <math>\bar{p}</math>
+
| width="20%" style="border-right:1px solid black" | <math>\operatorname{e}</math>
−
| bar ''p''
+
| width="4%"  | <math>\{\!</math>
−
|-
+
| width="16%" | <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),</math>
−
| style="text-align:center" | <math>p'\!</math>
+
| width="20%" | <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),</math>
−
| ''p'' prime,<p> ''p'' complement
+
| width="20%" | <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),</math>
−
|-
+
| width="16%" | <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime})</math>
−
| style="text-align:center" | <math>!p\!</math>
+
| width="4%"  | <math>\}\!</math>
−
| bang ''p''
+
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{f}</math>
 +
| <math>\{\!</math>
 +
| <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),</math>
 +
| <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),</math>
 +
| <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),</math>
 +
| <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime})</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{g}</math>
 +
| <math>\{\!</math>
 +
| <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),</math>
 +
| <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),</math>
 +
| <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),</math>
 +
| <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime})</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{h}</math>
 +
| <math>\{\!</math>
 +
| <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),</math>
 +
| <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),</math>
 +
| <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),</math>
 +
| <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime})</math>
 +
| <math>\}\!</math>
 
|}
 
|}
 +
 
<br>
 
<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''".
+
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
 +
|+ <math>\text{Table 34.1}~~\text{Multiplicative Presentation of the Group}~Z_4(\cdot)</math>
 +
|- style="height:50px"
 +
| width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
 +
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{1}</math>
 +
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{a}</math>
 +
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{b}</math>
 +
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{c}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{1}</math>
 +
| <math>\operatorname{1}</math>
 +
| <math>\operatorname{a}</math>
 +
| <math>\operatorname{b}</math>
 +
| <math>\operatorname{c}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{a}</math>
 +
| <math>\operatorname{a}</math>
 +
| <math>\operatorname{b}</math>
 +
| <math>\operatorname{c}</math>
 +
| <math>\operatorname{1}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{b}</math>
 +
| <math>\operatorname{b}</math>
 +
| <math>\operatorname{c}</math>
 +
| <math>\operatorname{1}</math>
 +
| <math>\operatorname{a}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{c}</math>
 +
| <math>\operatorname{c}</math>
 +
| <math>\operatorname{1}</math>
 +
| <math>\operatorname{a}</math>
 +
| <math>\operatorname{b}</math>
 +
|}
  
−
* 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''.
+
<br>
  
−
* 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.
+
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
 +
|+ <math>\text{Table 34.2}~~\text{Regular Representation of the Group}~Z_4(\cdot)</math>
 +
|- style="height:50px"
 +
| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
 +
| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
 +
|- style="height:50px"
 +
| width="20%" style="border-right:1px solid black" | <math>\operatorname{1}</math>
 +
| width="4%"  | <math>\{\!</math>
 +
| width="16%" | <math>(\operatorname{1}, \operatorname{1}),</math>
 +
| width="20%" | <math>(\operatorname{a}, \operatorname{a}),</math>
 +
| width="20%" | <math>(\operatorname{b}, \operatorname{b}),</math>
 +
| width="16%" | <math>(\operatorname{c}, \operatorname{c})</math>
 +
| width="4%" | <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{a}</math>
 +
| <math>\{\!</math>
 +
| <math>(\operatorname{1}, \operatorname{a}),</math>
 +
| <math>(\operatorname{a}, \operatorname{b}),</math>
 +
| <math>(\operatorname{b}, \operatorname{c}),</math>
 +
| <math>(\operatorname{c}, \operatorname{1})</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{b}</math>
 +
| <math>\{\!</math>
 +
| <math>(\operatorname{1}, \operatorname{b}),</math>
 +
| <math>(\operatorname{a}, \operatorname{c}),</math>
 +
| <math>(\operatorname{b}, \operatorname{1}),</math>
 +
| <math>(\operatorname{c}, \operatorname{a})</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{c}</math>
 +
| <math>\{\!</math>
 +
| <math>(\operatorname{1}, \operatorname{c}),</math>
 +
| <math>(\operatorname{a}, \operatorname{1}),</math>
 +
| <math>(\operatorname{b}, \operatorname{a}),</math>
 +
| <math>(\operatorname{c}, \operatorname{b})</math>
 +
| <math>\}\!</math>
 +
|}
 +
 
 +
<br>
  
−
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]].
+
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
 +
|+ <math>\text{Table 35.1}~~\text{Additive Presentation of the Group}~Z_4(+)</math>
 +
|- style="height:50px"
 +
| width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>+\!</math>
 +
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{0}</math>
 +
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{1}</math>
 +
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{2}</math>
 +
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{3}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{0}</math>
 +
| <math>\operatorname{0}</math>
 +
| <math>\operatorname{1}</math>
 +
| <math>\operatorname{2}</math>
 +
| <math>\operatorname{3}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{1}</math>
 +
| <math>\operatorname{1}</math>
 +
| <math>\operatorname{2}</math>
 +
| <math>\operatorname{3}</math>
 +
| <math>\operatorname{0}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{2}</math>
 +
| <math>\operatorname{2}</math>
 +
| <math>\operatorname{3}</math>
 +
| <math>\operatorname{0}</math>
 +
| <math>\operatorname{1}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{3}</math>
 +
| <math>\operatorname{3}</math>
 +
| <math>\operatorname{0}</math>
 +
| <math>\operatorname{1}</math>
 +
| <math>\operatorname{2}</math>
 +
|}
  
−
Algebraically, logical negation corresponds to the ''complement'' in a [[Boolean algebra]] (for classical logic) or a [[Heyting algebra]] (for intuitionistic logic).
+
<br>
  
−
====[[Logical conjunction]]====
+
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
 +
|+ <math>\text{Table 35.2}~~\text{Regular Representation of the Group}~Z_4(+)</math>
 +
|- style="height:50px"
 +
| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
 +
| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
 +
|- style="height:50px"
 +
| width="20%" style="border-right:1px solid black" | <math>\operatorname{0}</math>
 +
| width="4%"  | <math>\{\!</math>
 +
| width="16%" | <math>(\operatorname{0}, \operatorname{0}),</math>
 +
| width="20%" | <math>(\operatorname{1}, \operatorname{1}),</math>
 +
| width="20%" | <math>(\operatorname{2}, \operatorname{2}),</math>
 +
| width="16%" | <math>(\operatorname{3}, \operatorname{3})</math>
 +
| width="4%"  | <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{1}</math>
 +
| <math>\{\!</math>
 +
| <math>(\operatorname{0}, \operatorname{1}),</math>
 +
| <math>(\operatorname{1}, \operatorname{2}),</math>
 +
| <math>(\operatorname{2}, \operatorname{3}),</math>
 +
| <math>(\operatorname{3}, \operatorname{0})</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{2}</math>
 +
| <math>\{\!</math>
 +
| <math>(\operatorname{0}, \operatorname{2}),</math>
 +
| <math>(\operatorname{1}, \operatorname{3}),</math>
 +
| <math>(\operatorname{2}, \operatorname{0}),</math>
 +
| <math>(\operatorname{3}, \operatorname{1})</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{3}</math>
 +
| <math>\{\!</math>
 +
| <math>(\operatorname{0}, \operatorname{3}),</math>
 +
| <math>(\operatorname{1}, \operatorname{0}),</math>
 +
| <math>(\operatorname{2}, \operatorname{1}),</math>
 +
| <math>(\operatorname{3}, \operatorname{2})</math>
 +
| <math>\}\!</math>
 +
|}
  
−
'''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.
+
<br>
  
−
The [[truth table]] of '''p AND q''' (also written as '''p &and; q''', '''p & q''', or '''p<math>\cdot</math>q''') is as follows:
+
==Higher Order Propositions==
  
−
{| 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>
 
<br>
  
−
====[[Logical disjunction]]====
+
<table align="center" cellpadding="4" cellspacing="0" style="text-align:center; width:90%">
 +
 
 +
<caption><font size="+2"><math>\text{Table 1.} ~~ \text{Higher Order Propositions} ~ (n = 1)</math></font></caption>
 +
 
 +
<tr>
 +
<td style="border-bottom:2px solid black" align="right"><math>x:</math></td>
 +
<td style="border-bottom:2px solid black"><math>1 ~ 0</math></td>
 +
<td style="border-bottom:2px solid black; border-right:2px solid black"><math>f</math></td>
 +
<td style="border-bottom:2px solid black"><math>m_{0}</math></td>
 +
<td style="border-bottom:2px solid black"><math>m_{1}</math></td>
 +
<td style="border-bottom:2px solid black"><math>m_{2}</math></td>
 +
<td style="border-bottom:2px solid black"><math>m_{3}</math></td>
 +
<td style="border-bottom:2px solid black"><math>m_{4}</math></td>
 +
<td style="border-bottom:2px solid black"><math>m_{5}</math></td>
 +
<td style="border-bottom:2px solid black"><math>m_{6}</math></td>
 +
<td style="border-bottom:2px solid black"><math>m_{7}</math></td>
 +
<td style="border-bottom:2px solid black"><math>m_{8}</math></td>
 +
<td style="border-bottom:2px solid black"><math>m_{9}</math></td>
 +
<td style="border-bottom:2px solid black"><math>m_{10}</math></td>
 +
<td style="border-bottom:2px solid black"><math>m_{11}</math></td>
 +
<td style="border-bottom:2px solid black"><math>m_{12}</math></td>
 +
<td style="border-bottom:2px solid black"><math>m_{13}</math></td>
 +
<td style="border-bottom:2px solid black"><math>m_{14}</math></td>
 +
<td style="border-bottom:2px solid black"><math>m_{15}</math></td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{0}</math></td>
 +
<td><math>0 ~ 0</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{(~)}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td></tr>
  
−
'''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.
+
<tr>
 +
<td><math>f_{1}</math></td>
 +
<td><math>0 ~ 1</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{(} x \texttt{)}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td></tr>
  
−
The [[truth table]] of '''p OR q''' (also written as '''p &or; q''') is as follows:
+
<tr>
 +
<td><math>f_{2}</math></td>
 +
<td><math>1 ~ 0</math></td>
 +
<td style="border-right:2px solid black"><math>x</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{3}</math></td>
 +
<td><math>1 ~ 1</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{((~))}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
</table>
  
−
{| 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>
 
<br>
  
−
====[[Logical equality]]====
+
<table align="center" border="1" cellpadding="4" cellspacing="0" style="text-align:center; width:90%">
 +
 
 +
<caption><font size="+2"><math>\text{Table 2.} ~~ \text{Interpretive Categories for Higher Order Propositions} ~ (n = 1)</math></font></caption>
  
−
'''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.
+
<tr>
 +
<td style="border-bottom:2px solid black; border-right:2px solid black">Measure</td>
 +
<td style="border-bottom:2px solid black">Happening</td>
 +
<td style="border-bottom:2px solid black">Exactness</td>
 +
<td style="border-bottom:2px solid black">Existence</td>
 +
<td style="border-bottom:2px solid black">Linearity</td>
 +
<td style="border-bottom:2px solid black">Uniformity</td>
 +
<td style="border-bottom:2px solid black">Information</td></tr>
  
−
The [[truth table]] of '''p EQ q''' (also written as '''p = q''', '''p &harr; q''', or '''p &equiv; q''') is as follows:
+
<tr>
 +
<td style="border-right:2px solid black"><math>m_{0}</math></td>
 +
<td>Nothing happens</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td></tr>
  
−
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%"
+
<tr>
−
|+ '''Logical Equality'''
+
<td style="border-right:2px solid black"><math>m_{1}</math></td>
−
|- style="background:aliceblue"
+
<td>&nbsp;</td>
−
! style="width:15%" | p
+
<td>Just false</td>
−
! style="width:15%" | q
+
<td>Nothing exists</td>
−
! style="width:15%" | p = q
+
<td>&nbsp;</td>
−
|-
+
<td>&nbsp;</td>
−
| F || F || T
+
<td>&nbsp;</td></tr>
−
|-
 
−
| F || T || F
 
−
|-
 
−
| T || F || F
 
−
|-
 
−
| T || T || T
 
−
|}
 
−
<br>
 
  
−
====[[Exclusive disjunction]]====
+
<tr>
 +
<td style="border-right:2px solid black"><math>m_{2}</math></td>
 +
<td>&nbsp;</td>
 +
<td>Just not <math>x</math></td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td></tr>
  
−
'''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.
+
<tr>
 +
<td style="border-right:2px solid black"><math>m_{3}</math></td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>Nothing is <math>x</math></td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td></tr>
  
−
The [[truth table]] of '''p XOR q''' (also written as '''p + q''', '''p &oplus; q''', or '''p &ne; q''') is as follows:
+
<tr>
 +
<td style="border-right:2px solid black"><math>m_{4}</math></td>
 +
<td>&nbsp;</td>
 +
<td>Just <math>x</math></td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td></tr>
  
−
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%"
+
<tr>
−
|+ '''Exclusive Disjunction'''
+
<td style="border-right:2px solid black"><math>m_{5}</math></td>
−
|- style="background:aliceblue"
+
<td>&nbsp;</td>
−
! style="width:15%" | p
+
<td>&nbsp;</td>
−
! style="width:15%" | q
+
<td>Everything is <math>x</math></td>
−
! style="width:15%" | p XOR q
+
<td><math>f</math> is linear</td>
−
|-
+
<td>&nbsp;</td>
−
| F || F || F
+
<td>&nbsp;</td></tr>
−
|-
+
 
−
| F || T || T
+
<tr>
−
|-
+
<td style="border-right:2px solid black"><math>m_{6}</math></td>
−
| T || F || T
+
<td>&nbsp;</td>
−
|-
+
<td>&nbsp;</td>
−
| T || T || F
+
<td>&nbsp;</td>
−
|}
+
<td>&nbsp;</td>
−
<br>
+
<td><math>f</math> is not uniform</td>
 +
<td><math>f</math> is informed</td></tr>
 +
 
 +
<tr>
 +
<td style="border-right:2px solid black"><math>m_{7}</math></td>
 +
<td>&nbsp;</td>
 +
<td>Not just true</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td></tr>
 +
 
 +
<tr>
 +
<td style="border-right:2px solid black"><math>m_{8}</math></td>
 +
<td>&nbsp;</td>
 +
<td>Just true</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td></tr>
 +
 
 +
<tr>
 +
<td style="border-right:2px solid black"><math>m_{9}</math></td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td><math>f</math> is uniform</td>
 +
<td><math>f</math> is not informed</td></tr>
 +
 
 +
<tr>
 +
<td style="border-right:2px solid black"><math>m_{10}</math></td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>Something is not <math>x</math></td>
 +
<td><math>f</math> is not linear</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td></tr>
 +
 
 +
<tr>
 +
<td style="border-right:2px solid black"><math>m_{11}</math></td>
 +
<td>&nbsp;</td>
 +
<td>Not just <math>x</math></td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td></tr>
  
−
The following equivalents can then be deduced:
+
<tr>
 +
<td style="border-right:2px solid black"><math>m_{12}</math></td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>Something is <math>x</math></td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td></tr>
  
−
: <math>\begin{matrix}
+
<tr>
−
p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\
+
<td style="border-right:2px solid black"><math>m_{13}</math></td>
−
\\
+
<td>&nbsp;</td>
−
      & = & (p \lor q) & \land & (\lnot p \lor \lnot q) \\
+
<td>Not just not <math>x</math></td>
−
\\
+
<td>&nbsp;</td>
−
      & = & (p \lor q) & \land & \lnot (p \land q)
+
<td>&nbsp;</td>
−
\end{matrix}</math>
+
<td>&nbsp;</td>
 +
<td>&nbsp;</td></tr>
  
−
'''Generalized''' or '''n-ary''' XOR is true when the number of 1-bits  is odd.
+
<tr>
 +
<td style="border-right:2px solid black"><math>m_{14}</math></td>
 +
<td>&nbsp;</td>
 +
<td>Not just false</td>
 +
<td>Something exists</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td></tr>
  
−
<pre>
+
<tr>
−
A + B = (A &#8743; !B) &#8744; (!A &#8743; B)
+
<td style="border-right:2px solid black"><math>m_{15}</math></td>
−
      = {(A &#8743; !B) &#8744; !A} &#8743; {(A &#8743; !B) &#8744; B}
+
<td>Anything happens</td>
−
      = {(A &#8744; !A) &#8743; (!B &#8744; !A)} &#8743; {(A &#8744; B) &#8743; (!B &#8744; B)}
+
<td>&nbsp;</td>
−
      = (!A &#8744; !B) &#8743; (A &#8744; B)
+
<td>&nbsp;</td>
−
      = !(A &#8743; B) &#8743; (A &#8744; B)
+
<td>&nbsp;</td>
−
</pre>
+
<td>&nbsp;</td>
 +
<td>&nbsp;</td></tr>
  
−
<pre>
+
</table>
−
p + q = (p &#8743; !q)  &#8744; (!p &#8743; B)
+
 
−
+
<br>
−
      = {(p &#8743; !q) &#8744; !p} &#8743; {(p &#8743; !q) &#8744; q}
+
 
−
+
<table align="center" cellpadding="1" cellspacing="0" style="background:white; color:black; text-align:center; width:90%">
−
      = {(p &#8744; !q) &#8743; (!q &#8744; !p)} &#8743; {(p &#8744; q) &#8743; (!q &#8744; q)}
+
 
−
+
<caption><font size="+2"><math>\text{Table 3.} ~~ \text{Higher Order Propositions} ~ (n = 2)</math></font></caption>
−
      = (!p &#8744; !q) &#8743; (p &#8744; q)
+
 
−
+
<tr>
−
      = !(p &#8743; q) &#8743; (p &#8744; q)
+
<td style="border-bottom:2px solid black" align="right"><math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td>
−
</pre>
+
<td style="border-bottom:2px solid black">
 +
<math>\begin{matrix}1100\\1010\end{matrix}</math></td>
 +
<td style="border-bottom:2px solid black; border-right:2px solid black"><math>f</math></td>
 +
<td style="border-bottom:2px solid black"><math>\underset{0}{m}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\underset{1}{m}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\underset{2}{m}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\underset{3}{m}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\underset{4}{m}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\underset{5}{m}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\underset{6}{m}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\underset{7}{m}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\underset{8}{m}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\underset{9}{m}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\underset{10}{m}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\underset{11}{m}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\underset{12}{m}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\underset{13}{m}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\underset{14}{m}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\underset{15}{m}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\underset{16}{m}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\underset{17}{m}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\underset{18}{m}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\underset{19}{m}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\underset{20}{m}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\underset{21}{m}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\underset{22}{m}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\underset{23}{m}</math></td>
 +
</tr>
 +
 
 +
<tr>
 +
<td><math>f_{0}</math></td>
 +
<td><math>0000</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{(~)}</math></td>
 +
<td>0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td>0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td>0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td>0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td>0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td>0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td>0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td>0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td>0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td>0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td>0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td>0</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{1}</math></td>
 +
<td><math>0001</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td>
 +
<td>0</td><td>0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td>0</td><td>0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td>0</td><td>0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td>0</td><td>0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td>0</td><td>0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td>0</td><td>0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{2}</math></td>
 +
<td><math>0010</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{3}</math></td>
 +
<td><math>0011</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)}</math></td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{4}</math></td>
 +
<td><math>0100</math></td>
 +
<td style="border-right:2px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{5}</math></td>
 +
<td><math>0101</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{(} v \texttt{)}</math></td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{6}</math></td>
 +
<td><math>0110</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{7}</math></td>
 +
<td><math>0111</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{8}</math></td>
 +
<td><math>1000</math></td>
 +
<td style="border-right:2px solid black"><math>u ~ v</math></td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{9}</math></td>
 +
<td><math>1001</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{10}</math></td>
 +
<td><math>1010</math></td>
 +
<td style="border-right:2px solid black"><math>v</math></td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{11}</math></td>
 +
<td><math>1011</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{12}</math></td>
 +
<td><math>1100</math></td>
 +
<td style="border-right:2px solid black"><math>u</math></td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{13}</math></td>
 +
<td><math>1101</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{14}</math></td>
 +
<td><math>1110</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{15}</math></td>
 +
<td><math>1111</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{((~))}</math></td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td>
 +
<td>0</td><td>0</td><td>0</td><td>0</td></tr>
 +
 
 +
</table>
  
−
<pre>
+
<br>
−
p + q = (p &#8743; ~q) &#8744; (~p &#8743; q)
+
 
−
+
<table align="center" cellpadding="1" cellspacing="0" style="text-align:center; width:90%">
−
      = ((p &#8743; ~q) &#8744; ~p) &#8743; ((p &#8743; ~q) &#8744; q)
+
 
−
+
<caption><font size="+2"><math>\text{Table 4.} ~~ \text{Qualifiers of the Implication Ordering:} ~ \alpha_{i} f = \Upsilon (f_{i}, f) = \Upsilon (f_{i} \Rightarrow f)</math></font></caption>
−
      = ((p &#8744; ~q) &#8743; (~q &#8744; ~p)) &#8743; ((p &#8744; q) &#8743; (~q &#8744; q))
+
 
−
+
<tr>
−
      = (~p &#8744; ~q) &#8743; (p &#8744; q)
+
<td style="border-bottom:2px solid black" align="right">
−
+
<math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td>
−
      = ~(p &#8743; q) &#8743; (p &#8744; q)
+
<td style="border-bottom:2px solid black">
 +
<math>\begin{matrix}1100\\1010\end{matrix}</math></td>
 +
<td style="border-bottom:2px solid black; border-right:2px solid black"><math>f</math></td>
 +
<td style="border-bottom:2px solid black"><math>\alpha_{15}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\alpha_{14}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\alpha_{13}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\alpha_{12}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\alpha_{11}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\alpha_{10}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\alpha_{9}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\alpha_{8}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\alpha_{7}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\alpha_{6}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\alpha_{5}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\alpha_{4}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\alpha_{3}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\alpha_{2}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\alpha_{1}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\alpha_{0}</math></td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{0}</math></td>
 +
<td><math>0000</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{(~)}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{1}</math></td>
 +
<td><math>0001</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{2}</math></td>
 +
<td><math>0010</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{3}</math></td>
 +
<td><math>0011</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{4}</math></td>
 +
<td><math>0100</math></td>
 +
<td style="border-right:2px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{5}</math></td>
 +
<td><math>0101</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{(} v \texttt{)}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{6}</math></td>
 +
<td><math>0110</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{7}</math></td>
 +
<td><math>0111</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{8}</math></td>
 +
<td><math>1000</math></td>
 +
<td style="border-right:2px solid black"><math>u ~ v</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{9}</math></td>
 +
<td><math>1001</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{10}</math></td>
 +
<td><math>1010</math></td>
 +
<td style="border-right:2px solid black"><math>v</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{11}</math></td>
 +
<td><math>1011</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{12}</math></td>
 +
<td><math>1100</math></td>
 +
<td style="border-right:2px solid black"><math>u</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{13}</math></td>
 +
<td><math>1101</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{14}</math></td>
 +
<td><math>1110</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{15}</math></td>
 +
<td><math>1111</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{((~))}</math></td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
</table>
 +
 
 +
<br>
 +
 
 +
<table align="center" cellpadding="1" cellspacing="0" style="text-align:center; width:90%">
 +
 
 +
<caption><font size="+2"><math>\text{Table 5.} ~~ \text{Qualifiers of the Implication Ordering:} ~ \beta_i f = \Upsilon (f, f_i) = \Upsilon (f \Rightarrow f_i)</math></font></caption>
 +
 
 +
<tr>
 +
<td style="border-bottom:2px solid black" align="right">
 +
<math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td>
 +
<td style="border-bottom:2px solid black">
 +
<math>\begin{matrix}1100\\1010\end{matrix}</math></td>
 +
 
 +
<td style="border-bottom:2px solid black; border-right:2px solid black"><math>f</math></td>
 +
<td style="border-bottom:2px solid black"><math>\beta_{0}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\beta_{1}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\beta_{2}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\beta_{3}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\beta_{4}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\beta_{5}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\beta_{6}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\beta_{7}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\beta_{8}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\beta_{9}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\beta_{10}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\beta_{11}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\beta_{12}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\beta_{13}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\beta_{14}</math></td>
 +
<td style="border-bottom:2px solid black"><math>\beta_{15}</math></td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{0}</math></td>
 +
<td><math>0000</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{(~)}</math></td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{1}</math></td>
 +
<td><math>0001</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{2}</math></td>
 +
<td><math>0010</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{3}</math></td>
 +
<td><math>0011</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{4}</math></td>
 +
<td><math>0100</math></td>
 +
<td style="border-right:2px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{5}</math></td>
 +
<td><math>0101</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{(} v \texttt{)}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{6}</math></td>
 +
<td><math>0110</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{7}</math></td>
 +
<td><math>0111</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{8}</math></td>
 +
<td><math>1000</math></td>
 +
<td style="border-right:2px solid black"><math>u ~ v</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{9}</math></td>
 +
<td><math>1001</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{10}</math></td>
 +
<td><math>1010</math></td>
 +
<td style="border-right:2px solid black"><math>v</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{11}</math></td>
 +
<td><math>1011</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{12}</math></td>
 +
<td><math>1100</math></td>
 +
<td style="border-right:2px solid black"><math>u</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{13}</math></td>
 +
<td><math>1101</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{14}</math></td>
 +
<td><math>1110</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{15}</math></td>
 +
<td><math>1111</math></td>
 +
<td style="border-right:2px solid black"><math>\texttt{((~))}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
</table>
 +
 
 +
<br>
 +
 
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 +
|+ <math>\text{Table 7.} ~~ \text{Syllogistic Premisses as Higher Order Indicator Functions}</math>
 +
|
 +
<math>\begin{array}{clcl}
 +
\mathrm{A}
 +
& \mathrm{Universal~Affirmative}
 +
& \mathrm{All} ~ u ~ \mathrm{is} ~ v
 +
& \mathrm{Indicator~of} ~ u \texttt{(} v \texttt{)} = 0
 +
\\
 +
\mathrm{E}
 +
& \mathrm{Universal~Negative}
 +
& \mathrm{All} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}
 +
& \mathrm{Indicator~of} ~ u \cdot v = 0
 +
\\
 +
\mathrm{I}
 +
& \mathrm{Particular~Affirmative}
 +
& \mathrm{Some} ~ u ~ \mathrm{is} ~ v
 +
& \mathrm{Indicator~of} ~ u \cdot v = 1
 +
\\
 +
\mathrm{O}
 +
& \mathrm{Particular~Negative}
 +
& \mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}
 +
& \mathrm{Indicator~of} ~ u \texttt{(} v \texttt{)} = 1
 +
\end{array}</math>
 +
|}
 +
 
 +
<br>
 +
 
 +
<table align="center" cellpadding="4" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:90%">
 +
 
 +
<caption><font size="+2"><math>\text{Table 8.} ~~ \text{Simple Qualifiers of Propositions (Version 1)}</math></font></caption>
 +
 
 +
<tr>
 +
<td width="4%" style="border-bottom:1px solid black" align="right">
 +
<math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td>
 +
<td width="6%" style="border-bottom:1px solid black">
 +
<math>\begin{matrix}1100\\1010\end{matrix}</math></td>
 +
<td width="10%" style="border-bottom:1px solid black; border-right:1px solid black">
 +
<math>f</math></td>
 +
<td width="10%" style="border-bottom:1px solid black">
 +
<math>\begin{smallmatrix}
 +
\texttt{(} \ell_{11} \texttt{)}
 +
\\
 +
\mathrm{No} ~ u
 +
\\
 +
\mathrm{is} ~ v
 +
\end{smallmatrix}</math></td>
 +
<td width="10%" style="border-bottom:1px solid black">
 +
<math>\begin{smallmatrix}
 +
\texttt{(} \ell_{10} \texttt{)}
 +
\\
 +
\mathrm{No} ~ u
 +
\\
 +
\mathrm{is} ~ \texttt{(} v \texttt{)}
 +
\end{smallmatrix}</math></td>
 +
<td width="10%" style="border-bottom:1px solid black">
 +
<math>\begin{smallmatrix}
 +
\texttt{(} \ell_{01} \texttt{)}
 +
\\
 +
\mathrm{No} ~ \texttt{(} u \texttt{)}
 +
\\
 +
\mathrm{is} ~ v
 +
\end{smallmatrix}</math></td>
 +
<td width="10%" style="border-bottom:1px solid black">
 +
<math>\begin{smallmatrix}
 +
\texttt{(} \ell_{00} \texttt{)}
 +
\\
 +
\mathrm{No} ~ \texttt{(} u \texttt{)}
 +
\\
 +
\mathrm{is} ~ \texttt{(} v \texttt{)}
 +
\end{smallmatrix}</math></td>
 +
<td width="10%" style="border-bottom:1px solid black">
 +
<math>\begin{smallmatrix}
 +
\ell_{00}
 +
\\
 +
\mathrm{Some} ~ \texttt{(} u \texttt{)}
 +
\\
 +
\mathrm{is} ~ \texttt{(} v \texttt{)}
 +
\end{smallmatrix}</math></td>
 +
<td width="10%" style="border-bottom:1px solid black">
 +
<math>\begin{smallmatrix}
 +
\ell_{01}
 +
\\
 +
\mathrm{Some} ~ \texttt{(} u \texttt{)}
 +
\\
 +
\mathrm{is} ~ v
 +
\end{smallmatrix}</math></td>
 +
<td width="10%" style="border-bottom:1px solid black">
 +
<math>\begin{smallmatrix}
 +
\ell_{10}
 +
\\
 +
\mathrm{Some} ~ u
 +
\\
 +
\mathrm{is} ~ \texttt{(} v \texttt{)}
 +
\end{smallmatrix}</math></td>
 +
<td width="10%" style="border-bottom:1px solid black">
 +
<math>\begin{smallmatrix}
 +
\ell_{11}
 +
\\
 +
\mathrm{Some} ~ u
 +
\\
 +
\mathrm{is} ~ v
 +
\end{smallmatrix}</math></td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{0}</math></td>
 +
<td><math>0000</math></td>
 +
<td style="border-right:1px solid black"><math>\texttt{(~)}</math></td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{1}</math></td>
 +
<td><math>0001</math></td>
 +
<td style="border-right:1px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{2}</math></td>
 +
<td><math>0010</math></td>
 +
<td style="border-right:1px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{3}</math></td>
 +
<td><math>0011</math></td>
 +
<td style="border-right:1px solid black"><math>\texttt{(} u \texttt{)}</math></td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{4}</math></td>
 +
<td><math>0100</math></td>
 +
<td style="border-right:1px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{5}</math></td>
 +
<td><math>0101</math></td>
 +
<td style="border-right:1px solid black"><math>\texttt{(} v \texttt{)}</math></td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{6}</math></td>
 +
<td><math>0110</math></td>
 +
<td style="border-right:1px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{7}</math></td>
 +
<td><math>0111</math></td>
 +
<td style="border-right:1px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{8}</math></td>
 +
<td><math>1000</math></td>
 +
<td style="border-right:1px solid black"><math>u ~ v</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{9}</math></td>
 +
<td><math>1001</math></td>
 +
<td style="border-right:1px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{10}</math></td>
 +
<td><math>1010</math></td>
 +
<td style="border-right:1px solid black"><math>v</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{11}</math></td>
 +
<td><math>1011</math></td>
 +
<td style="border-right:1px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{12}</math></td>
 +
<td><math>1100</math></td>
 +
<td style="border-right:1px solid black"><math>u</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{13}</math></td>
 +
<td><math>1101</math></td>
 +
<td style="border-right:1px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{14}</math></td>
 +
<td><math>1110</math></td>
 +
<td style="border-right:1px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{15}</math></td>
 +
<td><math>1111</math></td>
 +
<td style="border-right:1px solid black"><math>\texttt{((~))}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
</table>
 +
 
 +
<br>
 +
 
 +
<table align="center" cellpadding="4" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:90%">
 +
 
 +
<caption><font size="+2"><math>\text{Table 9.} ~~ \text{Simple Qualifiers of Propositions (Version 2)}</math></font></caption>
 +
 
 +
<tr>
 +
<td width="4%" style="border-bottom:1px solid black" align="right">
 +
<math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td>
 +
<td width="6%" style="border-bottom:1px solid black">
 +
<math>\begin{matrix}1100\\1010\end{matrix}</math></td>
 +
<td width="10%" style="border-bottom:1px solid black; border-right:1px solid black">
 +
<math>f</math></td>
 +
<td width="10%" style="border-bottom:1px solid black">
 +
<math>\begin{smallmatrix}
 +
\texttt{(} \ell_{11} \texttt{)}
 +
\\
 +
\mathrm{No} ~ u
 +
\\
 +
\mathrm{is} ~ v
 +
\end{smallmatrix}</math></td>
 +
<td width="10%" style="border-bottom:1px solid black">
 +
<math>\begin{smallmatrix}
 +
\texttt{(} \ell_{10} \texttt{)}
 +
\\
 +
\mathrm{No} ~ u
 +
\\
 +
\mathrm{is} ~ \texttt{(} v \texttt{)}
 +
\end{smallmatrix}</math></td>
 +
<td width="10%" style="border-bottom:1px solid black">
 +
<math>\begin{smallmatrix}
 +
\texttt{(} \ell_{01} \texttt{)}
 +
\\
 +
\mathrm{No} ~ \texttt{(} u \texttt{)}
 +
\\
 +
\mathrm{is} ~ v
 +
\end{smallmatrix}</math></td>
 +
<td width="10%" style="border-bottom:1px solid black">
 +
<math>\begin{smallmatrix}
 +
\texttt{(} \ell_{00} \texttt{)}
 +
\\
 +
\mathrm{No} ~ \texttt{(} u \texttt{)}
 +
\\
 +
\mathrm{is} ~ \texttt{(} v \texttt{)}
 +
\end{smallmatrix}</math></td>
 +
<td width="10%" style="border-bottom:1px solid black">
 +
<math>\begin{smallmatrix}
 +
\ell_{00}
 +
\\
 +
\mathrm{Some} ~ \texttt{(} u \texttt{)}
 +
\\
 +
\mathrm{is} ~ \texttt{(} v \texttt{)}
 +
\end{smallmatrix}</math></td>
 +
<td width="10%" style="border-bottom:1px solid black">
 +
<math>\begin{smallmatrix}
 +
\ell_{01}
 +
\\
 +
\mathrm{Some} ~ \texttt{(} u \texttt{)}
 +
\\
 +
\mathrm{is} ~ v
 +
\end{smallmatrix}</math></td>
 +
<td width="10%" style="border-bottom:1px solid black">
 +
<math>\begin{smallmatrix}
 +
\ell_{10}
 +
\\
 +
\mathrm{Some} ~ u
 +
\\
 +
\mathrm{is} ~ \texttt{(} v \texttt{)}
 +
\end{smallmatrix}</math></td>
 +
<td width="10%" style="border-bottom:1px solid black">
 +
<math>\begin{smallmatrix}
 +
\ell_{11}
 +
\\
 +
\mathrm{Some} ~ u
 +
\\
 +
\mathrm{is} ~ v
 +
\end{smallmatrix}</math></td></tr>
 +
 
 +
<tr>
 +
<td style="border-bottom:1px solid black"><math>f_{0}</math></td>
 +
<td style="border-bottom:1px solid black"><math>0000</math></td>
 +
<td style="border-bottom:1px solid black; border-right:1px solid black"><math>\texttt{(~)}</math></td>
 +
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
 +
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
 +
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
 +
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
 +
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
 +
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
 +
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
 +
<td style="border-bottom:1px solid black; background:white; color:black">0</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{1}</math></td>
 +
<td><math>0001</math></td>
 +
<td style="border-right:1px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{2}</math></td>
 +
<td><math>0010</math></td>
 +
<td style="border-right:1px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{4}</math></td>
 +
<td><math>0100</math></td>
 +
<td style="border-right:1px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td></tr>
 +
 
 +
<tr>
 +
<td style="border-bottom:1px solid black"><math>f_{8}</math></td>
 +
<td style="border-bottom:1px solid black"><math>1000</math></td>
 +
<td style="border-bottom:1px solid black; border-right:1px solid black"><math>u ~ v</math></td>
 +
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
 +
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
 +
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
 +
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
 +
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
 +
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
 +
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
 +
<td style="border-bottom:1px solid black; background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{3}</math></td>
 +
<td><math>0011</math></td>
 +
<td style="border-right:1px solid black"><math>\texttt{(} u \texttt{)}</math></td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td></tr>
 +
 
 +
<tr>
 +
<td style="border-bottom:1px solid black"><math>f_{12}</math></td>
 +
<td style="border-bottom:1px solid black"><math>1100</math></td>
 +
<td style="border-bottom:1px solid black; border-right:1px solid black"><math>u</math></td>
 +
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
 +
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
 +
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
 +
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
 +
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
 +
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
 +
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
 +
<td style="border-bottom:1px solid black; background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{6}</math></td>
 +
<td><math>0110</math></td>
 +
<td style="border-right:1px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td></tr>
 +
 
 +
<tr>
 +
<td style="border-bottom:1px solid black"><math>f_{9}</math></td>
 +
<td style="border-bottom:1px solid black"><math>1001</math></td>
 +
<td style="border-bottom:1px solid black; border-right:1px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td>
 +
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
 +
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
 +
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
 +
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
 +
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
 +
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
 +
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
 +
<td style="border-bottom:1px solid black; background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{5}</math></td>
 +
<td><math>0101</math></td>
 +
<td style="border-right:1px solid black"><math>\texttt{(} v \texttt{)}</math></td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td></tr>
 +
 
 +
<tr>
 +
<td style="border-bottom:1px solid black"><math>f_{10}</math></td>
 +
<td style="border-bottom:1px solid black"><math>1010</math></td>
 +
<td style="border-bottom:1px solid black; border-right:1px solid black"><math>v</math></td>
 +
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
 +
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
 +
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
 +
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
 +
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
 +
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
 +
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
 +
<td style="border-bottom:1px solid black; background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{7}</math></td>
 +
<td><math>0111</math></td>
 +
<td style="border-right:1px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{11}</math></td>
 +
<td><math>1011</math></td>
 +
<td style="border-right:1px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{13}</math></td>
 +
<td><math>1101</math></td>
 +
<td style="border-right:1px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td style="border-bottom:1px solid black"><math>f_{14}</math></td>
 +
<td style="border-bottom:1px solid black"><math>1110</math></td>
 +
<td style="border-bottom:1px solid black; border-right:1px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td>
 +
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
 +
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
 +
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
 +
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
 +
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
 +
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
 +
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
 +
<td style="border-bottom:1px solid black; background:black; color:white">1</td></tr>
 +
 
 +
<tr>
 +
<td><math>f_{15}</math></td>
 +
<td><math>1111</math></td>
 +
<td style="border-right:1px solid black"><math>\texttt{((~))}</math></td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:white; color:black">0</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td>
 +
<td style="background:black; color:white">1</td></tr>
 +
 
 +
</table>
 +
 
 +
<br>
 +
 
 +
<table align="center" cellpadding="4" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:90%">
 +
 
 +
<caption><font size="+2"><math>\text{Table 10.} ~~ \text{Relation of Quantifiers to Higher Order Propositions}</math></font></caption>
 +
 
 +
<tr>
 +
<td style="border-bottom:1px solid black"><math>\mathrm{Mnemonic}</math></td>
 +
<td style="border-bottom:1px solid black"><math>\mathrm{Category}</math></td>
 +
<td style="border-bottom:1px solid black"><math>\mathrm{Classical~Form}</math></td>
 +
<td style="border-bottom:1px solid black"><math>\mathrm{Alternate~Form}</math></td>
 +
<td style="border-bottom:1px solid black"><math>\mathrm{Symmetric~Form}</math></td>
 +
<td style="border-bottom:1px solid black"><math>\mathrm{Operator}</math></td></tr>
 +
 
 +
<tr>
 +
<td><math>\begin{matrix}
 +
\mathrm{E}
 +
\\
 +
\mathrm{Exclusive}
 +
\end{matrix}</math></td>
 +
<td><math>\begin{matrix}
 +
\mathrm{Universal}
 +
\\
 +
\mathrm{Negative}
 +
\end{matrix}</math></td>
 +
<td><math>\mathrm{All} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
 +
<td>&nbsp;</td>
 +
<td><math>\mathrm{No} ~ u ~ \mathrm{is} ~ v</math></td>
 +
<td><math>\texttt{(} \ell_{11} \texttt{)}</math></td></tr>
 +
 
 +
<tr>
 +
<td style="border-bottom:1px solid black">
 +
<math>\begin{matrix}
 +
\mathrm{A}
 +
\\
 +
\mathrm{Absolute}
 +
\end{matrix}</math></td>
 +
<td style="border-bottom:1px solid black">
 +
<math>\begin{matrix}
 +
\mathrm{Universal}
 +
\\
 +
\mathrm{Affirmative}
 +
\end{matrix}</math></td>
 +
<td style="border-bottom:1px solid black"><math>\mathrm{All} ~ u ~ \mathrm{is} ~ v</math></td>
 +
<td style="border-bottom:1px solid black">&nbsp;</td>
 +
<td style="border-bottom:1px solid black"><math>\mathrm{No} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
 +
<td style="border-bottom:1px solid black"><math>\texttt{(} \ell_{10} \texttt{)}</math></td></tr>
 +
 
 +
<tr>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td><math>\mathrm{All} ~ v ~ \mathrm{is} ~ u</math></td>
 +
<td><math>\mathrm{No} ~ v ~ \mathrm{is} ~ \texttt{(} u \texttt{)}</math></td>
 +
<td><math>\mathrm{No} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td>
 +
<td><math>\texttt{(} \ell_{01} \texttt{)}</math></td></tr>
 +
 
 +
<tr>
 +
<td style="border-bottom:1px solid black">&nbsp;</td>
 +
<td style="border-bottom:1px solid black">&nbsp;</td>
 +
<td style="border-bottom:1px solid black"><math>\mathrm{All} ~ \texttt{(} v \texttt{)} ~ \mathrm{is} ~ u</math></td>
 +
<td style="border-bottom:1px solid black"><math>\mathrm{No} ~ \texttt{(} v \texttt{)} ~ \mathrm{is} ~ \texttt{(} u \texttt{)}</math></td>
 +
<td style="border-bottom:1px solid black"><math>\mathrm{No} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
 +
<td style="border-bottom:1px solid black"><math>\texttt{(} \ell_{00} \texttt{)}</math></td></tr>
 +
 
 +
<tr>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
 +
<td>&nbsp;</td>
 +
<td><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
 +
<td><math>\ell_{00}</math></td></tr>
 +
 
 +
<tr>
 +
<td style="border-bottom:1px solid black">&nbsp;</td>
 +
<td style="border-bottom:1px solid black">&nbsp;</td>
 +
<td style="border-bottom:1px solid black"><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td>
 +
<td style="border-bottom:1px solid black">&nbsp;</td>
 +
<td style="border-bottom:1px solid black"><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td>
 +
<td style="border-bottom:1px solid black"><math>\ell_{01}</math></td></tr>
 +
 
 +
<tr>
 +
<td><math>\begin{matrix}
 +
\mathrm{O}
 +
\\
 +
\mathrm{Obtrusive}
 +
\end{matrix}</math></td>
 +
<td><math>\begin{matrix}
 +
\mathrm{Particular}
 +
\\
 +
\mathrm{Negative}
 +
\end{matrix}</math></td>
 +
<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
 +
<td>&nbsp;</td>
 +
<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
 +
<td><math>\ell_{10}</math></td></tr>
 +
 
 +
<tr>
 +
<td><math>\begin{matrix}
 +
\mathrm{I}
 +
\\
 +
\mathrm{Indefinite}
 +
\end{matrix}</math></td>
 +
<td><math>\begin{matrix}
 +
\mathrm{Particular}
 +
\\
 +
\mathrm{Affirmative}
 +
\end{matrix}</math></td>
 +
<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ v</math></td>
 +
<td>&nbsp;</td>
 +
<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ v</math></td>
 +
<td><math>\ell_{11}</math></td></tr>
 +
 
 +
</table>
 +
 
 +
<br>
 +
 
 +
==Inquiry Driven Systems==
 +
 
 +
===Table 1.  Sign Relation of Interpreter ''A''===
 +
 
 +
<pre>
 +
Table 1.  Sign Relation of Interpreter A
 +
o---------------o---------------o---------------o
 +
| Object        | Sign          | Interpretant  |
 +
o---------------o---------------o---------------o
 +
| A            | "A"          | "A"          |
 +
| A            | "A"          | "i"          |
 +
| A            | "i"          | "A"          |
 +
| A            | "i"          | "i"          |
 +
| B            | "B"          | "B"          |
 +
| B            | "B"          | "u"          |
 +
| B            | "u"          | "B"          |
 +
| B            | "u"          | "u"          |
 +
o---------------o---------------o---------------o
 
</pre>
 
</pre>
−
 
+
 
−
: <math>\begin{matrix}
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
−
p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\
+
|+ Table 1.  Sign Relation of Interpreter ''A''
−
& = & ((p \land \lnot q) \lor \lnot p) & \and & ((p \land \lnot q) \lor q) \\
+
|- style="background:paleturquoise"
−
& = & ((p \lor \lnot q) \land (\lnot q \lor \lnot p)) & \land & ((p \lor q) \land (\lnot q \lor q)) \\
+
! style="width:20%" | Object
−
& = & (\lnot p \lor \lnot q) & \land & (p \lor q) \\
+
! style="width:20%" | Sign
−
& = & \lnot (p \land q) & \land & (p \lor q)
+
! 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>
 +
 
 +
===Table 2.  Sign Relation of Interpreter ''B''===
 +
 
 +
<pre>
 +
Table 2.  Sign Relation of Interpreter B
 +
o---------------o---------------o---------------o
 +
| Object        | Sign          | Interpretant  |
 +
o---------------o---------------o---------------o
 +
| A            | "A"          | "A"          |
 +
| A            | "A"          | "u"          |
 +
| A            | "u"          | "A"          |
 +
| A            | "u"          | "u"          |
 +
| B            | "B"          | "B"          |
 +
| B            | "B"          | "i"          |
 +
| B            | "i"          | "B"          |
 +
| B            | "i"          | "i"          |
 +
o---------------o---------------o---------------o
 +
</pre>
 +
 
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
 +
|+ Table 2.  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>
 +
 
 +
===Table 3.  Semiotic Partition of Interpreter ''A''===
 +
 
 +
<pre>
 +
Table 3.  A's Semiotic Partition
 +
o-------------------------------o
 +
|      "A"            "i"      |
 +
o-------------------------------o
 +
|      "u"            "B"      |
 +
o-------------------------------o
 +
</pre>
 +
 
 +
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
 +
|+ Table 3.  Semiotic Partition of Interpreter ''A''
 +
|
 +
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| width="50%" | "A"
 +
| width="50%" | "i"
 +
|}
 +
|-
 +
|
 +
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| width="50%" | "u"
 +
| width="50%" | "B"
 +
|}
 +
|}
 +
<br>
 +
 
 +
===Table 4.  Semiotic Partition of Interpreter ''B''===
 +
 
 +
<pre>
 +
Table 4.  B's Semiotic Partition
 +
o---------------o---------------o
 +
|      "A"      |      "i"      |
 +
|              |              |
 +
|      "u"      |      "B"      |
 +
o---------------o---------------o
 +
</pre>
 +
 
 +
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
 +
|+ Table 4.  Semiotic Partition of Interpreter ''B''
 +
|
 +
{| align="center" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:50%"
 +
| "A"
 +
|-
 +
| "u"
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:50%"
 +
| "i"
 +
|-
 +
| "B"
 +
|}
 +
|}
 +
<br>
 +
 
 +
===Table 5.  Alignments of Capacities===
 +
 
 +
<pre>
 +
Table 5.  Alignments of Capacities
 +
o-------------------o-----------------------------o
 +
|      Formal      |          Formative          |
 +
o-------------------o-----------------------------o
 +
|    Objective    |        Instrumental        |
 +
|      Passive      |          Active            |
 +
o-------------------o--------------o--------------o
 +
|    Afforded      |  Possessed  |  Exercised  |
 +
o-------------------o--------------o--------------o
 +
</pre>
 +
 
 +
===Table 6.  Alignments of Capacities in Aristotle===
 +
 
 +
<pre>
 +
Table 6.  Alignments of Capacities in Aristotle
 +
o-------------------o-----------------------------o
 +
|      Matter      |            Form            |
 +
o-------------------o-----------------------------o
 +
|  Potentiality    |          Actuality          |
 +
|    Receptivity    |  Possession  |  Exercise  |
 +
|      Life        |    Sleep    |    Waking    |
 +
|        Wax        |        Impression          |
 +
|        Axe        |    Edge      |  Cutting    |
 +
|        Eye        |  Vision    |    Seeing    |
 +
|      Body        |            Soul            |
 +
o-------------------o-----------------------------o
 +
|      Ship?      |          Sailor?          |
 +
o-------------------o-----------------------------o
 +
</pre>
 +
 
 +
===Table 7.  Synthesis of Alignments===
 +
 
 +
<pre>
 +
Table 7.  Synthesis of Alignments
 +
o-------------------o-----------------------------o
 +
|      Formal      |          Formative          |
 +
o-------------------o-----------------------------o
 +
|    Objective    |        Instrumental        |
 +
|      Passive      |          Active            |
 +
|    Afforded      |  Possessed  |  Exercised  |
 +
|      To Hold      |  To Have    |    To Use    |
 +
|    Receptivity    |  Possession  |  Exercise  |
 +
|  Potentiality    |          Actuality          |
 +
|      Matter      |            Form            |
 +
o-------------------o-----------------------------o
 +
</pre>
 +
 
 +
===Table 8.  Boolean Product===
 +
 
 +
<pre>
 +
Table 8.  Boolean Product
 +
o---------o---------o---------o
 +
|  %*%  %  %0%  |  %1%  |
 +
o=========o=========o=========o
 +
|  %0%  %  %0%  |  %0%  |
 +
o---------o---------o---------o
 +
|  %1%  %  %0%  |  %1%  |
 +
o---------o---------o---------o
 +
</pre>
 +
 
 +
===Table 9.  Boolean Sum===
 +
 
 +
<pre>
 +
Table 9.  Boolean Sum
 +
o---------o---------o---------o
 +
|  %+%  %  %0%  |  %1%  |
 +
o=========o=========o=========o
 +
|  %0%  %  %0%  |  %1%  |
 +
o---------o---------o---------o
 +
|  %1%  %  %1%  |  %0%  |
 +
o---------o---------o---------o
 +
</pre>
 +
 
 +
==Logical Tables==
 +
 
 +
===Table Templates===
 +
 
 +
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
 +
|+ Table 1.  Two Variable Template
 +
|- style="background:paleturquoise"
 +
|
 +
{| align="right" style="background:paleturquoise; text-align:right"
 +
| u :
 +
|-
 +
| v :
 +
|}
 +
|
 +
{| style="background:paleturquoise"
 +
| 1 1 0 0
 +
|-
 +
| 1 0 1 0
 +
|}
 +
|
 +
{| style="background:paleturquoise"
 +
| f
 +
|-
 +
| &nbsp;
 +
|}
 +
|
 +
{| style="background:paleturquoise"
 +
| f
 +
|-
 +
| &nbsp;
 +
|}
 +
|
 +
{| style="background:paleturquoise"
 +
| f
 +
|-
 +
| &nbsp;
 +
|}
 +
|-
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| f<sub>0</sub>
 +
|-
 +
| f<sub>1</sub>
 +
|-
 +
| f<sub>2</sub>
 +
|-
 +
| f<sub>3</sub>
 +
|-
 +
| f<sub>4</sub>
 +
|-
 +
| f<sub>5</sub>
 +
|-
 +
| f<sub>6</sub>
 +
|-
 +
| f<sub>7</sub>
 +
|}
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| 0 0 0 0
 +
|-
 +
| 0 0 0 1
 +
|-
 +
| 0 0 1 0
 +
|-
 +
| 0 0 1 1
 +
|-
 +
| 0 1 0 0
 +
|-
 +
| 0 1 0 1
 +
|-
 +
| 0 1 1 0
 +
|-
 +
| 0 1 1 1
 +
|}
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| f<sub>0</sub>
 +
|-
 +
| f<sub>1</sub>
 +
|-
 +
| f<sub>2</sub>
 +
|-
 +
| f<sub>3</sub>
 +
|-
 +
| f<sub>4</sub>
 +
|-
 +
| f<sub>5</sub>
 +
|-
 +
| f<sub>6</sub>
 +
|-
 +
| f<sub>7</sub>
 +
|}
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| f<sub>0</sub>
 +
|-
 +
| f<sub>1</sub>
 +
|-
 +
| f<sub>2</sub>
 +
|-
 +
| f<sub>3</sub>
 +
|-
 +
| f<sub>4</sub>
 +
|-
 +
| f<sub>5</sub>
 +
|-
 +
| f<sub>6</sub>
 +
|-
 +
| f<sub>7</sub>
 +
|}
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| f<sub>0</sub>
 +
|-
 +
| f<sub>1</sub>
 +
|-
 +
| f<sub>2</sub>
 +
|-
 +
| f<sub>3</sub>
 +
|-
 +
| f<sub>4</sub>
 +
|-
 +
| f<sub>5</sub>
 +
|-
 +
| f<sub>6</sub>
 +
|-
 +
| f<sub>7</sub>
 +
|}
 +
|-
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| f<sub>8</sub>
 +
|-
 +
| f<sub>9</sub>
 +
|-
 +
| f<sub>10</sub>
 +
|-
 +
| f<sub>11</sub>
 +
|-
 +
| f<sub>12</sub>
 +
|-
 +
| f<sub>13</sub>
 +
|-
 +
| f<sub>14</sub>
 +
|-
 +
| f<sub>15</sub>
 +
|}
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| 1 0 0 0
 +
|-
 +
| 1 0 0 1
 +
|-
 +
| 1 0 1 0
 +
|-
 +
| 1 0 1 1
 +
|-
 +
| 1 1 0 0
 +
|-
 +
| 1 1 0 1
 +
|-
 +
| 1 1 1 0
 +
|-
 +
| 1 1 1 1
 +
|}
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| f<sub>8</sub>
 +
|-
 +
| f<sub>9</sub>
 +
|-
 +
| f<sub>10</sub>
 +
|-
 +
| f<sub>11</sub>
 +
|-
 +
| f<sub>12</sub>
 +
|-
 +
| f<sub>13</sub>
 +
|-
 +
| f<sub>14</sub>
 +
|-
 +
| f<sub>15</sub>
 +
|}
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| f<sub>8</sub>
 +
|-
 +
| f<sub>9</sub>
 +
|-
 +
| f<sub>10</sub>
 +
|-
 +
| f<sub>11</sub>
 +
|-
 +
| f<sub>12</sub>
 +
|-
 +
| f<sub>13</sub>
 +
|-
 +
| f<sub>14</sub>
 +
|-
 +
| f<sub>15</sub>
 +
|}
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| f<sub>8</sub>
 +
|-
 +
| f<sub>9</sub>
 +
|-
 +
| f<sub>10</sub>
 +
|-
 +
| f<sub>11</sub>
 +
|-
 +
| f<sub>12</sub>
 +
|-
 +
| f<sub>13</sub>
 +
|-
 +
| f<sub>14</sub>
 +
|-
 +
| f<sub>15</sub>
 +
|}
 +
|}
 +
<br>
 +
 
 +
<font face="courier new">
 +
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
 +
|+ Table 2.  Two Variable Template
 +
|- style="background:paleturquoise"
 +
|
 +
{| align="right" style="background:paleturquoise; text-align:right"
 +
| u :
 +
|-
 +
| v :
 +
|}
 +
|
 +
{| style="background:paleturquoise"
 +
| 1100
 +
|-
 +
| 1010
 +
|}
 +
|
 +
{| style="background:paleturquoise"
 +
| f
 +
|-
 +
| &nbsp;
 +
|}
 +
|
 +
{| style="background:paleturquoise"
 +
| f
 +
|-
 +
| &nbsp;
 +
|}
 +
|
 +
{| style="background:paleturquoise"
 +
| f
 +
|-
 +
| &nbsp;
 +
|}
 +
|-
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| f<sub>0</sub>
 +
|-
 +
| f<sub>1</sub>
 +
|-
 +
| f<sub>2</sub>
 +
|-
 +
| f<sub>3</sub>
 +
|-
 +
| f<sub>4</sub>
 +
|-
 +
| f<sub>5</sub>
 +
|-
 +
| f<sub>6</sub>
 +
|-
 +
| f<sub>7</sub>
 +
|}
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| 0000
 +
|-
 +
| 0001
 +
|-
 +
| 0010
 +
|-
 +
| 0011
 +
|-
 +
| 0100
 +
|-
 +
| 0101
 +
|-
 +
| 0110
 +
|-
 +
| 0111
 +
|}
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| ()
 +
|-
 +
| &nbsp;(u)(v)&nbsp;
 +
|-
 +
| &nbsp;(u)&nbsp;v&nbsp;&nbsp;
 +
|-
 +
| &nbsp;(u)&nbsp;&nbsp;&nbsp;&nbsp;
 +
|-
 +
| &nbsp;&nbsp;u&nbsp;(v)&nbsp;
 +
|-
 +
| &nbsp;&nbsp;&nbsp;&nbsp;(v)&nbsp;
 +
|-
 +
| &nbsp;(u,&nbsp;v)&nbsp;
 +
|-
 +
| &nbsp;(u&nbsp;&nbsp;v)&nbsp;
 +
|}
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| f<sub>0</sub>
 +
|-
 +
| f<sub>1</sub>
 +
|-
 +
| f<sub>2</sub>
 +
|-
 +
| f<sub>3</sub>
 +
|-
 +
| f<sub>4</sub>
 +
|-
 +
| f<sub>5</sub>
 +
|-
 +
| f<sub>6</sub>
 +
|-
 +
| f<sub>7</sub>
 +
|}
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| f<sub>0</sub>
 +
|-
 +
| f<sub>1</sub>
 +
|-
 +
| f<sub>2</sub>
 +
|-
 +
| f<sub>3</sub>
 +
|-
 +
| f<sub>4</sub>
 +
|-
 +
| f<sub>5</sub>
 +
|-
 +
| f<sub>6</sub>
 +
|-
 +
| f<sub>7</sub>
 +
|}
 +
|-
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| f<sub>8</sub>
 +
|-
 +
| f<sub>9</sub>
 +
|-
 +
| f<sub>10</sub>
 +
|-
 +
| f<sub>11</sub>
 +
|-
 +
| f<sub>12</sub>
 +
|-
 +
| f<sub>13</sub>
 +
|-
 +
| f<sub>14</sub>
 +
|-
 +
| f<sub>15</sub>
 +
|}
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| 1000
 +
|-
 +
| 1001
 +
|-
 +
| 1010
 +
|-
 +
| 1011
 +
|-
 +
| 1100
 +
|-
 +
| 1101
 +
|-
 +
| 1110
 +
|-
 +
| 1111
 +
|}
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| &nbsp;&nbsp;u&nbsp;&nbsp;v&nbsp;&nbsp;
 +
|-
 +
| ((u,&nbsp;v))
 +
|-
 +
| &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;v&nbsp;&nbsp;
 +
|-
 +
| &nbsp;(u&nbsp;(v))
 +
|-
 +
| &nbsp;&nbsp;u&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
 +
|-
 +
| ((u)&nbsp;v)&nbsp;
 +
|-
 +
| ((u)(v))
 +
|-
 +
| (())
 +
|}
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| f<sub>8</sub>
 +
|-
 +
| f<sub>9</sub>
 +
|-
 +
| f<sub>10</sub>
 +
|-
 +
| f<sub>11</sub>
 +
|-
 +
| f<sub>12</sub>
 +
|-
 +
| f<sub>13</sub>
 +
|-
 +
| f<sub>14</sub>
 +
|-
 +
| f<sub>15</sub>
 +
|}
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| f<sub>8</sub>
 +
|-
 +
| f<sub>9</sub>
 +
|-
 +
| f<sub>10</sub>
 +
|-
 +
| f<sub>11</sub>
 +
|-
 +
| f<sub>12</sub>
 +
|-
 +
| f<sub>13</sub>
 +
|-
 +
| f<sub>14</sub>
 +
|-
 +
| f<sub>15</sub>
 +
|}
 +
|}
 +
</font><br>
 +
 
 +
===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.
 +
 
 +
<pre>
 +
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)
 +
</pre>
 +
 
 +
<pre>
 +
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)
 +
</pre>
 +
 
 +
<pre>
 +
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)
 +
</pre>
 +
 
 +
: <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>
 +
 
 +
====[[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>
 +
 
 +
==Relational Tables==
 +
 
 +
===Factorization===
 +
 
 +
{| align="center" style="text-align:center; width:60%"
 +
|
 +
{| align="center" style="text-align:center; width:100%"
 +
| <math>\text{Table 7.  Plural Denotation}\!</math>
 +
|}
 +
|-
 +
|
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:100%"
 +
|- style="background:#f0f0ff"
 +
| width="33%" | <math>\text{Object}\!</math>
 +
| width="33%" | <math>\text{Sign}\!</math>
 +
| width="33%" | <math>\text{Interpretant}\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
o_1 \\ o_2 \\ o_3 \\ \ldots \\ o_k \\ \ldots
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
s \\ s \\ s \\ \ldots \\ s \\ \ldots
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\ldots \\ \ldots \\ \ldots \\ \ldots \\ \ldots \\ \ldots
 +
\end{matrix}</math>
 +
|}
 +
|}
 +
 
 +
<br>
 +
 
 +
{| align="center" style="text-align:center; width:60%"
 +
|
 +
{| align="center" style="text-align:center; width:100%"
 +
| <math>\text{Table 8.  Sign Relation}~ L</math>
 +
|}
 +
|-
 +
|
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:100%"
 +
|- style="background:#f0f0ff"
 +
| width="33%" | <math>\text{Object}\!</math>
 +
| width="33%" | <math>\text{Sign}\!</math>
 +
| width="33%" | <math>\text{Interpretant}\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
o_1 \\ o_2 \\ o_3
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
s \\ s \\ s
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\ldots \\ \ldots \\ \ldots
 
\end{matrix}</math>
 
\end{matrix}</math>
−
 
−
====[[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>
 
−
 
−
==Relational Tables==
 
  
 
===Sign Relations===
 
===Sign Relations===

Latest revision as of 03:22, 26 April 2012

Cactus Language

Ascii Tables

o-------------------o
|                   |
|         @         |
|                   |
o-------------------o
|                   |
|         o         |
|         |         |
|         @         |
|                   |
o-------------------o
|                   |
|         a         |
|         @         |
|                   |
o-------------------o
|                   |
|         a         |
|         o         |
|         |         |
|         @         |
|                   |
o-------------------o
|                   |
|       a b c       |
|         @         |
|                   |
o-------------------o
|                   |
|       a b c       |
|       o o o       |
|        \|/        |
|         o         |
|         |         |
|         @         |
|                   |
o-------------------o
|                   |
|         a   b     |
|         o---o     |
|         |         |
|         @         |
|                   |
o-------------------o
|                   |
|       a   b       |
|       o---o       |
|        \ /        |
|         @         |
|                   |
o-------------------o
|                   |
|       a   b       |
|       o---o       |
|        \ /        |
|         o         |
|         |         |
|         @         |
|                   |
o-------------------o
|                   |
|      a  b  c      |
|      o--o--o      |
|       \   /       |
|        \ /        |
|         @         |
|                   |
o-------------------o
|                   |
|      a  b  c      |
|      o  o  o      |
|      |  |  |      |
|      o--o--o      |
|       \   /       |
|        \ /        |
|         @         |
|                   |
o-------------------o
|                   |
|         b  c      |
|         o  o      |
|      a  |  |      |
|      o--o--o      |
|       \   /       |
|        \ /        |
|         @         |
|                   |
o-------------------o

Table 13.  The Existential Interpretation
o----o-------------------o-------------------o-------------------o
| Ex |   Cactus Graph    | Cactus Expression |    Existential    |
|    |                   |                   |  Interpretation   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|  1 |         @         |        " "        |       true.       |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |         o         |                   |                   |
|    |         |         |                   |                   |
|  2 |         @         |        ( )        |      untrue.      |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |         a         |                   |                   |
|  3 |         @         |         a         |         a.        |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |         a         |                   |                   |
|    |         o         |                   |                   |
|    |         |         |                   |                   |
|  4 |         @         |        (a)        |       not a.      |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |       a b c       |                   |                   |
|  5 |         @         |       a b c       |   a and b and c.  |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |       a b c       |                   |                   |
|    |       o o o       |                   |                   |
|    |        \|/        |                   |                   |
|    |         o         |                   |                   |
|    |         |         |                   |                   |
|  6 |         @         |    ((a)(b)(c))    |    a or b or c.   |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |                   |                   |    a implies b.   |
|    |         a   b     |                   |                   |
|    |         o---o     |                   |    if a then b.   |
|    |         |         |                   |                   |
|  7 |         @         |     ( a (b))      |    no a sans b.   |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |       a   b       |                   |                   |
|    |       o---o       |                   | a exclusive-or b. |
|    |        \ /        |                   |                   |
|  8 |         @         |     ( a , b )     | a not equal to b. |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |       a   b       |                   |                   |
|    |       o---o       |                   |                   |
|    |        \ /        |                   |                   |
|    |         o         |                   | a if & only if b. |
|    |         |         |                   |                   |
|  9 |         @         |    (( a , b ))    | a equates with b. |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |      a  b  c      |                   |                   |
|    |      o--o--o      |                   |                   |
|    |       \   /       |                   |                   |
|    |        \ /        |                   |  just one false   |
| 10 |         @         |   ( a , b , c )   |  out of a, b, c.  |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |      a  b  c      |                   |                   |
|    |      o  o  o      |                   |                   |
|    |      |  |  |      |                   |                   |
|    |      o--o--o      |                   |                   |
|    |       \   /       |                   |                   |
|    |        \ /        |                   |   just one true   |
| 11 |         @         |   ((a),(b),(c))   |   among a, b, c.  |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |                   |                   |   genus a over    |
|    |         b  c      |                   |   species b, c.   |
|    |         o  o      |                   |                   |
|    |      a  |  |      |                   |   partition a     |
|    |      o--o--o      |                   |   among b & c.    |
|    |       \   /       |                   |                   |
|    |        \ /        |                   |   whole pie a:    |
| 12 |         @         |   ( a ,(b),(c))   |   slices b, c.    |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o

Table 14.  The Entitative Interpretation
o----o-------------------o-------------------o-------------------o
| En |   Cactus Graph    | Cactus Expression |    Entitative     |
|    |                   |                   |  Interpretation   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|  1 |         @         |        " "        |      untrue.      |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |         o         |                   |                   |
|    |         |         |                   |                   |
|  2 |         @         |        ( )        |       true.       |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |         a         |                   |                   |
|  3 |         @         |         a         |         a.        |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |         a         |                   |                   |
|    |         o         |                   |                   |
|    |         |         |                   |                   |
|  4 |         @         |        (a)        |       not a.      |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |       a b c       |                   |                   |
|  5 |         @         |       a b c       |    a or b or c.   |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |       a b c       |                   |                   |
|    |       o o o       |                   |                   |
|    |        \|/        |                   |                   |
|    |         o         |                   |                   |
|    |         |         |                   |                   |
|  6 |         @         |    ((a)(b)(c))    |   a and b and c.  |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |                   |                   |    a implies b.   |
|    |                   |                   |                   |
|    |         o a       |                   |    if a then b.   |
|    |         |         |                   |                   |
|  7 |         @ b       |      (a) b        |    not a, or b.   |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |       a   b       |                   |                   |
|    |       o---o       |                   | a if & only if b. |
|    |        \ /        |                   |                   |
|  8 |         @         |     ( a , b )     | a equates with b. |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |       a   b       |                   |                   |
|    |       o---o       |                   |                   |
|    |        \ /        |                   |                   |
|    |         o         |                   | a exclusive-or b. |
|    |         |         |                   |                   |
|  9 |         @         |    (( a , b ))    | a not equal to b. |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |      a  b  c      |                   |                   |
|    |      o--o--o      |                   |                   |
|    |       \   /       |                   |                   |
|    |        \ /        |                   | not just one true |
| 10 |         @         |   ( a , b , c )   | out of a, b, c.   |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |      a  b  c      |                   |                   |
|    |      o--o--o      |                   |                   |
|    |       \   /       |                   |                   |
|    |        \ /        |                   |                   |
|    |         o         |                   |                   |
|    |         |         |                   |   just one true   |
| 11 |         @         |  (( a , b , c ))  |   among a, b, c.  |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |      a            |                   |                   |
|    |      o            |                   |   genus a over    |
|    |      |  b  c      |                   |   species b, c.   |
|    |      o--o--o      |                   |                   |
|    |       \   /       |                   |   partition a     |
|    |        \ /        |                   |   among b & c.    |
|    |         o         |                   |                   |
|    |         |         |                   |   whole pie a:    |
| 12 |         @         |  (((a), b , c ))  |   slices b, c.    |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o

Table 15.  Existential & Entitative Interpretations of Cactus Structures
o-----------------o-----------------o-----------------o-----------------o
|  Cactus Graph   |  Cactus String  |  Existential    |   Entitative    |
|                 |                 | Interpretation  | Interpretation  |
o-----------------o-----------------o-----------------o-----------------o
|                 |                 |                 |                 |
|        @        |       " "       |      true       |      false      |
|                 |                 |                 |                 |
o-----------------o-----------------o-----------------o-----------------o
|                 |                 |                 |                 |
|        o        |                 |                 |                 |
|        |        |                 |                 |                 |
|        @        |       ( )       |      false      |      true       |
|                 |                 |                 |                 |
o-----------------o-----------------o-----------------o-----------------o
|                 |                 |                 |                 |
|   C_1 ... C_k   |                 |                 |                 |
|        @        |   C_1 ... C_k   | C_1 & ... & C_k | C_1 v ... v C_k |
|                 |                 |                 |                 |
o-----------------o-----------------o-----------------o-----------------o
|                 |                 |                 |                 |
|  C_1 C_2   C_k  |                 |  Just one       |  Not just one   |
|   o---o-...-o   |                 |                 |                 |
|    \       /    |                 |  of the C_j,    |  of the C_j,    |
|     \     /     |                 |                 |                 |
|      \   /      |                 |  j = 1 to k,    |  j = 1 to k,    |
|       \ /       |                 |                 |                 |
|        @        | (C_1, ..., C_k) |  is not true.   |  is true.       |
|                 |                 |                 |                 |
o-----------------o-----------------o-----------------o-----------------o

Wiki TeX Tables


\(\text{Table A.}~~\text{Existential Interpretation}\)
\(\text{Cactus Graph}\!\) \(\text{Cactus Expression}\!\) \(\text{Interpretation}\!\)
Cactus Node Big Fat.jpg \({}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}\) \(\operatorname{true}.\)
Cactus Spike Big Fat.jpg \(\texttt{(~)}\) \(\operatorname{false}.\)
Cactus A Big.jpg \(a\!\) \(a.\!\)
Cactus (A) Big.jpg \(\texttt{(} a \texttt{)}\)

\(\begin{matrix} \tilde{a} \'"`UNIQ-MathJax1-QINU`"' '''Generalized''' or '''n-ary''' XOR is true when the number of 1-bits is odd. '"`UNIQ--pre-0000001A-QINU`"' '"`UNIQ--pre-0000001B-QINU`"' '"`UNIQ--pre-0000001C-QINU`"' '"`UNIQ-MathJax2-QINU`"' ===='"`UNIQ--h-39--QINU`"'[[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 → q''') and the logical implication '''p implies q''' (symbolized as '''p ⇒ 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 ⇒ q |- | F || F || T |- | F || T || T |- | T || F || F |- | T || T || T |} <br> ===='"`UNIQ--h-40--QINU`"'[[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 | q''' or '''p ↑ 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 ↑ q |- | F || F || T |- | F || T || T |- | T || F || T |- | T || T || F |} <br> ===='"`UNIQ--h-41--QINU`"'[[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 ⊥ q''' or '''p ↓ 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 ↓ q |- | F || F || T |- | F || T || F |- | T || F || F |- | T || T || F |} <br> =='"`UNIQ--h-42--QINU`"'Relational Tables== ==='"`UNIQ--h-43--QINU`"'Factorization=== {| align="center" style="text-align:center; width:60%" | {| align="center" style="text-align:center; width:100%" | \(\text{Table 7. Plural Denotation}\!\)

|- |

\(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\)

\(\begin{matrix} o_1 \\ o_2 \\ o_3 \\ \ldots \\ o_k \\ \ldots \end{matrix}\)

\(\begin{matrix} s \\ s \\ s \\ \ldots \\ s \\ \ldots \end{matrix}\)

\(\begin{matrix} \ldots \\ \ldots \\ \ldots \\ \ldots \\ \ldots \\ \ldots \end{matrix}\)

|}


\(\text{Table 8. Sign Relation}~ L\)
\(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\)

\(\begin{matrix} o_1 \\ o_2 \\ o_3 \end{matrix}\)

\(\begin{matrix} s \\ s \\ s \end{matrix}\)

\(\begin{matrix} \ldots \\ \ldots \\ \ldots \end{matrix}\)

Sign Relations

  O = Object Domain
  S = Sign Domain
  I = Interpretant Domain


  O = {Ann, Bob} = {A, B}
  S = {"Ann", "Bob", "I", "You"} = {"A", "B", "i", "u"}
  I = {"Ann", "Bob", "I", "You"} = {"A", "B", "i", "u"}


LA = Sign Relation of Interpreter A
Object Sign 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"


LB = Sign Relation of Interpreter B
Object Sign 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"


Triadic Relations

Algebraic Examples

L0 = {(x, y, z) ∈ B3 : x + y + z = 0}
X Y Z
0 0 0
0 1 1
1 0 1
1 1 0


L1 = {(x, y, z) ∈ B3 : x + y + z = 1}
X Y Z
0 0 1
0 1 0
1 0 0
1 1 1


Semiotic Examples

LA = Sign Relation of Interpreter A
Object Sign 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"


LB = Sign Relation of Interpreter B
Object Sign 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"


Dyadic Projections

  LOS = projOS(L) = { (o, s) ∈ O × S : (o, s, i) ∈ L for some i ∈ I }
  LSO = projSO(L) = { (s, o) ∈ S × O : (o, s, i) ∈ L for some i ∈ I }
  LIS = projIS(L) = { (i, s) ∈ I × S : (o, s, i) ∈ L for some o ∈ O }
  LSI = projSI(L) = { (s, i) ∈ S × I : (o, s, i) ∈ L for some o ∈ O }
  LOI = projOI(L) = { (o, i) ∈ O × I : (o, s, i) ∈ L for some s ∈ S }
  LIO = projIO(L) = { (i, o) ∈ I × O : (o, s, i) ∈ L for some s ∈ S }


Method 1 : Subtitles as Captions

projOS(LA)
Object Sign
A "A"
A "i"
B "B"
B "u"
projOS(LB)
Object Sign
A "A"
A "u"
B "B"
B "i"


projSI(LA)
Sign Interpretant
"A" "A"
"A" "i"
"i" "A"
"i" "i"
"B" "B"
"B" "u"
"u" "B"
"u" "u"
projSI(LB)
Sign Interpretant
"A" "A"
"A" "u"
"u" "A"
"u" "u"
"B" "B"
"B" "i"
"i" "B"
"i" "i"


projOI(LA)
Object Interpretant
A "A"
A "i"
B "B"
B "u"
projOI(LB)
Object Interpretant
A "A"
A "u"
B "B"
B "i"


Method 2 : Subtitles as Top Rows

projOS(LA)
Object Sign
A "A"
A "i"
B "B"
B "u"
projOS(LB)
Object Sign
A "A"
A "u"
B "B"
B "i"


projSI(LA)
Sign Interpretant
"A" "A"
"A" "i"
"i" "A"
"i" "i"
"B" "B"
"B" "u"
"u" "B"
"u" "u"
projSI(LB)
Sign Interpretant
"A" "A"
"A" "u"
"u" "A"
"u" "u"
"B" "B"
"B" "i"
"i" "B"
"i" "i"


projOI(LA)
Object Interpretant
A "A"
A "i"
B "B"
B "u"
projOI(LB)
Object Interpretant
A "A"
A "u"
B "B"
B "i"


Relation Reduction

Method 1 : Subtitles as Captions

L0 = {(x, y, z) ∈ B3 : x + y + z = 0}
X Y Z
0 0 0
0 1 1
1 0 1
1 1 0


L1 = {(x, y, z) ∈ B3 : x + y + z = 1}
X Y Z
0 0 1
0 1 0
1 0 0
1 1 1


projXY(L0)
X Y
0 0
0 1
1 0
1 1
projXZ(L0)
X Z
0 0
0 1
1 1
1 0
projYZ(L0)
Y Z
0 0
1 1
0 1
1 0


projXY(L1)
X Y
0 0
0 1
1 0
1 1
projXZ(L1)
X Z
0 1
0 0
1 0
1 1
projYZ(L1)
Y Z
0 1
1 0
0 0
1 1


projXY(L0) = projXY(L1) projXZ(L0) = projXZ(L1) projYZ(L0) = projYZ(L1)


LA = Sign Relation of Interpreter A
Object Sign 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"


LB = Sign Relation of Interpreter B
Object Sign 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"


projXY(LA)
Object Sign
A "A"
A "i"
B "B"
B "u"
projXZ(LA)
Object Interpretant
A "A"
A "i"
B "B"
B "u"
projYZ(LA)
Sign Interpretant
"A" "A"
"A" "i"
"i" "A"
"i" "i"
"B" "B"
"B" "u"
"u" "B"
"u" "u"


projXY(LB)
Object Sign
A "A"
A "u"
B "B"
B "i"
projXZ(LB)
Object Interpretant
A "A"
A "u"
B "B"
B "i"
projYZ(LB)
Sign Interpretant
"A" "A"
"A" "u"
"u" "A"
"u" "u"
"B" "B"
"B" "i"
"i" "B"
"i" "i"


projXY(LA) ≠ projXY(LB) projXZ(LA) ≠ projXZ(LB) projYZ(LA) ≠ projYZ(LB)


Method 2 : Subtitles as Top Rows

L0 = {(x, y, z) ∈ B3 : x + y + z = 0}
X Y Z
0 0 0
0 1 1
1 0 1
1 1 0


L1 = {(x, y, z) ∈ B3 : x + y + z = 1}
X Y Z
0 0 1
0 1 0
1 0 0
1 1 1


projXY(L0)
X Y
0 0
0 1
1 0
1 1
projXZ(L0)
X Z
0 0
0 1
1 1
1 0
projYZ(L0)
Y Z
0 0
1 1
0 1
1 0


projXY(L1)
X Y
0 0
0 1
1 0
1 1
projXZ(L1)
X Z
0 1
0 0
1 0
1 1
projYZ(L1)
Y Z
0 1
1 0
0 0
1 1


projXY(L0) = projXY(L1) projXZ(L0) = projXZ(L1) projYZ(L0) = projYZ(L1)


LA = Sign Relation of Interpreter A
Object Sign 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"


LB = Sign Relation of Interpreter B
Object Sign 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"


projXY(LA)
Object Sign
A "A"
A "i"
B "B"
B "u"
projXZ(LA)
Object Interpretant
A "A"
A "i"
B "B"
B "u"
projYZ(LA)
Sign Interpretant
"A" "A"
"A" "i"
"i" "A"
"i" "i"
"B" "B"
"B" "u"
"u" "B"
"u" "u"


projXY(LB)
Object Sign
A "A"
A "u"
B "B"
B "i"
projXZ(LB)
Object Interpretant
A "A"
A "u"
B "B"
B "i"
projYZ(LB)
Sign Interpretant
"A" "A"
"A" "u"
"u" "A"
"u" "u"
"B" "B"
"B" "i"
"i" "B"
"i" "i"


projXY(LA) ≠ projXY(LB) projXZ(LA) ≠ projXZ(LB) projYZ(LA) ≠ projYZ(LB)


Formatted Text Display

So in a triadic fact, say, the example
A gives B to C
we make no distinction in the ordinary logic of relations between the 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:
A gives B to C 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

Binary Operations
x0 x1 2f0 2f1 2f2 2f3 2f4 2f5 2f6 2f7 2f8 2f9 2f10 2f11 2f12 2f13 2f14 2f15
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


Draft 1

TRUTH TABLES FOR THE BOOLEAN OPERATIONS OF ARITY UP TO 2
Constants
0f0 0f1
0 1
    
Unary Operations
x0 1f0 1f1 1f2 1f3
0 0 1 0 1
1 0 0 1 1
    
Binary Operations
x0 x1 2f0 2f1 2f2 2f3 2f4 2f5 2f6 2f7 2f8 2f9 2f10 2f11 2f12 2f13 2f14 2f15
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

Draft 2

TRUTH TABLES FOR THE BOOLEAN OPERATIONS OF ARITY UP TO 2
Constants
0f0 0f1
0 1
    
Unary Operations
x0 1f0 1f1 1f2 1f3
0 0 1 0 1
1 0 0 1 1
    
Binary Operations
x0 x1 2f0 2f1 2f2 2f3 2f4 2f5 2f6 2f7 2f8 2f9 2f10 2f11 2f12 2f13 2f14 2f15
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

Inquiry and Analogy

Test Patterns

1 0 1 0 1 0 1 0
0 1 0 1 0 1 0 1


1 0 1 0 1 0 1 0
0 1 0 1 0 1 0 1


1 0 1 0 1 0 1 0
0 1 0 1 0 1 0 1


Table 10

Table 10. Higher Order Propositions (n = 1)
\(x\): 1 0 \(f\) \(m_0\) \(m_1\) \(m_2\) \(m_3\) \(m_4\) \(m_5\) \(m_6\) \(m_7\) \(m_8\) \(m_9\) \(m_{10}\) \(m_{11}\) \(m_{12}\) \(m_{13}\) \(m_{14}\) \(m_{15}\)
\(f_0\) 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


Table 10. Higher Order Propositions (n = 1)
\(x:\) 1 0 \(f\!\) \(m_0\) \(m_1\) \(m_2\) \(m_3\) \(m_4\) \(m_5\) \(m_6\) \(m_7\) \(m_8\) \(m_9\) \(m_{10}\) \(m_{11}\) \(m_{12}\) \(m_{13}\) \(m_{14}\) \(m_{15}\)
\(f_0\) 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


Table 11

Table 11. Interpretive Categories for Higher Order Propositions (n = 1)
Measure Happening Exactness Existence Linearity Uniformity Information
\(m_0\!\) Nothing happens          
\(m_1\!\)   Just false Nothing exists      
\(m_2\!\)   Just not \(x\!\)        
\(m_3\!\)     Nothing is \(x\!\)      
\(m_4\!\)   Just \(x\!\)        
\(m_5\!\)     Everything is \(x\!\) \(f\!\) is linear    
\(m_6\!\)         \(f\!\) is not uniform \(f\!\) is informed
\(m_7\!\)   Not just true        
\(m_8\!\)   Just true        
\(m_9\!\)         \(f\!\) is uniform \(f\!\) is not informed
\(m_{10}\!\)     Something is not \(x\!\) \(f\!\) is not linear    
\(m_{11}\!\)   Not just \(x\!\)        
\(m_{12}\!\)     Something is \(x\!\)      
\(m_{13}\!\)   Not just not \(x\!\)        
\(m_{14}\!\)   Not just false Something exists      
\(m_{15}\!\) Anything happens          


Table 12

Table 12. Higher Order Propositions (n = 2)
\(x:\)
\(y:\) 		1100
1010 		\(f\!\) \(m_0\) \(m_1\) \(m_2\) \(m_3\) \(m_4\) \(m_5\) \(m_6\) \(m_7\) \(m_8\) \(m_9\) \(m_{10}\) \(m_{11}\) \(m_{12}\) \(m_{13}\) \(m_{14}\) \(m_{15}\) \(m_{16}\) \(m_{17}\) \(m_{18}\) \(m_{19}\) \(m_{20}\) \(m_{21}\) \(m_{22}\) \(m_{23}\)
\(f_0\) 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
\(f_1\) 0001 \((x)(y)\!\)     1 1 0 0 1 1 0 0 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 0 0 0 0 1 1 1 1
\(f_3\) 0011 \((x)\!\)                 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0
\(f_4\) 0100 \(x (y)\!\)                                 1 1 1 1 1 1 1 1
\(f_5\) 0101 \((y)\!\)                                                
\(f_6\) 0110 \((x, y)\!\)                                                
\(f_7\) 0111 \((x y)\!\)                                                
\(f_8\) 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 \(((~))\!\)                                                


Table 12. Higher Order Propositions (n = 2)
\(u:\)
\(v:\) 		1100
1010 		\(f\!\) \(m_0\) \(m_1\) \(m_2\) \(m_3\) \(m_4\) \(m_5\) \(m_6\) \(m_7\) \(m_8\) \(m_9\) \(m_{10}\) \(m_{11}\) \(m_{12}\) \(m_{13}\) \(m_{14}\) \(m_{15}\) \(m_{16}\) \(m_{17}\) \(m_{18}\) \(m_{19}\) \(m_{20}\) \(m_{21}\) \(m_{22}\) \(m_{23}\)
\(f_0\) 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
\(f_1\) 0001 \((u)(v)\!\) 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
\(f_2\) 0010 \((u) v\!\) 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
\(f_3\) 0011 \((u)\!\) 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0
\(f_4\) 0100 \(u (v)\!\) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
\(f_5\) 0101 \((v)\!\) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
\(f_6\) 0110 \((u, v)\!\) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
\(f_7\) 0111 \((u v)\!\) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
\(f_8\) 1000 \(u v\!\) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
\(f_9\) 1001 \(((u, v))\!\) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
\(f_{10}\) 1010 \(v\!\) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
\(f_{11}\) 1011 \((u (v))\!\) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
\(f_{12}\) 1100 \(u\!\) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
\(f_{13}\) 1101 \(((u) v)\!\) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
\(f_{14}\) 1110 \(((u)(v))\!\) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
\(f_{15}\) 1111 \(((~))\!\) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0


Table 13

Table 13. Qualifiers of Implication Ordering:  \(\alpha_i f = \Upsilon (f_i, f) = \Upsilon (f_i \Rightarrow f)\)
\(u:\)
\(v:\) 		1100
1010 		\(f\!\) \(\alpha_0\) \(\alpha_1\) \(\alpha_2\) \(\alpha_3\) \(\alpha_4\) \(\alpha_5\) \(\alpha_6\) \(\alpha_7\) \(\alpha_8\) \(\alpha_9\) \(\alpha_{10}\) \(\alpha_{11}\) \(\alpha_{12}\) \(\alpha_{13}\) \(\alpha_{14}\) \(\alpha_{15}\)
\(f_0\) 0000 \((~)\) 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
\(f_1\) 0001 \((u)(v)\!\) 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
\(f_2\) 0010 \((u) v\!\) 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
\(f_3\) 0011 \((u)\!\) 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
\(f_4\) 0100 \(u (v)\!\) 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
\(f_5\) 0101 \((v)\!\) 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0
\(f_6\) 0110 \((u, v)\!\) 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0
\(f_7\) 0111 \((u v)\!\) 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0
\(f_8\) 1000 \(u v\!\) 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
\(f_9\) 1001 \(((u, v))\!\) 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0
\(f_{10}\) 1010 \(v\!\) 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0
\(f_{11}\) 1011 \((u (v))\!\) 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0
\(f_{12}\) 1100 \(u\!\) 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0
\(f_{13}\) 1101 \(((u) v)\!\) 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0
\(f_{14}\) 1110 \(((u)(v))\!\) 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
\(f_{15}\) 1111 \(((~))\) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1


Table 14

Table 14. Qualifiers of Implication Ordering:  \(\beta_i f = \Upsilon (f, f_i) = \Upsilon (f \Rightarrow f_i)\)
\(u:\)
\(v:\) 		1100
1010 		\(f\!\) \(\beta_0\) \(\beta_1\) \(\beta_2\) \(\beta_3\) \(\beta_4\) \(\beta_5\) \(\beta_6\) \(\beta_7\) \(\beta_8\) \(\beta_9\) \(\beta_{10}\) \(\beta_{11}\) \(\beta_{12}\) \(\beta_{13}\) \(\beta_{14}\) \(\beta_{15}\)
\(f_0\) 0000 \((~)\) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
\(f_1\) 0001 \((u)(v)\!\) 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
\(f_2\) 0010 \((u) v\!\) 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
\(f_3\) 0011 \((u)\!\) 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1
\(f_4\) 0100 \(u (v)\!\) 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
\(f_5\) 0101 \((v)\!\) 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1
\(f_6\) 0110 \((u, v)\!\) 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1
\(f_7\) 0111 \((u v)\!\) 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
\(f_8\) 1000 \(u v\!\) 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
\(f_9\) 1001 \(((u, v))\!\) 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1
\(f_{10}\) 1010 \(v\!\) 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1
\(f_{11}\) 1011 \((u (v))\!\) 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1
\(f_{12}\) 1100 \(u\!\) 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1
\(f_{13}\) 1101 \(((u) v)\!\) 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1
\(f_{14}\) 1110 \(((u)(v))\!\) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1
\(f_{15}\) 1111 \(((~))\!\) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1


Figure 15

Table 16

Table 16. Syllogistic Premisses as Higher Order Indicator Functions

\(\begin{array}{clcl} \mathrm{A} & \mathrm{Universal~Affirmative} & \mathrm{All}\ u\ \mathrm{is}\ v & \mathrm{Indicator~of}\ u (v) = 0 \\ \mathrm{E} & \mathrm{Universal~Negative} & \mathrm{All}\ u\ \mathrm{is}\ (v) & \mathrm{Indicator~of}\ u \cdot v = 0 \\ \mathrm{I} & \mathrm{Particular~Affirmative} & \mathrm{Some}\ u\ \mathrm{is}\ v & \mathrm{Indicator~of}\ u \cdot v = 1 \\ \mathrm{O} & \mathrm{Particular~Negative} & \mathrm{Some}\ u\ \mathrm{is}\ (v) & \mathrm{Indicator~of}\ u (v) = 1 \\ \end{array}\)


Table 17

Table 17. Simple Qualifiers of Propositions (Version 1)
\(u:\)
\(v:\) 		1100
1010 		\(f\!\) \((\ell_{11})\)
\(\text{No } u \)
\(\text{is } v \) 		\((\ell_{10})\)
\(\text{No } u \)
\(\text{is }(v)\) 		\((\ell_{01})\)
\(\text{No }(u)\)
\(\text{is } v \) 		\((\ell_{00})\)
\(\text{No }(u)\)
\(\text{is }(v)\) 		\( \ell_{00} \)
\(\text{Some }(u)\)
\(\text{is }(v)\) 		\( \ell_{01} \)
\(\text{Some }(u)\)
\(\text{is } v \) 		\( \ell_{10} \)
\(\text{Some } u \)
\(\text{is }(v)\) 		\( \ell_{11} \)
\(\text{Some } u \)
\(\text{is } v \)
\(f_0\) 0000 \((~)\) 1 1 1 1 0 0 0 0
\(f_1\) 0001 \((u)(v)\!\) 1 1 1 0 1 0 0 0
\(f_2\) 0010 \((u) v\!\) 1 1 0 1 0 1 0 0
\(f_3\) 0011 \((u)\!\) 1 1 0 0 1 1 0 0
\(f_4\) 0100 \(u (v)\!\) 1 0 1 1 0 0 1 0
\(f_5\) 0101 \((v)\!\) 1 0 1 0 1 0 1 0
\(f_6\) 0110 \((u, v)\!\) 1 0 0 1 0 1 1 0
\(f_7\) 0111 \((u v)\!\) 1 0 0 0 1 1 1 0
\(f_8\) 1000 \(u v\!\) 0 1 1 1 0 0 0 1
\(f_9\) 1001 \(((u, v))\!\) 0 1 1 0 1 0 0 1
\(f_{10}\) 1010 \(v\!\) 0 1 0 1 0 1 0 1
\(f_{11}\) 1011 \((u (v))\!\) 0 1 0 0 1 1 0 1
\(f_{12}\) 1100 \(u\!\) 0 0 1 1 0 0 1 1
\(f_{13}\) 1101 \(((u) v)\!\) 0 0 1 0 1 0 1 1
\(f_{14}\) 1110 \(((u)(v))\!\) 0 0 0 1 0 1 1 1
\(f_{15}\) 1111 \(((~))\) 0 0 0 0 1 1 1 1


Table 18

Table 18. Simple Qualifiers of Propositions (Version 2)
\(u:\)
\(v:\) 		1100
1010 		\(f\!\) \((\ell_{11})\)
\(\text{No } u \)
\(\text{is } v \) 		\((\ell_{10})\)
\(\text{No } u \)
\(\text{is }(v)\) 		\((\ell_{01})\)
\(\text{No }(u)\)
\(\text{is } v \) 		\((\ell_{00})\)
\(\text{No }(u)\)
\(\text{is }(v)\) 		\( \ell_{00} \)
\(\text{Some }(u)\)
\(\text{is }(v)\) 		\( \ell_{01} \)
\(\text{Some }(u)\)
\(\text{is } v \) 		\( \ell_{10} \)
\(\text{Some } u \)
\(\text{is }(v)\) 		\( \ell_{11} \)
\(\text{Some } u \)
\(\text{is } v \)
\(f_0\) 0000 \((~)\) 1 1 1 1 0 0 0 0
\(f_1\) 0001 \((u)(v)\!\) 1 1 1 0 1 0 0 0
\(f_2\) 0010 \((u) v\!\) 1 1 0 1 0 1 0 0
\(f_4\) 0100 \(u (v)\!\) 1 0 1 1 0 0 1 0
\(f_8\) 1000 \(u v\!\) 0 1 1 1 0 0 0 1
\(f_3\) 0011 \((u)\!\) 1 1 0 0 1 1 0 0
\(f_{12}\) 1100 \(u\!\) 0 0 1 1 0 0 1 1
\(f_6\) 0110 \((u, v)\!\) 1 0 0 1 0 1 1 0
\(f_9\) 1001 \(((u, v))\!\) 0 1 1 0 1 0 0 1
\(f_5\) 0101 \((v)\!\) 1 0 1 0 1 0 1 0
\(f_{10}\) 1010 \(v\!\) 0 1 0 1 0 1 0 1
\(f_7\) 0111 \((u v)\!\) 1 0 0 0 1 1 1 0
\(f_{11}\) 1011 \((u (v))\!\) 0 1 0 0 1 1 0 1
\(f_{13}\) 1101 \(((u) v)\!\) 0 0 1 0 1 0 1 1
\(f_{14}\) 1110 \(((u)(v))\!\) 0 0 0 1 0 1 1 1
\(f_{15}\) 1111 \(((~))\) 0 0 0 0 1 1 1 1


Table 19

Table 19. Relation of Quantifiers to Higher Order Propositions
\(\text{Mnemonic}\) \(\text{Category}\) \(\text{Classical Form}\) \(\text{Alternate Form}\) \(\text{Symmetric Form}\) \(\text{Operator}\)
\(\text{E}\!\)
\(\text{Exclusive}\) 		\(\text{Universal}\)
\(\text{Negative}\) 		\(\text{All}\ u\ \text{is}\ (v)\)   \(\text{No}\ u\ \text{is}\ v \) \((\ell_{11})\)
\(\text{A}\!\)
\(\text{Absolute}\) 		\(\text{Universal}\)
\(\text{Affirmative}\) 		\(\text{All}\ u\ \text{is}\ v \)   \(\text{No}\ u\ \text{is}\ (v)\) \((\ell_{10})\)
    \(\text{All}\ v\ \text{is}\ u \) \(\text{No}\ v\ \text{is}\ (u)\) \(\text{No}\ (u)\ \text{is}\ v \) \((\ell_{01})\)
    \(\text{All}\ (v)\ \text{is}\ u \) \(\text{No}\ (v)\ \text{is}\ (u)\) \(\text{No}\ (u)\ \text{is}\ (v)\) \((\ell_{00})\)
    \(\text{Some}\ (u)\ \text{is}\ (v)\)   \(\text{Some}\ (u)\ \text{is}\ (v)\) \(\ell_{00}\!\)
    \(\text{Some}\ (u)\ \text{is}\ v\)   \(\text{Some}\ (u)\ \text{is}\ v\) \(\ell_{01}\!\)
\(\text{O}\!\)
\(\text{Obtrusive}\) 		\(\text{Particular}\)
\(\text{Negative}\) 		\(\text{Some}\ u\ \text{is}\ (v)\)   \(\text{Some}\ u\ \text{is}\ (v)\) \(\ell_{10}\!\)
\(\text{I}\!\)
\(\text{Indefinite}\) 		\(\text{Particular}\)
\(\text{Affirmative}\) 		\(\text{Some}\ u\ \text{is}\ v\)   \(\text{Some}\ u\ \text{is}\ v\) \(\ell_{11}\!\)


Retrieved from "https://mywikibiz.com/index.php?title=User:Jon_Awbrey/TABLE&oldid=156537"