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

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{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
 
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
−
|+ <math>\text{Table 0.}~~\text{Propositional Forms on Two Variables}</math>
+
|+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math>
 
|- style="background:#f0f0ff"
 
|- style="background:#f0f0ff"
 
| style="width:15%" |
 
| style="width:15%" |
Line 1,438: Line 1,438:
 
| <math>\text{true}\!</math>
 
| <math>\text{true}\!</math>
 
| <math>1\!</math>
 
| <math>1\!</math>
 +
|}
 +
 +
<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;
 +
|-
 +
| f<sub>0</sub>
 +
| f<sub>0000</sub>
 +
| 0 0 0 0
 +
| (&nbsp;)
 +
| false
 +
| 0
 +
|-
 +
|
 +
{| 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>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<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"
 +
|
 +
<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>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p>&not;x &and; &not;y</p>
 +
<p>&not;x &and; y</p>
 +
<p>x &and; &not;y</p>
 +
<p>x &and; y</p>
 +
|}
 +
|-
 +
|
 +
{| align="center"
 +
|
 +
<p>f<sub>3</sub></p>
 +
<p>f<sub>12</sub></p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<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>
 +
|}
 +
|-
 +
|
 +
{| 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>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p>x not equal to y</p>
 +
<p>x equal to y</p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p>x &ne; y</p>
 +
<p>x = y</p>
 +
|}
 +
|-
 +
|
 +
{| align="center"
 +
|
 +
<p>f<sub>5</sub></p>
 +
<p>f<sub>10</sub></p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p>f<sub>0101</sub></p>
 +
<p>f<sub>1010</sub></p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p>0 1 0 1</p>
 +
<p>1 0 1 0</p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<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
 
|}
 
|}
  

Revision as of 22:18, 29 May 2009

Differential Logic

Ascii Tables

Table A1.  Propositional Forms On Two Variables
o---------o---------o---------o----------o------------------o----------o
| L_1     | L_2     | L_3     | L_4      | L_5              | L_6      |
|         |         |         |          |                  |          |
| Decimal | Binary  | Vector  | Cactus   | English          | Ordinary |
o---------o---------o---------o----------o------------------o----------o
|         |       x : 1 1 0 0 |          |                  |          |
|         |       y : 1 0 1 0 |          |                  |          |
o---------o---------o---------o----------o------------------o----------o
|         |         |         |          |                  |          |
| f_0     | f_0000  | 0 0 0 0 |    ()    | false            |    0     |
|         |         |         |          |                  |          |
| f_1     | f_0001  | 0 0 0 1 |  (x)(y)  | neither x nor y  | ~x & ~y  |
|         |         |         |          |                  |          |
| f_2     | f_0010  | 0 0 1 0 |  (x) y   | y and not x      | ~x &  y  |
|         |         |         |          |                  |          |
| f_3     | f_0011  | 0 0 1 1 |  (x)     | not x            | ~x       |
|         |         |         |          |                  |          |
| f_4     | f_0100  | 0 1 0 0 |   x (y)  | x and not y      |  x & ~y  |
|         |         |         |          |                  |          |
| f_5     | f_0101  | 0 1 0 1 |     (y)  | not y            |      ~y  |
|         |         |         |          |                  |          |
| f_6     | f_0110  | 0 1 1 0 |  (x, y)  | x not equal to y |  x +  y  |
|         |         |         |          |                  |          |
| f_7     | f_0111  | 0 1 1 1 |  (x  y)  | not both x and y | ~x v ~y  |
|         |         |         |          |                  |          |
| f_8     | f_1000  | 1 0 0 0 |   x  y   | x and y          |  x &  y  |
|         |         |         |          |                  |          |
| f_9     | f_1001  | 1 0 0 1 | ((x, y)) | x equal to y     |  x =  y  |
|         |         |         |          |                  |          |
| f_10    | f_1010  | 1 0 1 0 |      y   | y                |       y  |
|         |         |         |          |                  |          |
| f_11    | f_1011  | 1 0 1 1 |  (x (y)) | not x without y  |  x => y  |
|         |         |         |          |                  |          |
| f_12    | f_1100  | 1 1 0 0 |   x      | x                |  x       |
|         |         |         |          |                  |          |
| f_13    | f_1101  | 1 1 0 1 | ((x) y)  | not y without x  |  x <= y  |
|         |         |         |          |                  |          |
| f_14    | f_1110  | 1 1 1 0 | ((x)(y)) | x or y           |  x v  y  |
|         |         |         |          |                  |          |
| f_15    | f_1111  | 1 1 1 1 |   (())   | true             |    1     |
|         |         |         |          |                  |          |
o---------o---------o---------o----------o------------------o----------o

Table A2.  Propositional Forms On Two Variables
o---------o---------o---------o----------o------------------o----------o
| L_1     | L_2     | L_3     | L_4      | L_5              | L_6      |
|         |         |         |          |                  |          |
| Decimal | Binary  | Vector  | Cactus   | English          | Ordinary |
o---------o---------o---------o----------o------------------o----------o
|         |       x : 1 1 0 0 |          |                  |          |
|         |       y : 1 0 1 0 |          |                  |          |
o---------o---------o---------o----------o------------------o----------o
|         |         |         |          |                  |          |
| f_0     | f_0000  | 0 0 0 0 |    ()    | false            |    0     |
|         |         |         |          |                  |          |
o---------o---------o---------o----------o------------------o----------o
|         |         |         |          |                  |          |
| f_1     | f_0001  | 0 0 0 1 |  (x)(y)  | neither x nor y  | ~x & ~y  |
|         |         |         |          |                  |          |
| f_2     | f_0010  | 0 0 1 0 |  (x) y   | y and not x      | ~x &  y  |
|         |         |         |          |                  |          |
| f_4     | f_0100  | 0 1 0 0 |   x (y)  | x and not y      |  x & ~y  |
|         |         |         |          |                  |          |
| f_8     | f_1000  | 1 0 0 0 |   x  y   | x and y          |  x &  y  |
|         |         |         |          |                  |          |
o---------o---------o---------o----------o------------------o----------o
|         |         |         |          |                  |          |
| f_3     | f_0011  | 0 0 1 1 |  (x)     | not x            | ~x       |
|         |         |         |          |                  |          |
| f_12    | f_1100  | 1 1 0 0 |   x      | x                |  x       |
|         |         |         |          |                  |          |
o---------o---------o---------o----------o------------------o----------o
|         |         |         |          |                  |          |
| f_6     | f_0110  | 0 1 1 0 |  (x, y)  | x not equal to y |  x +  y  |
|         |         |         |          |                  |          |
| f_9     | f_1001  | 1 0 0 1 | ((x, y)) | x equal to y     |  x =  y  |
|         |         |         |          |                  |          |
o---------o---------o---------o----------o------------------o----------o
|         |         |         |          |                  |          |
| 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

Table A3.  Ef Expanded Over Differential Features {dx, dy}
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
|      |     f      |   T_11 f   |   T_10 f   |   T_01 f   |   T_00 f   |
|      |            |            |            |            |            |
|      |            | Ef| dx dy  | Ef| dx(dy) | Ef| (dx)dy | Ef|(dx)(dy)|
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_1  |   (x)(y)   |    x  y    |    x (y)   |   (x) y    |   (x)(y)   |
|      |            |            |            |            |            |
| f_2  |   (x) y    |    x (y)   |    x  y    |   (x)(y)   |   (x) y    |
|      |            |            |            |            |            |
| f_4  |    x (y)   |   (x) y    |   (x)(y)   |    x  y    |    x (y)   |
|      |            |            |            |            |            |
| f_8  |    x  y    |   (x)(y)   |   (x) y    |    x (y)   |    x  y    |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_3  |   (x)      |    x       |    x       |   (x)      |   (x)      |
|      |            |            |            |            |            |
| f_12 |    x       |   (x)      |   (x)      |    x       |    x       |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_6  |   (x, y)   |   (x, y)   |  ((x, y))  |  ((x, y))  |   (x, y)   |
|      |            |            |            |            |            |
| f_9  |  ((x, y))  |  ((x, y))  |   (x, y)   |   (x, y)   |  ((x, y))  |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_5  |      (y)   |       y    |      (y)   |       y    |      (y)   |
|      |            |            |            |            |            |
| f_10 |       y    |      (y)   |       y    |      (y)   |       y    |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_7  |   (x  y)   |  ((x)(y))  |  ((x) y)   |   (x (y))  |   (x  y)   |
|      |            |            |            |            |            |
| f_11 |   (x (y))  |  ((x) y)   |  ((x)(y))  |   (x  y)   |   (x (y))  |
|      |            |            |            |            |            |
| f_13 |  ((x) y)   |   (x (y))  |   (x  y)   |  ((x)(y))  |  ((x) y)   |
|      |            |            |            |            |            |
| f_14 |  ((x)(y))  |   (x  y)   |   (x (y))  |  ((x) y)   |  ((x)(y))  |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_15 |    (())    |    (())    |    (())    |    (())    |    (())    |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|                   |            |            |            |            |
| Fixed Point Total |      4     |      4     |      4     |     16     |
|                   |            |            |            |            |
o-------------------o------------o------------o------------o------------o

Table A4.  Df Expanded Over Differential Features {dx, dy}
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
|      |     f      | Df| dx dy  | Df| dx(dy) | Df| (dx)dy | Df|(dx)(dy)|
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_1  |   (x)(y)   |  ((x, y))  |    (y)     |    (x)     |     ()     |
|      |            |            |            |            |            |
| f_2  |   (x) y    |   (x, y)   |     y      |    (x)     |     ()     |
|      |            |            |            |            |            |
| f_4  |    x (y)   |   (x, y)   |    (y)     |     x      |     ()     |
|      |            |            |            |            |            |
| f_8  |    x  y    |  ((x, y))  |     y      |     x      |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_3  |   (x)      |    (())    |    (())    |     ()     |     ()     |
|      |            |            |            |            |            |
| f_12 |    x       |    (())    |    (())    |     ()     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_6  |   (x, y)   |     ()     |    (())    |    (())    |     ()     |
|      |            |            |            |            |            |
| f_9  |  ((x, y))  |     ()     |    (())    |    (())    |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_5  |      (y)   |    (())    |     ()     |    (())    |     ()     |
|      |            |            |            |            |            |
| f_10 |       y    |    (())    |     ()     |    (())    |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_7  |   (x  y)   |  ((x, y))  |     y      |     x      |     ()     |
|      |            |            |            |            |            |
| f_11 |   (x (y))  |   (x, y)   |    (y)     |     x      |     ()     |
|      |            |            |            |            |            |
| f_13 |  ((x) y)   |   (x, y)   |     y      |    (x)     |     ()     |
|      |            |            |            |            |            |
| f_14 |  ((x)(y))  |  ((x, y))  |    (y)     |    (x)     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_15 |    (())    |     ()     |     ()     |     ()     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o

Table A5.  Ef Expanded Over Ordinary Features {x, y}
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
|      |     f      |  Ef | xy   | Ef | x(y)  | Ef | (x)y  | Ef | (x)(y)|
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_1  |   (x)(y)   |   dx  dy   |   dx (dy)  |  (dx) dy   |  (dx)(dy)  |
|      |            |            |            |            |            |
| f_2  |   (x) y    |   dx (dy)  |   dx  dy   |  (dx)(dy)  |  (dx) dy   |
|      |            |            |            |            |            |
| f_4  |    x (y)   |  (dx) dy   |  (dx)(dy)  |   dx  dy   |   dx (dy)  |
|      |            |            |            |            |            |
| f_8  |    x  y    |  (dx)(dy)  |  (dx) dy   |   dx (dy)  |   dx  dy   |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_3  |   (x)      |   dx       |   dx       |  (dx)      |  (dx)      |
|      |            |            |            |            |            |
| f_12 |    x       |  (dx)      |  (dx)      |   dx       |   dx       |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_6  |   (x, y)   |  (dx, dy)  | ((dx, dy)) | ((dx, dy)) |  (dx, dy)  |
|      |            |            |            |            |            |
| f_9  |  ((x, y))  | ((dx, dy)) |  (dx, dy)  |  (dx, dy)  | ((dx, dy)) |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_5  |      (y)   |       dy   |      (dy)  |       dy   |      (dy)  |
|      |            |            |            |            |            |
| f_10 |       y    |      (dy)  |       dy   |      (dy)  |       dy   |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_7  |   (x  y)   | ((dx)(dy)) | ((dx) dy)  |  (dx (dy)) |  (dx  dy)  |
|      |            |            |            |            |            |
| f_11 |   (x (y))  | ((dx) dy)  | ((dx)(dy)) |  (dx  dy)  |  (dx (dy)) |
|      |            |            |            |            |            |
| f_13 |  ((x) y)   |  (dx (dy)) |  (dx  dy)  | ((dx)(dy)) | ((dx) dy)  |
|      |            |            |            |            |            |
| f_14 |  ((x)(y))  |  (dx  dy)  |  (dx (dy)) | ((dx) dy)  | ((dx)(dy)) |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_15 |    (())    |    (())    |    (())    |    (())    |    (())    |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o

Table A6.  Df Expanded Over Ordinary Features {x, y}
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
|      |     f      |  Df | xy   | Df | x(y)  | Df | (x)y  | Df | (x)(y)|
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_1  |   (x)(y)   |   dx  dy   |   dx (dy)  |  (dx) dy   | ((dx)(dy)) |
|      |            |            |            |            |            |
| f_2  |   (x) y    |   dx (dy)  |   dx  dy   | ((dx)(dy)) |  (dx) dy   |
|      |            |            |            |            |            |
| f_4  |    x (y)   |  (dx) dy   | ((dx)(dy)) |   dx  dy   |   dx (dy)  |
|      |            |            |            |            |            |
| f_8  |    x  y    | ((dx)(dy)) |  (dx) dy   |   dx (dy)  |   dx  dy   |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_3  |   (x)      |   dx       |   dx       |   dx       |   dx       |
|      |            |            |            |            |            |
| f_12 |    x       |   dx       |   dx       |   dx       |   dx       |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_6  |   (x, y)   |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |
|      |            |            |            |            |            |
| f_9  |  ((x, y))  |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_5  |      (y)   |       dy   |       dy   |       dy   |       dy   |
|      |            |            |            |            |            |
| f_10 |       y    |       dy   |       dy   |       dy   |       dy   |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_7  |   (x  y)   | ((dx)(dy)) |  (dx) dy   |   dx (dy)  |   dx  dy   |
|      |            |            |            |            |            |
| f_11 |   (x (y))  |  (dx) dy   | ((dx)(dy)) |   dx  dy   |   dx (dy)  |
|      |            |            |            |            |            |
| f_13 |  ((x) y)   |   dx (dy)  |   dx  dy   | ((dx)(dy)) |  (dx) dy   |
|      |            |            |            |            |            |
| f_14 |  ((x)(y))  |   dx  dy   |   dx (dy)  |  (dx) dy   | ((dx)(dy)) |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_15 |    (())    |     ()     |     ()     |     ()     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o

Wiki Tables : New Versions

Propositional Forms on Two Variables


Table A1.  Propositional Forms on Two Variables
L1 L2 L3 L4 L5 L6
  x : 1 1 0 0      
  y : 1 0 1 0      
f0 f0000 0 0 0 0 ( ) false 0
f1 f0001 0 0 0 1 (x)(y) neither x nor y ¬x ∧ ¬y
f2 f0010 0 0 1 0 (x) y y and not x ¬x ∧ y
f3 f0011 0 0 1 1 (x) not x ¬x
f4 f0100 0 1 0 0 x (y) x and not y x ∧ ¬y
f5 f0101 0 1 0 1 (y) not y ¬y
f6 f0110 0 1 1 0 (x, y) x not equal to y x ≠ y
f7 f0111 0 1 1 1 (x y) not both x and y ¬x ∨ ¬y
f8 f1000 1 0 0 0 x y x and y x ∧ y
f9 f1001 1 0 0 1 ((x, y)) x equal to y x = y
f10 f1010 1 0 1 0 y y y
f11 f1011 1 0 1 1 (x (y)) not x without y x ⇒ y
f12 f1100 1 1 0 0 x x x
f13 f1101 1 1 0 1 ((x) y) not y without x x ⇐ y
f14 f1110 1 1 1 0 ((x)(y)) x or y x ∨ y
f15 f1111 1 1 1 1 (( )) true 1


Table A2.  Propositional Forms on Two Variables
L1 L2 L3 L4 L5 L6
  x : 1 1 0 0      
  y : 1 0 1 0      
f0 f0000 0 0 0 0 ( ) false 0

f1

f2

f4

f8

f0001

f0010

f0100

f1000

0 0 0 1

0 0 1 0

0 1 0 0

1 0 0 0

(x)(y)

(x) y

x (y)

x y

neither x nor y

not x but y

x but not y

x and y

¬x ∧ ¬y

¬x ∧ y

x ∧ ¬y

x ∧ y

f3

f12

f0011

f1100

0 0 1 1

1 1 0 0

(x)

x

not x

x

¬x

x

f6

f9

f0110

f1001

0 1 1 0

1 0 0 1

(x, y)

((x, y))

x not equal to y

x equal to y

x ≠ y

x = y

f5

f10

f0101

f1010

0 1 0 1

1 0 1 0

(y)

y

not y

y

¬y

y

f7

f11

f13

f14

f0111

f1011

f1101

f1110

0 1 1 1

1 0 1 1

1 1 0 1

1 1 1 0

(x y)

(x (y))

((x) y)

((x)(y))

not both x and y

not x without y

not y without x

x or y

¬x ∨ ¬y

x ⇒ y

x ⇐ y

x ∨ y

f15 f1111 1 1 1 1 (( )) true 1


Differential Propositions


Table 14.  Differential Propositions
  A : 1 1 0 0      
  dA : 1 0 1 0      
f0 g0 0 0 0 0 ( ) False 0

 
 
 
 

g1
g2
g4
g8

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

(A)(dA)
(A) dA
A (dA)
A dA

Neither A nor dA
Not A but dA
A but not dA
A and dA

¬A ∧ ¬dA
¬A ∧ dA
A ∧ ¬dA
A ∧ dA

f1
f2

g3
g12

0 0 1 1
1 1 0 0

(A)
A

Not A
A

¬A
A

 
 

g6
g9

0 1 1 0
1 0 0 1

(A, dA)
((A, dA))

A not equal to dA
A equal to dA

A ≠ dA
A = dA

 
 

g5
g10

0 1 0 1
1 0 1 0

(dA)
dA

Not dA
dA

¬dA
dA

 
 
 
 

g7
g11
g13
g14

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

(A dA)
(A (dA))
((A) dA)
((A)(dA))

Not both A and dA
Not A without dA
Not dA without A
A or dA

¬A ∨ ¬dA
A ⇒ dA
A ⇐ dA
A ∨ dA

f3 g15 1 1 1 1 (( )) True 1


Wiki Tables : Old Versions

Propositional Forms on Two Variables


Table 1. Propositional Forms on Two Variables
L1 L2 L3 L4 L5 L6
  x : 1 1 0 0      
  y : 1 0 1 0      
f0 f0000 0 0 0 0 ( ) false 0
f1 f0001 0 0 0 1 (x)(y) neither x nor y ¬x ∧ ¬y
f2 f0010 0 0 1 0 (x) y y and not x ¬x ∧ y
f3 f0011 0 0 1 1 (x) not x ¬x
f4 f0100 0 1 0 0 x (y) x and not y x ∧ ¬y
f5 f0101 0 1 0 1 (y) not y ¬y
f6 f0110 0 1 1 0 (x, y) x not equal to y x ≠ y
f7 f0111 0 1 1 1 (x y) not both x and y ¬x ∨ ¬y
f8 f1000 1 0 0 0 x y x and y x ∧ y
f9 f1001 1 0 0 1 ((x, y)) x equal to y x = y
f10 f1010 1 0 1 0 y y y
f11 f1011 1 0 1 1 (x (y)) not x without y x → y
f12 f1100 1 1 0 0 x x x
f13 f1101 1 1 0 1 ((x) y) not y without x x ← y
f14 f1110 1 1 1 0 ((x)(y)) x or y x ∨ y
f15 f1111 1 1 1 1 (( )) true 1


Differential Propositions


Table 14. Differential Propositions
  A : 1 1 0 0      
  dA : 1 0 1 0      
f0 g0 0 0 0 0 ( ) False 0

 
 
 
 

g1
g2
g4
g8

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

(A)(dA)
(A) dA
A (dA)
A dA

Neither A nor dA
Not A but dA
A but not dA
A and dA

¬A ∧ ¬dA
¬A ∧ dA
A ∧ ¬dA
A ∧ dA

f1
f2

g3
g12

0 0 1 1
1 1 0 0

(A)
A

Not A
A

¬A
A

 
 

g6
g9

0 1 1 0
1 0 0 1

(A, dA)
((A, dA))

A not equal to dA
A equal to dA

A ≠ dA
A = dA

 
 

g5
g10

0 1 0 1
1 0 1 0

(dA)
dA

Not dA
dA

¬dA
dA

 
 
 
 

g7
g11
g13
g14

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

(A dA)
(A (dA))
((A) dA)
((A)(dA))

Not both A and dA
Not A without dA
Not dA without A
A or dA

¬A ∨ ¬dA
A → dA
A ← dA
A ∨ dA

f3 g15 1 1 1 1 (( )) True 1


Wiki TeX Tables


\(\text{Table A1.}~~\text{Propositional Forms on Two Variables}\)

\(\mathcal{L}_1\)

\(\text{Decimal}\)

\(\mathcal{L}_2\)

\(\text{Binary}\)

\(\mathcal{L}_3\)

\(\text{Vector}\)

\(\mathcal{L}_4\)

\(\text{Cactus}\)

\(\mathcal{L}_5\)

\(\text{English}\)

\(\mathcal{L}_6\)

\(\text{Ordinary}\)

  \(x\colon\!\) \(1~1~0~0\!\)      
  \(y\colon\!\) \(1~0~1~0\!\)      
\(f_{0}\!\) \(f_{0000}\!\) \(0~0~0~0\!\) \((~)\!\) \(\text{false}\!\) \(0\!\)
\(f_{1}\!\) \(f_{0001}\!\) \(0~0~0~1\!\) \((x)(y)\!\) \(\text{neither}~ x ~\text{nor}~ y\!\) \(\lnot x \land \lnot y\!\)
\(f_{2}\!\) \(f_{0010}\!\) \(0~0~1~0\!\) \((x)~y\!\) \(y ~\text{without}~ x\!\) \(\lnot x \land y\!\)
\(f_{3}\!\) \(f_{0011}\!\) \(0~0~1~1\!\) \((x)\!\) \(\text{not}~ x\!\) \(\lnot x\!\)
\(f_{4}\!\) \(f_{0100}\!\) \(0~1~0~0\!\) \(x~(y)\!\) \(x ~\text{without}~ y\!\) \(x \land \lnot y\!\)
\(f_{5}\!\) \(f_{0101}\!\) \(0~1~0~1\!\) \((y)\!\) \(\text{not}~ y\!\) \(\lnot y\!\)
\(f_{6}\!\) \(f_{0110}\!\) \(0~1~1~0\!\) \((x,~y)\!\) \(x ~\text{not equal to}~ y\!\) \(x \ne y\!\)
\(f_{7}\!\) \(f_{0111}\!\) \(0~1~1~1\!\) \((x~y)\!\) \(\text{not both}~ x ~\text{and}~ y\!\) \(\lnot x \lor \lnot y\!\)
\(f_{8}\!\) \(f_{1000}\!\) \(1~0~0~0\!\) \(x~y\!\) \(x ~\text{and}~ y\!\) \(x \land y\!\)
\(f_{9}\!\) \(f_{1001}\!\) \(1~0~0~1\!\) \(((x,~y))\!\) \(x ~\text{equal to}~ y\!\) \(x = y\!\)
\(f_{10}\!\) \(f_{1010}\!\) \(1~0~1~0\!\) \(y\!\) \(y\!\) \(y\!\)
\(f_{11}\!\) \(f_{1011}\!\) \(1~0~1~1\!\) \((x~(y))\!\) \(\text{not}~ x ~\text{without}~ y\!\) \(x \Rightarrow y\!\)
\(f_{12}\!\) \(f_{1100}\!\) \(1~1~0~0\!\) \(x\!\) \(x\!\) \(x\!\)
\(f_{13}\!\) \(f_{1101}\!\) \(1~1~0~1\!\) \(((x)~y)\!\) \(\text{not}~ y ~\text{without}~ x\!\) \(x \Leftarrow y\!\)
\(f_{14}\!\) \(f_{1110}\!\) \(1~1~1~0\!\) \(((x)(y))\!\) \(x ~\text{or}~ y\!\) \(x \lor y\!\)
\(f_{15}\!\) \(f_{1111}\!\) \(1~1~1~1\!\) \(((~))\!\) \(\text{true}\!\) \(1\!\)


Table A2.  Propositional Forms on Two Variables
L1 L2 L3 L4 L5 L6
  x : 1 1 0 0      
  y : 1 0 1 0      
f0 f0000 0 0 0 0 ( ) false 0

f1

f2

f4

f8

f0001

f0010

f0100

f1000

0 0 0 1

0 0 1 0

0 1 0 0

1 0 0 0

(x)(y)

(x) y

x (y)

x y

neither x nor y

not x but y

x but not y

x and y

¬x ∧ ¬y

¬x ∧ y

x ∧ ¬y

x ∧ y

f3

f12

f0011

f1100

0 0 1 1

1 1 0 0

(x)

x

not x

x

¬x

x

f6

f9

f0110

f1001

0 1 1 0

1 0 0 1

(x, y)

((x, y))

x not equal to y

x equal to y

x ≠ y

x = y

f5

f10

f0101

f1010

0 1 0 1

1 0 1 0

(y)

y

not y

y

¬y

y

f7

f11

f13

f14

f0111

f1011

f1101

f1110

0 1 1 1

1 0 1 1

1 1 0 1

1 1 1 0

(x y)

(x (y))

((x) y)

((x)(y))

not both x and y

not x without y

not y without x

x or y

¬x ∨ ¬y

x ⇒ y

x ⇐ y

x ∨ y

f15 f1111 1 1 1 1 (( )) true 1


TeX Tables

\tableofcontents

\subsection{Table A1.  Propositional Forms on Two Variables}

Table A1 lists equivalent expressions for the Boolean functions of two variables in a number of different notational systems.

\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}

Inquiry Driven Systems

Table 1. Sign Relation of Interpreter A

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
Table 1. 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"


Table 2. Sign Relation of Interpreter B

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
Table 2. 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"


Table 3. Semiotic Partition of Interpreter A

Table 3.  A's Semiotic Partition
o-------------------------------o
|      "A"             "i"      |
o-------------------------------o
|      "u"             "B"      |
o-------------------------------o
Table 3. Semiotic Partition of Interpreter A
"A" "i"
"u" "B"


Table 4. Semiotic Partition of Interpreter B

Table 4.  B's Semiotic Partition
o---------------o---------------o
|      "A"      |      "i"      |
|               |               |
|      "u"      |      "B"      |
o---------------o---------------o
Table 4. Semiotic Partition of Interpreter B
"A"
"u"
"i"
"B"


Table 5. Alignments of Capacities

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

Table 6. Alignments of Capacities in Aristotle

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

Table 7. Synthesis of Alignments

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

Table 8. Boolean Product

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

Table 9. Boolean Sum

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

Logical Tables

Table Templates

Table 1. Two Variable Template
u :
v :
1 1 0 0
1 0 1 0
f
 
f
 
f
 
f0
f1
f2
f3
f4
f5
f6
f7
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
f0
f1
f2
f3
f4
f5
f6
f7
f0
f1
f2
f3
f4
f5
f6
f7
f0
f1
f2
f3
f4
f5
f6
f7
f8
f9
f10
f11
f12
f13
f14
f15
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
f8
f9
f10
f11
f12
f13
f14
f15
f8
f9
f10
f11
f12
f13
f14
f15
f8
f9
f10
f11
f12
f13
f14
f15


Table 2. Two Variable Template
u :
v :
1100
1010
f
 
f
 
f
 
f0
f1
f2
f3
f4
f5
f6
f7
0000
0001
0010
0011
0100
0101
0110
0111
()
 (u)(v) 
 (u) v  
 (u)    
  u (v) 
    (v) 
 (u, v) 
 (u  v) 
f0
f1
f2
f3
f4
f5
f6
f7
f0
f1
f2
f3
f4
f5
f6
f7
f8
f9
f10
f11
f12
f13
f14
f15
1000
1001
1010
1011
1100
1101
1110
1111
  u  v  
((u, v))
     v  
 (u (v))
  u     
((u) v) 
((u)(v))
(())
f8
f9
f10
f11
f12
f13
f14
f15
f8
f9
f10
f11
f12
f13
f14
f15


Higher Order Propositions

Table 7. Higher Order Propositions (n = 1)
\ 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
F0 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
F1 0 1 (x) 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
F2 1 0 x 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
F3 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1


Table 8. Interpretive Categories for Higher Order Propositions (n = 1)
Measure Happening Exactness Existence Linearity Uniformity Information
m0 nothing happens          
m1   just false nothing exists      
m2   just not x        
m3     nothing is x      
m4   just x        
m5     everything is x F is linear    
m6         F is not uniform F is informed
m7   not just true        
m8   just true        
m9         F is uniform F is not informed
m10     something is not x F is not linear    
m11   not just x        
m12     something is x      
m13   not just not x        
m14   not just false something exists      
m15 anything happens          


Table 9. Higher Order Propositions (n = 2)
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
y : 1010   0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
f0 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
f1 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
f2 0010 (x) y         1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
f3 0011 (x)                 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0
f4 0100 x (y)                                 1 1 1 1 1 1 1 1
f5 0101 (y)                                                
f6 0110 (x, y)                                                
f7 0111 (x y)                                                
f8 1000 x y                                                
f9 1001 ((x, y))                                                
f10 1010 y                                                
f11 1011 (x (y))                                                
f12 1100 x                                                
f13 1101 ((x) y)                                                
f14 1110 ((x)(y))                                                
f15 1111 (( ))                                                


Table 10. Qualifiers of Implication Ordering: αi f = Υ(fi ⇒ f)
x : 1100 f α α α α α α α α α α α α α α α α
y : 1010   15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
f0 0000 ( )                               1
f1 0001 (x)(y)                             1 1
f2 0010 (x) y                           1   1
f3 0011 (x)                         1 1 1 1
f4 0100 x (y)                       1       1
f5 0101 (y)                     1 1     1 1
f6 0110 (x, y)                   1   1   1   1
f7 0111 (x y)                 1 1 1 1 1 1 1 1
f8 1000 x y               1               1
f9 1001 ((x, y))             1 1             1 1
f10 1010 y           1   1           1   1
f11 1011 (x (y))         1 1 1 1         1 1 1 1
f12 1100 x       1       1       1       1
f13 1101 ((x) y)     1 1     1 1     1 1     1 1
f14 1110 ((x)(y))   1   1   1   1   1   1   1   1
f15 1111 (( )) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1


Table 11. Qualifiers of Implication Ordering: βi f = Υ(f ⇒ fi)
x : 1100 f β β β β β β β β β β β β β β β β
y : 1010   0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
f0 0000 ( ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
f1 0001 (x)(y)   1   1   1   1   1   1   1   1
f2 0010 (x) y     1 1     1 1     1 1     1 1
f3 0011 (x)       1       1       1       1
f4 0100 x (y)         1 1 1 1         1 1 1 1
f5 0101 (y)           1   1           1   1
f6 0110 (x, y)             1 1             1 1
f7 0111 (x y)               1               1
f8 1000 x y                 1 1 1 1 1 1 1 1
f9 1001 ((x, y))                   1   1   1   1
f10 1010 y                     1 1     1 1
f11 1011 (x (y))                       1       1
f12 1100 x                         1 1 1 1
f13 1101 ((x) y)                           1   1
f14 1110 ((x)(y))                             1 1
f15 1111 (( ))                               1


Table 13. Syllogistic Premisses as Higher Order Indicator Functions
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


Table 14. Relation of Quantifiers to Higher Order Propositions
Mnemonic Category Classical Form Alternate Form Symmetric Form Operator
E
Exclusive 		Universal
Negative 		All x is (y)   No x is y (L11)
A
Absolute 		Universal
Affirmative 		All x is y   No x is (y) (L10)
    All y is x No y is (x) No (x) is y (L01)
    All (y) is x No (y) is (x) No (x) is (y) (L00)
    Some (x) is (y)   Some (x) is (y) L00
    Some (x) is y   Some (x) is y L01
O
Obtrusive 		Particular
Negative 		Some x is (y)   Some x is (y) L10
I
Indefinite 		Particular
Affirmative 		Some x is y   Some x is y L11


Table 15. Simple Qualifiers of Propositions (n = 2)
x : 1100 f (L11) (L10) (L01) (L00) L00 L01 L10 L11
y : 1010   no x
is y 		no x
is (y) 		no (x)
is y 		no (x)
is (y) 		some (x)
is (y) 		some (x)
is y 		some x
is (y) 		some x
is y
f0 0000 ( ) 1 1 1 1 0 0 0 0
f1 0001 (x)(y) 1 1 1 0 1 0 0 0
f2 0010 (x) y 1 1 0 1 0 1 0 0
f3 0011 (x) 1 1 0 0 1 1 0 0
f4 0100 x (y) 1 0 1 1 0 0 1 0
f5 0101 (y) 1 0 1 0 1 0 1 0
f6 0110 (x, y) 1 0 0 1 0 1 1 0
f7 0111 (x y) 1 0 0 0 1 1 1 0
f8 1000 x y 0 1 1 1 0 0 0 1
f9 1001 ((x, y)) 0 1 1 0 1 0 0 1
f10 1010 y 0 1 0 1 0 1 0 1
f11 1011 (x (y)) 0 1 0 0 1 1 0 1
f12 1100 x 0 0 1 1 0 0 1 1
f13 1101 ((x) y) 0 0 1 0 1 0 1 1
f14 1110 ((x)(y)) 0 0 0 1 0 1 1 1
f15 1111 (( )) 0 0 0 0 1 1 1 1 


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



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



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



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



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



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



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



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


Zeroth Order Logic

Table 1. Propositional Forms on Two Variables
L1 L2 L3 L4 L5 L6
  x : 1 1 0 0      
  y : 1 0 1 0      
f0 f0000 0 0 0 0 ( ) false 0
f1 f0001 0 0 0 1 (x)(y) neither x nor y ¬x ∧ ¬y
f2 f0010 0 0 1 0 (x) y y and not x ¬x ∧ y
f3 f0011 0 0 1 1 (x) not x ¬x
f4 f0100 0 1 0 0 x (y) x and not y x ∧ ¬y
f5 f0101 0 1 0 1 (y) not y ¬y
f6 f0110 0 1 1 0 (x, y) x not equal to y x ≠ y
f7 f0111 0 1 1 1 (x y) not both x and y ¬x ∨ ¬y
f8 f1000 1 0 0 0 x y x and y x ∧ y
f9 f1001 1 0 0 1 ((x, y)) x equal to y x = y
f10 f1010 1 0 1 0 y y y
f11 f1011 1 0 1 1 (x (y)) not x without y x → y
f12 f1100 1 1 0 0 x x x
f13 f1101 1 1 0 1 ((x) y) not y without x x ← y
f14 f1110 1 1 1 0 ((x)(y)) x or y x ∨ y
f15 f1111 1 1 1 1 (( )) true 1


Table 1. Propositional Forms on Two Variables
L1 L2 L3 L4 L5 L6
  x : 1 1 0 0      
  y : 1 0 1 0      
f0 f0000 0 0 0 0 ( ) false 0
f1 f0001 0 0 0 1 (x)(y) neither x nor y ¬x ∧ ¬y
f2 f0010 0 0 1 0 (x) y y and not x ¬x ∧ y
f3 f0011 0 0 1 1 (x) not x ¬x
f4 f0100 0 1 0 0 x (y) x and not y x ∧ ¬y
f5 f0101 0 1 0 1 (y) not y ¬y
f6 f0110 0 1 1 0 (x, y) x not equal to y x ≠ y
f7 f0111 0 1 1 1 (x y) not both x and y ¬x ∨ ¬y
f8 f1000 1 0 0 0 x y x and y x ∧ y
f9 f1001 1 0 0 1 ((x, y)) x equal to y x = y
f10 f1010 1 0 1 0 y y y
f11 f1011 1 0 1 1 (x (y)) not x without y x → y
f12 f1100 1 1 0 0 x x x
f13 f1101 1 1 0 1 ((x) y) not y without x x ← y
f14 f1110 1 1 1 0 ((x)(y)) x or y x ∨ y
f15 f1111 1 1 1 1 (( )) true 1


Template Draft

Propositional Forms on Two Variables
L1 L2 L3 L4 L5 L6 Name
  x : 1 1 0 0        
  y : 1 0 1 0        
f0 f0000 0 0 0 0 ( ) false 0 Falsity
f1 f0001 0 0 0 1 (x)(y) neither x nor y ¬x ∧ ¬y NNOR
f2 f0010 0 0 1 0 (x) y y and not x ¬x ∧ y Insuccede
f3 f0011 0 0 1 1 (x) not x ¬x Not One
f4 f0100 0 1 0 0 x (y) x and not y x ∧ ¬y Imprecede
f5 f0101 0 1 0 1 (y) not y ¬y Not Two
f6 f0110 0 1 1 0 (x, y) x not equal to y x ≠ y Inequality
f7 f0111 0 1 1 1 (x y) not both x and y ¬x ∨ ¬y NAND
f8 f1000 1 0 0 0 x y x and y x ∧ y Conjunction
f9 f1001 1 0 0 1 ((x, y)) x equal to y x = y Equality
f10 f1010 1 0 1 0 y y y Two
f11 f1011 1 0 1 1 (x (y)) not x without y x → y Implication
f12 f1100 1 1 0 0 x x x One
f13 f1101 1 1 0 1 ((x) y) not y without x x ← y Involution
f14 f1110 1 1 1 0 ((x)(y)) x or y x ∨ y Disjunction
f15 f1111 1 1 1 1 (( )) true 1 Tautology


Truth Tables

Logical negation

Logical negation is an 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 ¬p) is as follows:

Logical Negation
p ¬p
F T
T F


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:

Variant Notations
Notation Vocalization
\(\bar{p}\) bar p
\(p'\!\) p prime,

p complement

\(!p\!\) bang p


No matter how it is notated or symbolized, the logical negation ¬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, ¬(¬p) ⇔ p.
  • Within a system of intuitionistic logic, however, ¬¬p is a weaker statement than p. On the other hand, the logical equivalence ¬¬¬p ⇔ ¬p remains valid.

Logical negation can be defined in terms of other logical operations. For example, ~p can be defined as p → F, where → is material implication and F is absolute falsehood. Conversely, one can define F as p & ~p for any proposition p, where & 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 → q can be defined as ~p ∨ q, where ∨ 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 operation on two logical values, typically the values of two propositions, 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 ∧ q, p & q, or p\(\cdot\)q) is as follows:

Logical Conjunction
p q p ∧ q
F F F
F T F
T F F
T T T


Logical disjunction

Logical disjunction is an operation on two logical values, typically the values of two propositions, 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 ∨ q) is as follows:

Logical Disjunction
p q p ∨ q
F F F
F T T
T F T
T T T


Logical equality

Logical equality is an operation on two logical values, typically the values of two propositions, 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 ↔ q, or p ≡ q) is as follows:

Logical Equality
p q p = q
F F T
F T F
T F F
T T T


Exclusive disjunction

Exclusive disjunction is an operation on two logical values, typically the values of two propositions, 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 ⊕ q, or p ≠ q) is as follows:

Exclusive Disjunction
p q p XOR q
F F F
F T T
T F T
T T F


The following equivalents can then be deduced:

\[\begin{matrix} p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\ \\ & = & (p \lor q) & \land & (\lnot p \lor \lnot q) \\ \\ & = & (p \lor q) & \land & \lnot (p \land q) \end{matrix}\]

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

 A + B = (A ∧ !B) ∨ (!A ∧ B)
       = {(A ∧ !B) ∨ !A} ∧ {(A ∧ !B) ∨ B}
       = {(A ∨ !A) ∧ (!B ∨ !A)} ∧ {(A ∨ B) ∧ (!B ∨ B)}
       = (!A ∨ !B) ∧ (A ∨ B)
       = !(A ∧ B) ∧ (A ∨ B)

 p + q = (p ∧ !q)  ∨ (!p ∧ B)
 
       = {(p ∧ !q) ∨ !p} ∧ {(p ∧ !q) ∨ q}
 
       = {(p ∨ !q) ∧ (!q ∨ !p)} ∧ {(p ∨ q) ∧ (!q ∨ q)}
 
       = (!p ∨ !q) ∧ (p ∨ q)
 
       = !(p ∧ q)  ∧ (p ∨ q)

 p + q = (p ∧ ~q)  ∨ (~p ∧ q)
 
       = ((p ∧ ~q) ∨ ~p) ∧ ((p ∧ ~q) ∨ q)
 
       = ((p ∨ ~q) ∧ (~q ∨ ~p)) ∧ ((p ∨ q) ∧ (~q ∨ q))
 
       = (~p ∨ ~q) ∧ (p ∨ q)
 
       = ~(p ∧ q)  ∧ (p ∨ q)

\[\begin{matrix} p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\ & = & ((p \land \lnot q) \lor \lnot p) & \and & ((p \land \lnot q) \lor q) \\ & = & ((p \lor \lnot q) \land (\lnot q \lor \lnot p)) & \land & ((p \lor q) \land (\lnot q \lor q)) \\ & = & (\lnot p \lor \lnot q) & \land & (p \lor q) \\ & = & \lnot (p \land q) & \land & (p \lor q) \end{matrix}\]

Logical implication

The material conditional and logical implication are both associated with an operation on two logical values, typically the values of two propositions, 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:

Logical Implication
p q p ⇒ q
F F T
F T T
T F F
T T T


Logical NAND

The NAND operation is a logical operation on two logical values, typically the values of two propositions, 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:

Logical NAND
p q p ↑ q
F F T
F T T
T F T
T T F


Logical NNOR

The NNOR operation is a logical operation on two logical values, typically the values of two propositions, 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:

Logical NOR
p q p ↓ q
F F T
F T F
T F F
T T F


Relational Tables

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
\(\text{Mnemonic}\) \(\text{Category}\) \(\text{Classical Form}\) \(\text{Alternate Form}\) \(\text{Symmetric Form}\) \(\text{Operator}\)
\(\text{E}\!\)
\(\text{Exclusive}\) 		\(\text{Universal}\)
\(\text{Negative}\) 		\(\text{All}\ u\ \text{is}\ (v)\)   \(\text{No}\ u\ \text{is}\ v \) \((\ell_{11})\)
\(\text{A}\!\)
\(\text{Absolute}\) 		\(\text{Universal}\)
\(\text{Affirmative}\) 		\(\text{All}\ u\ \text{is}\ v \)   \(\text{No}\ u\ \text{is}\ (v)\) \((\ell_{10})\)
    \(\text{All}\ v\ \text{is}\ u \) \(\text{No}\ v\ \text{is}\ (u)\) \(\text{No}\ (u)\ \text{is}\ v \) \((\ell_{01})\)
    \(\text{All}\ (v)\ \text{is}\ u \) \(\text{No}\ (v)\ \text{is}\ (u)\) \(\text{No}\ (u)\ \text{is}\ (v)\) \((\ell_{00})\)
    \(\text{Some}\ (u)\ \text{is}\ (v)\)   \(\text{Some}\ (u)\ \text{is}\ (v)\) \(\ell_{00}\!\)
    \(\text{Some}\ (u)\ \text{is}\ v\)   \(\text{Some}\ (u)\ \text{is}\ v\) \(\ell_{01}\!\)
\(\text{O}\!\)
\(\text{Obtrusive}\) 		\(\text{Particular}\)
\(\text{Negative}\) 		\(\text{Some}\ u\ \text{is}\ (v)\)   \(\text{Some}\ u\ \text{is}\ (v)\) \(\ell_{10}\!\)
\(\text{I}\!\)
\(\text{Indefinite}\) 		\(\text{Particular}\)
\(\text{Affirmative}\) 		\(\text{Some}\ u\ \text{is}\ v\)   \(\text{Some}\ u\ \text{is}\ v\) \(\ell_{11}\!\)


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