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===Wiki TeX Tables===
 
===Wiki TeX Tables===
 +
 +
<br>
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 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
 +
|+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math>
 +
|- style="background:#f0f0ff"
 +
| style="width:15%" |
 +
<p><math>\mathcal{L}_1</math></p>
 +
<p><math>\text{Decimal}</math></p>
 +
| style="width:15%" |
 +
<p><math>\mathcal{L}_2</math></p>
 +
<p><math>\text{Binary}</math></p>
 +
| style="width:15%" |
 +
<p><math>\mathcal{L}_3</math></p>
 +
<p><math>\text{Vector}</math></p>
 +
| style="width:15%" |
 +
<p><math>\mathcal{L}_4</math></p>
 +
<p><math>\text{Cactus}</math></p>
 +
| style="width:25%" |
 +
<p><math>\mathcal{L}_5</math></p>
 +
<p><math>\text{English}</math></p>
 +
| style="width:15%" |
 +
<p><math>\mathcal{L}_6</math></p>
 +
<p><math>\text{Ordinary}</math></p>
 +
|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | <math>x\colon\!</math>
 +
| <math>1~1~0~0\!</math>
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | <math>y\colon\!</math>
 +
| <math>1~0~1~0\!</math>
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
|-
 +
| <math>f_{0}\!</math>
 +
| <math>f_{0000}\!</math>
 +
| <math>0~0~0~0\!</math>
 +
| <math>(~)\!</math>
 +
| <math>\text{false}\!</math>
 +
| <math>0\!</math>
 +
|-
 +
| <math>f_{1}\!</math>
 +
| <math>f_{0001}\!</math>
 +
| <math>0~0~0~1\!</math>
 +
| <math>(x)(y)\!</math>
 +
| <math>\text{neither}~ x ~\text{nor}~ y\!</math>
 +
| <math>\lnot x \land \lnot y\!</math>
 +
|-
 +
| <math>f_{2}\!</math>
 +
| <math>f_{0010}\!</math>
 +
| <math>0~0~1~0\!</math>
 +
| <math>(x)~y\!</math>
 +
| <math>y ~\text{without}~ x\!</math>
 +
| <math>\lnot x \land y\!</math>
 +
|-
 +
| <math>f_{3}\!</math>
 +
| <math>f_{0011}\!</math>
 +
| <math>0~0~1~1\!</math>
 +
| <math>(x)\!</math>
 +
| <math>\text{not}~ x\!</math>
 +
| <math>\lnot x\!</math>
 +
|-
 +
| <math>f_{4}\!</math>
 +
| <math>f_{0100}\!</math>
 +
| <math>0~1~0~0\!</math>
 +
| <math>x~(y)\!</math>
 +
| <math>x ~\text{without}~ y\!</math>
 +
| <math>x \land \lnot y\!</math>
 +
|-
 +
| <math>f_{5}\!</math>
 +
| <math>f_{0101}\!</math>
 +
| <math>0~1~0~1\!</math>
 +
| <math>(y)\!</math>
 +
| <math>\text{not}~ y\!</math>
 +
| <math>\lnot y\!</math>
 +
|-
 +
| <math>f_{6}\!</math>
 +
| <math>f_{0110}\!</math>
 +
| <math>0~1~1~0\!</math>
 +
| <math>(x,~y)\!</math>
 +
| <math>x ~\text{not equal to}~ y\!</math>
 +
| <math>x \ne y\!</math>
 +
|-
 +
| <math>f_{7}\!</math>
 +
| <math>f_{0111}\!</math>
 +
| <math>0~1~1~1\!</math>
 +
| <math>(x~y)\!</math>
 +
| <math>\text{not both}~ x ~\text{and}~ y\!</math>
 +
| <math>\lnot x \lor \lnot y\!</math>
 +
|-
 +
| <math>f_{8}\!</math>
 +
| <math>f_{1000}\!</math>
 +
| <math>1~0~0~0\!</math>
 +
| <math>x~y\!</math>
 +
| <math>x ~\text{and}~ y\!</math>
 +
| <math>x \land y\!</math>
 +
|-
 +
| <math>f_{9}\!</math>
 +
| <math>f_{1001}\!</math>
 +
| <math>1~0~0~1\!</math>
 +
| <math>((x,~y))\!</math>
 +
| <math>x ~\text{equal to}~ y\!</math>
 +
| <math>x = y\!</math>
 +
|-
 +
| <math>f_{10}\!</math>
 +
| <math>f_{1010}\!</math>
 +
| <math>1~0~1~0\!</math>
 +
| <math>y\!</math>
 +
| <math>y\!</math>
 +
| <math>y\!</math>
 +
|-
 +
| <math>f_{11}\!</math>
 +
| <math>f_{1011}\!</math>
 +
| <math>1~0~1~1\!</math>
 +
| <math>(x~(y))\!</math>
 +
| <math>\text{not}~ x ~\text{without}~ y\!</math>
 +
| <math>x \Rightarrow y\!</math>
 +
|-
 +
| <math>f_{12}\!</math>
 +
| <math>f_{1100}\!</math>
 +
| <math>1~1~0~0\!</math>
 +
| <math>x\!</math>
 +
| <math>x\!</math>
 +
| <math>x\!</math>
 +
|-
 +
| <math>f_{13}\!</math>
 +
| <math>f_{1101}\!</math>
 +
| <math>1~1~0~1\!</math>
 +
| <math>((x)~y)\!</math>
 +
| <math>\text{not}~ y ~\text{without}~ x\!</math>
 +
| <math>x \Leftarrow y\!</math>
 +
|-
 +
| <math>f_{14}\!</math>
 +
| <math>f_{1110}\!</math>
 +
| <math>1~1~1~0\!</math>
 +
| <math>((x)(y))\!</math>
 +
| <math>x ~\text{or}~ y\!</math>
 +
| <math>x \lor y\!</math>
 +
|-
 +
| <math>f_{15}\!</math>
 +
| <math>f_{1111}\!</math>
 +
| <math>1~1~1~1\!</math>
 +
| <math>((~))\!</math>
 +
| <math>\text{true}\!</math>
 +
| <math>1\!</math>
 +
|}
  
 
<br>
 
<br>

Revision as of 02:26, 30 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


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

\(\mathcal{L}_1\)

\(\text{Decimal}\)

\(\mathcal{L}_2\)

\(\text{Binary}\)

\(\mathcal{L}_3\)

\(\text{Vector}\)

\(\mathcal{L}_4\)

\(\text{Cactus}\)

\(\mathcal{L}_5\)

\(\text{English}\)

\(\mathcal{L}_6\)

\(\text{Ordinary}\)

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


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

\(\mathcal{L}_1\)

\(\text{Decimal}\)

\(\mathcal{L}_2\)

\(\text{Binary}\)

\(\mathcal{L}_3\)

\(\text{Vector}\)

\(\mathcal{L}_4\)

\(\text{Cactus}\)

\(\mathcal{L}_5\)

\(\text{English}\)

\(\mathcal{L}_6\)

\(\text{Ordinary}\)

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


\(\text{Table A2.}~~\text{Propositional Forms on Two Variables}\)

\(\mathcal{L}_1\)

\(\text{Decimal}\)

\(\mathcal{L}_2\)

\(\text{Binary}\)

\(\mathcal{L}_3\)

\(\text{Vector}\)

\(\mathcal{L}_4\)

\(\text{Cactus}\)

\(\mathcal{L}_5\)

\(\text{English}\)

\(\mathcal{L}_6\)

\(\text{Ordinary}\)

  \(x\colon\!\) \(1~1~0~0\!\)      
  \(y\colon\!\) \(1~0~1~0\!\)      
\(f_0\!\) \(f_{0000}\!\) \(0~0~0~0\) \((~)\) \(\text{false}\!\) \(0\!\)

\(\begin{matrix} f_1 \'"`UNIQ-MathJax1-QINU`"' '''Generalized''' or '''n-ary''' XOR is true when the number of 1-bits is odd. '"`UNIQ--pre-00000010-QINU`"' '"`UNIQ--pre-00000011-QINU`"' '"`UNIQ--pre-00000012-QINU`"' '"`UNIQ-MathJax2-QINU`"' ===='"`UNIQ--h-31--QINU`"'[[Logical implication]]==== The '''material conditional''' and '''logical implication''' are both associated with an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if the first operand is true and the second operand is false. The [[truth table]] associated with the material conditional '''if p then q''' (symbolized as '''p → q''') and the logical implication '''p implies q''' (symbolized as '''p ⇒ q''') is as follows: {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%" |+ '''Logical Implication''' |- style="background:aliceblue" ! style="width:15%" | p ! style="width:15%" | q ! style="width:15%" | p ⇒ q |- | F || F || T |- | F || T || T |- | T || F || F |- | T || T || T |} <br> ===='"`UNIQ--h-32--QINU`"'[[Logical NAND]]==== The '''NAND operation''' is a [[logical operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if both of its operands are true. In other words, it produces a value of ''true'' if and only if at least one of its operands is false. The [[truth table]] of '''p NAND q''' (also written as '''p | q''' or '''p ↑ q''') is as follows: {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%" |+ '''Logical NAND''' |- style="background:aliceblue" ! style="width:15%" | p ! style="width:15%" | q ! style="width:15%" | p ↑ q |- | F || F || T |- | F || T || T |- | T || F || T |- | T || T || F |} <br> ===='"`UNIQ--h-33--QINU`"'[[Logical NNOR]]==== The '''NNOR operation''' is a [[logical operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' if and only if both of its operands are false. In other words, it produces a value of ''false'' if and only if at least one of its operands is true. The [[truth table]] of '''p NNOR q''' (also written as '''p ⊥ q''' or '''p ↓ q''') is as follows: {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%" |+ '''Logical NOR''' |- style="background:aliceblue" ! style="width:15%" | p ! style="width:15%" | q ! style="width:15%" | p ↓ q |- | F || F || T |- | F || T || F |- | T || F || F |- | T || T || F |} <br> =='"`UNIQ--h-34--QINU`"'Relational Tables== ==='"`UNIQ--h-35--QINU`"'Sign Relations=== {| cellpadding="4" | width="20px" |   | align="center" | '''O''' || = || Object Domain |- | width="20px" |   | align="center" | '''S''' || = || Sign Domain |- | width="20px" |   | align="center" | '''I''' || = || Interpretant Domain |} <br> {| cellpadding="4" | width="20px" |   | align="center" | '''O''' | = | {Ann, Bob} | = | {A, B} |- | width="20px" |   | align="center" | '''S''' | = | {"Ann", "Bob", "I", "You"} | = | {"A", "B", "i", "u"} |- | width="20px" |   | align="center" | '''I''' | = | {"Ann", "Bob", "I", "You"} | = | {"A", "B", "i", "u"} |} <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A |- style="background:paleturquoise" ! style="width:20%" | Object ! style="width:20%" | Sign ! style="width:20%" | Interpretant |- | '''A''' || '''"A"''' || '''"A"''' |- | '''A''' || '''"A"''' || '''"i"''' |- | '''A''' || '''"i"''' || '''"A"''' |- | '''A''' || '''"i"''' || '''"i"''' |- | '''B''' || '''"B"''' || '''"B"''' |- | '''B''' || '''"B"''' || '''"u"''' |- | '''B''' || '''"u"''' || '''"B"''' |- | '''B''' || '''"u"''' || '''"u"''' |} <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B |- style="background:paleturquoise" ! style="width:20%" | Object ! style="width:20%" | Sign ! style="width:20%" | Interpretant |- | '''A''' || '''"A"''' || '''"A"''' |- | '''A''' || '''"A"''' || '''"u"''' |- | '''A''' || '''"u"''' || '''"A"''' |- | '''A''' || '''"u"''' || '''"u"''' |- | '''B''' || '''"B"''' || '''"B"''' |- | '''B''' || '''"B"''' || '''"i"''' |- | '''B''' || '''"i"''' || '''"B"''' |- | '''B''' || '''"i"''' || '''"i"''' |} <br> ==='"`UNIQ--h-36--QINU`"'Triadic Relations=== ===='"`UNIQ--h-37--QINU`"'Algebraic Examples==== {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |+ '''L'''<sub>0</sub> = {(''x'', ''y'', ''z'') ∈ '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 0} |- style="background:paleturquoise" ! X !! Y !! Z |- | '''0''' || '''0''' || '''0''' |- | '''0''' || '''1''' || '''1''' |- | '''1''' || '''0''' || '''1''' |- | '''1''' || '''1''' || '''0''' |} <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |+ '''L'''<sub>1</sub> = {(''x'', ''y'', ''z'') ∈ '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 1} |- style="background:paleturquoise" ! X !! Y !! Z |- | '''0''' || '''0''' || '''1''' |- | '''0''' || '''1''' || '''0''' |- | '''1''' || '''0''' || '''0''' |- | '''1''' || '''1''' || '''1''' |} <br> ===='"`UNIQ--h-38--QINU`"'Semiotic Examples==== {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A |- style="background:paleturquoise" ! style="width:20%" | Object ! style="width:20%" | Sign ! style="width:20%" | Interpretant |- | '''A''' || '''"A"''' || '''"A"''' |- | '''A''' || '''"A"''' || '''"i"''' |- | '''A''' || '''"i"''' || '''"A"''' |- | '''A''' || '''"i"''' || '''"i"''' |- | '''B''' || '''"B"''' || '''"B"''' |- | '''B''' || '''"B"''' || '''"u"''' |- | '''B''' || '''"u"''' || '''"B"''' |- | '''B''' || '''"u"''' || '''"u"''' |} <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B |- style="background:paleturquoise" ! style="width:20%" | Object ! style="width:20%" | Sign ! style="width:20%" | Interpretant |- | '''A''' || '''"A"''' || '''"A"''' |- | '''A''' || '''"A"''' || '''"u"''' |- | '''A''' || '''"u"''' || '''"A"''' |- | '''A''' || '''"u"''' || '''"u"''' |- | '''B''' || '''"B"''' || '''"B"''' |- | '''B''' || '''"B"''' || '''"i"''' |- | '''B''' || '''"i"''' || '''"B"''' |- | '''B''' || '''"i"''' || '''"i"''' |} <br> ==='"`UNIQ--h-39--QINU`"'Dyadic Projections=== {| cellpadding="4" | width="20px" |   | '''L'''<sub>OS</sub> | = | ''proj''<sub>OS</sub>('''L''') | = | { (''o'', ''s'') ∈ '''O''' × '''S''' : (''o'', ''s'', ''i'') ∈ '''L''' for some ''i'' ∈ '''I''' } |- | width="20px" |   | '''L'''<sub>SO</sub> | = | ''proj''<sub>SO</sub>('''L''') | = | { (''s'', ''o'') ∈ '''S''' × '''O''' : (''o'', ''s'', ''i'') ∈ '''L''' for some ''i'' ∈ '''I''' } |- | width="20px" |   | '''L'''<sub>IS</sub> | = | ''proj''<sub>IS</sub>('''L''') | = | { (''i'', ''s'') ∈ '''I''' × '''S''' : (''o'', ''s'', ''i'') ∈ '''L''' for some ''o'' ∈ '''O''' } |- | width="20px" |   | '''L'''<sub>SI</sub> | = | ''proj''<sub>SI</sub>('''L''') | = | { (''s'', ''i'') ∈ '''S''' × '''I''' : (''o'', ''s'', ''i'') ∈ '''L''' for some ''o'' ∈ '''O''' } |- | width="20px" |   | '''L'''<sub>OI</sub> | = | ''proj''<sub>OI</sub>('''L''') | = | { (''o'', ''i'') ∈ '''O''' × '''I''' : (''o'', ''s'', ''i'') ∈ '''L''' for some ''s'' ∈ '''S''' } |- | width="20px" |   | '''L'''<sub>IO</sub> | = | ''proj''<sub>IO</sub>('''L''') | = | { (''i'', ''o'') ∈ '''I''' × '''O''' : (''o'', ''s'', ''i'') ∈ '''L''' for some ''s'' ∈ '''S''' } |} <br> ===='"`UNIQ--h-40--QINU`"'Method 1 : Subtitles as Captions==== {| align="center" style="width:90%" | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ ''proj''<sub>OS</sub>('''L'''<sub>A</sub>) |- style="background:paleturquoise" ! style="width:50%" | Object ! style="width:50%" | Sign |- | '''A''' || '''"A"''' |- | '''A''' || '''"i"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"u"''' |} | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ ''proj''<sub>OS</sub>('''L'''<sub>B</sub>) |- style="background:paleturquoise" ! style="width:50%" | Object ! style="width:50%" | Sign |- | '''A''' || '''"A"''' |- | '''A''' || '''"u"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"i"''' |} |} <br> {| align="center" style="width:90%" | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ ''proj''<sub>SI</sub>('''L'''<sub>A</sub>) |- style="background:paleturquoise" ! style="width:50%" | Sign ! style="width:50%" | Interpretant |- | '''"A"''' || '''"A"''' |- | '''"A"''' || '''"i"''' |- | '''"i"''' || '''"A"''' |- | '''"i"''' || '''"i"''' |- | '''"B"''' || '''"B"''' |- | '''"B"''' || '''"u"''' |- | '''"u"''' || '''"B"''' |- | '''"u"''' || '''"u"''' |} | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ ''proj''<sub>SI</sub>('''L'''<sub>B</sub>) |- style="background:paleturquoise" ! style="width:50%" | Sign ! style="width:50%" | Interpretant |- | '''"A"''' || '''"A"''' |- | '''"A"''' || '''"u"''' |- | '''"u"''' || '''"A"''' |- | '''"u"''' || '''"u"''' |- | '''"B"''' || '''"B"''' |- | '''"B"''' || '''"i"''' |- | '''"i"''' || '''"B"''' |- | '''"i"''' || '''"i"''' |} |} <br> {| align="center" style="width:90%" | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ ''proj''<sub>OI</sub>('''L'''<sub>A</sub>) |- style="background:paleturquoise" ! style="width:50%" | Object ! style="width:50%" | Interpretant |- | '''A''' || '''"A"''' |- | '''A''' || '''"i"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"u"''' |} | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ ''proj''<sub>OI</sub>('''L'''<sub>B</sub>) |- style="background:paleturquoise" ! style="width:50%" | Object ! style="width:50%" | Interpretant |- | '''A''' || '''"A"''' |- | '''A''' || '''"u"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"i"''' |} |} <br> ===='"`UNIQ--h-41--QINU`"'Method 2 : Subtitles as Top Rows==== {| align="center" style="width:90%" | align="center" style="width:45%" | ''proj''<sub>OS</sub>('''L'''<sub>A</sub>) {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! style="width:50%" | Object ! style="width:50%" | Sign |- | '''A''' || '''"A"''' |- | '''A''' || '''"i"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"u"''' |} | align="center" style="width:45%" | ''proj''<sub>OS</sub>('''L'''<sub>B</sub>) {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! style="width:50%" | Object ! style="width:50%" | Sign |- | '''A''' || '''"A"''' |- | '''A''' || '''"u"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"i"''' |} |} <br> {| align="center" style="width:90%" | align="center" style="width:45%" | ''proj''<sub>SI</sub>('''L'''<sub>A</sub>) {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! style="width:50%" | Sign ! style="width:50%" | Interpretant |- | '''"A"''' || '''"A"''' |- | '''"A"''' || '''"i"''' |- | '''"i"''' || '''"A"''' |- | '''"i"''' || '''"i"''' |- | '''"B"''' || '''"B"''' |- | '''"B"''' || '''"u"''' |- | '''"u"''' || '''"B"''' |- | '''"u"''' || '''"u"''' |} | align="center" style="width:45%" | ''proj''<sub>SI</sub>('''L'''<sub>B</sub>) {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! style="width:50%" | Sign ! style="width:50%" | Interpretant |- | '''"A"''' || '''"A"''' |- | '''"A"''' || '''"u"''' |- | '''"u"''' || '''"A"''' |- | '''"u"''' || '''"u"''' |- | '''"B"''' || '''"B"''' |- | '''"B"''' || '''"i"''' |- | '''"i"''' || '''"B"''' |- | '''"i"''' || '''"i"''' |} |} <br> {| align="center" style="width:90%" | align="center" style="width:45%" | ''proj''<sub>OI</sub>('''L'''<sub>A</sub>) {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! style="width:50%" | Object ! style="width:50%" | Interpretant |- | '''A''' || '''"A"''' |- | '''A''' || '''"i"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"u"''' |} | align="center" style="width:45%" | ''proj''<sub>OI</sub>('''L'''<sub>B</sub>) {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! style="width:50%" | Object ! style="width:50%" | Interpretant |- | '''A''' || '''"A"''' |- | '''A''' || '''"u"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"i"''' |} |} <br> ==='"`UNIQ--h-42--QINU`"'Relation Reduction=== ===='"`UNIQ--h-43--QINU`"'Method 1 : Subtitles as Captions==== {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |+ '''L'''<sub>0</sub> = {(''x'', ''y'', ''z'') ∈ '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 0} |- style="background:paleturquoise" ! X !! Y !! Z |- | '''0''' || '''0''' || '''0''' |- | '''0''' || '''1''' || '''1''' |- | '''1''' || '''0''' || '''1''' |- | '''1''' || '''1''' || '''0''' |} <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |+ '''L'''<sub>1</sub> = {(''x'', ''y'', ''z'') ∈ '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 1} |- style="background:paleturquoise" ! X !! Y !! Z |- | '''0''' || '''0''' || '''1''' |- | '''0''' || '''1''' || '''0''' |- | '''1''' || '''0''' || '''0''' |- | '''1''' || '''1''' || '''1''' |} <br> {| align="center" style="width:90%" | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ proj<sub>''XY''</sub>('''L'''<sub>0</sub>) |- style="background:paleturquoise" ! X !! Y |- | '''0''' || '''0''' |- | '''0''' || '''1''' |- | '''1''' || '''0''' |- | '''1''' || '''1''' |} | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ proj<sub>''XZ''</sub>('''L'''<sub>0</sub>) |- style="background:paleturquoise" ! X !! Z |- | '''0''' || '''0''' |- | '''0''' || '''1''' |- | '''1''' || '''1''' |- | '''1''' || '''0''' |} | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ proj<sub>''YZ''</sub>('''L'''<sub>0</sub>) |- style="background:paleturquoise" ! Y !! Z |- | '''0''' || '''0''' |- | '''1''' || '''1''' |- | '''0''' || '''1''' |- | '''1''' || '''0''' |} |} <br> {| align="center" style="width:90%" | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ proj<sub>''XY''</sub>('''L'''<sub>1</sub>) |- style="background:paleturquoise" ! X !! Y |- | '''0''' || '''0''' |- | '''0''' || '''1''' |- | '''1''' || '''0''' |- | '''1''' || '''1''' |} | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ proj<sub>''XZ''</sub>('''L'''<sub>1</sub>) |- style="background:paleturquoise" ! X !! Z |- | '''0''' || '''1''' |- | '''0''' || '''0''' |- | '''1''' || '''0''' |- | '''1''' || '''1''' |} | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ proj<sub>''YZ''</sub>('''L'''<sub>1</sub>) |- style="background:paleturquoise" ! Y !! Z |- | '''0''' || '''1''' |- | '''1''' || '''0''' |- | '''0''' || '''0''' |- | '''1''' || '''1''' |} |} <br> {| align="center" cellpadding="4" style="text-align:center; width:90%" | proj<sub>''XY''</sub>('''L'''<sub>0</sub>) = proj<sub>''XY''</sub>('''L'''<sub>1</sub>) | proj<sub>''XZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''XZ''</sub>('''L'''<sub>1</sub>) | proj<sub>''YZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''YZ''</sub>('''L'''<sub>1</sub>) |} <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A |- style="background:paleturquoise" ! style="width:20%" | Object ! style="width:20%" | Sign ! style="width:20%" | Interpretant |- | '''A''' || '''"A"''' || '''"A"''' |- | '''A''' || '''"A"''' || '''"i"''' |- | '''A''' || '''"i"''' || '''"A"''' |- | '''A''' || '''"i"''' || '''"i"''' |- | '''B''' || '''"B"''' || '''"B"''' |- | '''B''' || '''"B"''' || '''"u"''' |- | '''B''' || '''"u"''' || '''"B"''' |- | '''B''' || '''"u"''' || '''"u"''' |} <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B |- style="background:paleturquoise" ! style="width:20%" | Object ! style="width:20%" | Sign ! style="width:20%" | Interpretant |- | '''A''' || '''"A"''' || '''"A"''' |- | '''A''' || '''"A"''' || '''"u"''' |- | '''A''' || '''"u"''' || '''"A"''' |- | '''A''' || '''"u"''' || '''"u"''' |- | '''B''' || '''"B"''' || '''"B"''' |- | '''B''' || '''"B"''' || '''"i"''' |- | '''B''' || '''"i"''' || '''"B"''' |- | '''B''' || '''"i"''' || '''"i"''' |} <br> {| align="center" style="width:90%" | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ proj<sub>''XY''</sub>('''L'''<sub>A</sub>) |- style="background:paleturquoise" ! style="width:50%" | Object ! style="width:50%" | Sign |- | '''A''' || '''"A"''' |- | '''A''' || '''"i"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"u"''' |} | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ proj<sub>''XZ''</sub>('''L'''<sub>A</sub>) |- style="background:paleturquoise" ! style="width:50%" | Object ! style="width:50%" | Interpretant |- | '''A''' || '''"A"''' |- | '''A''' || '''"i"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"u"''' |} | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ proj<sub>''YZ''</sub>('''L'''<sub>A</sub>) |- style="background:paleturquoise" ! style="width:50%" | Sign ! style="width:50%" | Interpretant |- | '''"A"''' || '''"A"''' |- | '''"A"''' || '''"i"''' |- | '''"i"''' || '''"A"''' |- | '''"i"''' || '''"i"''' |- | '''"B"''' || '''"B"''' |- | '''"B"''' || '''"u"''' |- | '''"u"''' || '''"B"''' |- | '''"u"''' || '''"u"''' |} |} <br> {| align="center" style="width:90%" | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ proj<sub>''XY''</sub>('''L'''<sub>B</sub>) |- style="background:paleturquoise" ! style="width:50%" | Object ! style="width:50%" | Sign |- | '''A''' || '''"A"''' |- | '''A''' || '''"u"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"i"''' |} | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ proj<sub>''XZ''</sub>('''L'''<sub>B</sub>) |- style="background:paleturquoise" ! style="width:50%" | Object ! style="width:50%" | Interpretant |- | '''A''' || '''"A"''' |- | '''A''' || '''"u"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"i"''' |} | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ proj<sub>''YZ''</sub>('''L'''<sub>B</sub>) |- style="background:paleturquoise" ! style="width:50%" | Sign ! style="width:50%" | Interpretant |- | '''"A"''' || '''"A"''' |- | '''"A"''' || '''"u"''' |- | '''"u"''' || '''"A"''' |- | '''"u"''' || '''"u"''' |- | '''"B"''' || '''"B"''' |- | '''"B"''' || '''"i"''' |- | '''"i"''' || '''"B"''' |- | '''"i"''' || '''"i"''' |} |} <br> {| align="center" cellpadding="4" style="text-align:center; width:90%" | proj<sub>''XY''</sub>('''L'''<sub>A</sub>) ≠ proj<sub>''XY''</sub>('''L'''<sub>B</sub>) | proj<sub>''XZ''</sub>('''L'''<sub>A</sub>) ≠ proj<sub>''XZ''</sub>('''L'''<sub>B</sub>) | proj<sub>''YZ''</sub>('''L'''<sub>A</sub>) ≠ proj<sub>''YZ''</sub>('''L'''<sub>B</sub>) |} <br> ===='"`UNIQ--h-44--QINU`"'Method 2 : Subtitles as Top Rows==== {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |+ '''L'''<sub>0</sub> = {(''x'', ''y'', ''z'') ∈ '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 0} |- style="background:paleturquoise" ! X !! Y !! Z |- | '''0''' || '''0''' || '''0''' |- | '''0''' || '''1''' || '''1''' |- | '''1''' || '''0''' || '''1''' |- | '''1''' || '''1''' || '''0''' |} <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |+ '''L'''<sub>1</sub> = {(''x'', ''y'', ''z'') ∈ '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 1} |- style="background:paleturquoise" ! X !! Y !! Z |- | '''0''' || '''0''' || '''1''' |- | '''0''' || '''1''' || '''0''' |- | '''1''' || '''0''' || '''0''' |- | '''1''' || '''1''' || '''1''' |} <br> {| align="center" style="width:90%" | align="center" | proj<sub>''XY''</sub>('''L'''<sub>0</sub>) {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! X !! Y |- | '''0''' || '''0''' |- | '''0''' || '''1''' |- | '''1''' || '''0''' |- | '''1''' || '''1''' |} | align="center" | proj<sub>''XZ''</sub>('''L'''<sub>0</sub>) {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! X !! Z |- | '''0''' || '''0''' |- | '''0''' || '''1''' |- | '''1''' || '''1''' |- | '''1''' || '''0''' |} | align="center" | proj<sub>''YZ''</sub>('''L'''<sub>0</sub>) {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! Y !! Z |- | '''0''' || '''0''' |- | '''1''' || '''1''' |- | '''0''' || '''1''' |- | '''1''' || '''0''' |} |} <br> {| align="center" style="width:90%" | align="center" | proj<sub>''XY''</sub>('''L'''<sub>1</sub>) {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! X !! Y |- | '''0''' || '''0''' |- | '''0''' || '''1''' |- | '''1''' || '''0''' |- | '''1''' || '''1''' |} | align="center" | proj<sub>''XZ''</sub>('''L'''<sub>1</sub>) {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! X !! Z |- | '''0''' || '''1''' |- | '''0''' || '''0''' |- | '''1''' || '''0''' |- | '''1''' || '''1''' |} | align="center" | proj<sub>''YZ''</sub>('''L'''<sub>1</sub>) {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! Y !! Z |- | '''0''' || '''1''' |- | '''1''' || '''0''' |- | '''0''' || '''0''' |- | '''1''' || '''1''' |} |} <br> {| align="center" cellpadding="4" style="text-align:center; width:90%" | proj<sub>''XY''</sub>('''L'''<sub>0</sub>) = proj<sub>''XY''</sub>('''L'''<sub>1</sub>) | proj<sub>''XZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''XZ''</sub>('''L'''<sub>1</sub>) | proj<sub>''YZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''YZ''</sub>('''L'''<sub>1</sub>) |} <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A |- style="background:paleturquoise" ! style="width:20%" | Object ! style="width:20%" | Sign ! style="width:20%" | Interpretant |- | '''A''' || '''"A"''' || '''"A"''' |- | '''A''' || '''"A"''' || '''"i"''' |- | '''A''' || '''"i"''' || '''"A"''' |- | '''A''' || '''"i"''' || '''"i"''' |- | '''B''' || '''"B"''' || '''"B"''' |- | '''B''' || '''"B"''' || '''"u"''' |- | '''B''' || '''"u"''' || '''"B"''' |- | '''B''' || '''"u"''' || '''"u"''' |} <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B |- style="background:paleturquoise" ! style="width:20%" | Object ! style="width:20%" | Sign ! style="width:20%" | Interpretant |- | '''A''' || '''"A"''' || '''"A"''' |- | '''A''' || '''"A"''' || '''"u"''' |- | '''A''' || '''"u"''' || '''"A"''' |- | '''A''' || '''"u"''' || '''"u"''' |- | '''B''' || '''"B"''' || '''"B"''' |- | '''B''' || '''"B"''' || '''"i"''' |- | '''B''' || '''"i"''' || '''"B"''' |- | '''B''' || '''"i"''' || '''"i"''' |} <br> {| align="center" style="width:90%" | align="center" style="width:30%" | proj<sub>''XY''</sub>('''L'''<sub>A</sub>) {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! style="width:50%" | Object ! style="width:50%" | Sign |- | '''A''' || '''"A"''' |- | '''A''' || '''"i"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"u"''' |} | align="center" style="width:30%" | proj<sub>''XZ''</sub>('''L'''<sub>A</sub>) {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! style="width:50%" | Object ! style="width:50%" | Interpretant |- | '''A''' || '''"A"''' |- | '''A''' || '''"i"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"u"''' |} | align="center" style="width:30%" | proj<sub>''YZ''</sub>('''L'''<sub>A</sub>) {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! style="width:50%" | Sign ! style="width:50%" | Interpretant |- | '''"A"''' || '''"A"''' |- | '''"A"''' || '''"i"''' |- | '''"i"''' || '''"A"''' |- | '''"i"''' || '''"i"''' |- | '''"B"''' || '''"B"''' |- | '''"B"''' || '''"u"''' |- | '''"u"''' || '''"B"''' |- | '''"u"''' || '''"u"''' |} |} <br> {| align="center" style="width:90%" | align="center" style="width:30%" | proj<sub>''XY''</sub>('''L'''<sub>B</sub>) {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! style="width:50%" | Object ! style="width:50%" | Sign |- | '''A''' || '''"A"''' |- | '''A''' || '''"u"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"i"''' |} | align="center" style="width:30%" | proj<sub>''XZ''</sub>('''L'''<sub>B</sub>) {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! style="width:50%" | Object ! style="width:50%" | Interpretant |- | '''A''' || '''"A"''' |- | '''A''' || '''"u"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"i"''' |} | align="center" style="width:30%" | proj<sub>''YZ''</sub>('''L'''<sub>B</sub>) {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! style="width:50%" | Sign ! style="width:50%" | Interpretant |- | '''"A"''' || '''"A"''' |- | '''"A"''' || '''"u"''' |- | '''"u"''' || '''"A"''' |- | '''"u"''' || '''"u"''' |- | '''"B"''' || '''"B"''' |- | '''"B"''' || '''"i"''' |- | '''"i"''' || '''"B"''' |- | '''"i"''' || '''"i"''' |} |} <br> {| align="center" cellpadding="4" style="text-align:center; width:90%" | proj<sub>''XY''</sub>('''L'''<sub>A</sub>) ≠ proj<sub>''XY''</sub>('''L'''<sub>B</sub>) | proj<sub>''XZ''</sub>('''L'''<sub>A</sub>) ≠ proj<sub>''XZ''</sub>('''L'''<sub>B</sub>) | proj<sub>''YZ''</sub>('''L'''<sub>A</sub>) ≠ proj<sub>''YZ''</sub>('''L'''<sub>B</sub>) |} <br> ==='"`UNIQ--h-45--QINU`"'Formatted Text Display=== : So in a triadic fact, say, the example <br> {| align="center" cellspacing="8" style="width:72%" | align="center" | ''A'' gives ''B'' to ''C'' |} : we make no distinction in the ordinary logic of relations between the ''[[subject (grammar)|subject]] [[nominative]]'', the ''[[direct object]]'', and the ''[[indirect object]]''. We say that the proposition has three ''logical subjects''. We regard it as a mere affair of English grammar that there are six ways of expressing this: <br> {| align="center" cellspacing="8" style="width:72%" | style="width:36%" | ''A'' gives ''B'' to ''C'' | style="width:36%" | ''A'' benefits ''C'' with ''B'' |- | ''B'' enriches ''C'' at expense of ''A'' | ''C'' receives ''B'' from ''A'' |- | ''C'' thanks ''A'' for ''B'' | ''B'' leaves ''A'' for ''C'' |} : These six sentences express one and the same indivisible phenomenon. (C.S. Peirce, "The Categories Defended", MS 308 (1903), EP 2, 170-171). =='"`UNIQ--h-46--QINU`"'Work Area== {| border="1" cellspacing="0" cellpadding="0" style="text-align:center" |+ Binary Operations |- ! style="width:2em" | x<sub>0</sub> ! style="width:2em" | x<sub>1</sub> | style="width:2em" | <sup>2</sup>f<sub>0</sub> | style="width:2em" | <sup>2</sup>f<sub>1</sub> | style="width:2em" | <sup>2</sup>f<sub>2</sub> | style="width:2em" | <sup>2</sup>f<sub>3</sub> | style="width:2em" | <sup>2</sup>f<sub>4</sub> | style="width:2em" | <sup>2</sup>f<sub>5</sub> | style="width:2em" | <sup>2</sup>f<sub>6</sub> | style="width:2em" | <sup>2</sup>f<sub>7</sub> | style="width:2em" | <sup>2</sup>f<sub>8</sub> | style="width:2em" | <sup>2</sup>f<sub>9</sub> | style="width:2em" | <sup>2</sup>f<sub>10</sub> | style="width:2em" | <sup>2</sup>f<sub>11</sub> | style="width:2em" | <sup>2</sup>f<sub>12</sub> | style="width:2em" | <sup>2</sup>f<sub>13</sub> | style="width:2em" | <sup>2</sup>f<sub>14</sub> | style="width:2em" | <sup>2</sup>f<sub>15</sub> |- | 0 || 0 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 |- | 1 || 0 || 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 |- | 0 || 1 || 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 |- | 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 |} <br> ==='"`UNIQ--h-47--QINU`"'Draft 1=== <center><table> <caption>TRUTH TABLES FOR THE BOOLEAN OPERATIONS OF ARITY UP TO 2</caption> <tr valign="top"> <td><table border="5" cellspacing="0"> <caption>Constants</caption> <tr><td></td> <td><sup>0</sup>f<sub>0</sub></td> <td><sup>0</sup>f<sub>1</sub></td> </tr> <tr><td></td> <td align="center">0</td> <td align="center">1</td> </tr></table></td><td>    </td> <td><table border="5" cellspacing="0"><caption>Unary Operations</caption><tr> <td>x<sub>0</sub></td> <td></td> <td><sup>1</sup>f<sub>0 </sub></td> <td><sup>1</sup>f<sub>1 </sub></td> <td><sup>1</sup>f<sub>2 </sub></td> <td><sup>1</sup>f<sub>3 </sub></td> </tr><tr> <td align="center">0</td> <td></td> <td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td> </tr> <tr> <td align="center">1</td> <td></td> <td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td> </tr></table></td><td>    </td> <td><table border="5" cellspacing="0"><caption>Binary Operations</caption><tr> <td>x<sub>0</sub></td> <td>x<sub>1</sub></td> <td></td> <td><sup>2</sup>f<sub>0</sub></td> <td><sup>2</sup>f<sub>1 </sub></td> <td><sup> 2</sup>f<sub>2 </sub></td> <td><sup>2</sup>f<sub>3 </sub></td> <td><sup>2</sup>f<sub>4 </sub></td> <td><sup>2</sup>f<sub>5 </sub></td> <td><sup>2</sup>f<sub>6 </sub></td> <td><sup>2</sup>f<sub>7 </sub></td> <td><sup>2</sup>f<sub>8 </sub></td> <td><sup>2</sup>f<sub>9 </sub></td> <td><sup>2</sup>f<sub>10</sub></td> <td><sup>2</sup>f<sub>11</sub></td> <td><sup>2</sup>f<sub>12</sub></td> <td><sup>2</sup>f<sub>13</sub></td> <td><sup>2</sup>f<sub>14</sub></td> <td><sup>2</sup>f<sub>15</sub></td> </tr><tr> <td align="center">0</td> <td align="center">0</td> <td></td> <td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td> </tr> <tr> <td align="center">1</td> <td align="center">0</td> <td></td> <td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td> <td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td> <td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td> <td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td> </tr> <tr> <td align="center">0</td> <td align="center">1</td> <td></td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> </tr> <tr> <td align="center">1</td> <td align="center">1</td> <td></td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> </tr> </table></td> </table></center> ==='"`UNIQ--h-48--QINU`"'Draft 2=== <center><table> <caption>TRUTH TABLES FOR THE BOOLEAN OPERATIONS OF ARITY UP TO 2</caption> <tr valign="top"> <td><table border="5" cellspacing="0"> <caption>Constants</caption> <tr><td></td> <td><sup>0</sup>f<sub>0</sub></td> <td><sup>0</sup>f<sub>1</sub></td> </tr> <tr><td></td> <td align="center">0</td> <td align="center">1</td> </tr></table></td><td>    </td> <td><table border="5" cellspacing="0"><caption>Unary Operations</caption><tr> <td>x<sub>0</sub></td> <td></td> <td><sup>1</sup>f<sub>0 </sub></td> <td><sup>1</sup>f<sub>1 </sub></td> <td><sup>1</sup>f<sub>2 </sub></td> <td><sup>1</sup>f<sub>3 </sub></td> </tr><tr> <td align="center">0</td> <td></td> <td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td> </tr> <tr> <td align="center">1</td> <td></td> <td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td> </tr></table></td><td>    </td> <td><table border="5" cellspacing="0"><caption>Binary Operations</caption><tr> <td>x<sub>0</sub></td> <td>x<sub>1</sub></td> <td></td> <td><sup>2</sup>f<sub>0</sub></td> <td><sup>2</sup>f<sub>1 </sub></td> <td><sup> 2</sup>f<sub>2 </sub></td> <td><sup>2</sup>f<sub>3 </sub></td> <td><sup>2</sup>f<sub>4 </sub></td> <td><sup>2</sup>f<sub>5 </sub></td> <td><sup>2</sup>f<sub>6 </sub></td> <td><sup>2</sup>f<sub>7 </sub></td> <td><sup>2</sup>f<sub>8 </sub></td> <td><sup>2</sup>f<sub>9 </sub></td> <td><sup>2</sup>f<sub>10</sub></td> <td><sup>2</sup>f<sub>11</sub></td> <td><sup>2</sup>f<sub>12</sub></td> <td><sup>2</sup>f<sub>13</sub></td> <td><sup>2</sup>f<sub>14</sub></td> <td><sup>2</sup>f<sub>15</sub></td> </tr><tr> <td align="center">0</td> <td align="center">0</td> <td></td> <td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td> </tr> <tr> <td align="center">1</td> <td align="center">0</td> <td></td> <td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td> <td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td> <td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td> <td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td> </tr> <tr> <td align="center">0</td> <td align="center">1</td> <td></td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> </tr> <tr> <td align="center">1</td> <td align="center">1</td> <td></td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> </tr> </table></td> </table></center> =='"`UNIQ--h-49--QINU`"'Inquiry and Analogy== ==='"`UNIQ--h-50--QINU`"'Test Patterns=== {| align="center" | style="background:black; color:white" | 1 | style="background:white; color:black" | 0 | style="background:black; color:white" | 1 | style="background:white; color:black" | 0 | style="background:black; color:white" | 1 | style="background:white; color:black" | 0 | style="background:black; color:white" | 1 | style="background:white; color:black" | 0 |- | style="background:white; color:black" | 0 | style="background:black; color:white" | 1 | style="background:white; color:black" | 0 | style="background:black; color:white" | 1 | style="background:white; color:black" | 0 | style="background:black; color:white" | 1 | style="background:white; color:black" | 0 | style="background:black; color:white" | 1 |}<br> {| align="center" | style="background:white; color:black" | 1 | style="background:black; color:white" | 0 | style="background:white; color:black" | 1 | style="background:black; color:white" | 0 | style="background:white; color:black" | 1 | style="background:black; color:white" | 0 | style="background:white; color:black" | 1 | style="background:black; color:white" | 0 |- | style="background:black; color:white" | 0 | style="background:white; color:black" | 1 | style="background:black; color:white" | 0 | style="background:white; color:black" | 1 | style="background:black; color:white" | 0 | style="background:white; color:black" | 1 | style="background:black; color:white" | 0 | style="background:white; color:black" | 1 |}<br> {| align="center" | style="background:black; color:white" | 1 | style="background:white; color:black" | 0 | style="background:black; color:white" | 1 | style="background:white; color:black" | 0 | style="background:black; color:white" | 1 | style="background:white; color:black" | 0 | style="background:black; color:white" | 1 | style="background:white; color:black" | 0 |- | style="background:white; color:black" | 0 | style="background:black; color:white" | 1 | style="background:white; color:black" | 0 | style="background:black; color:white" | 1 | style="background:white; color:black" | 0 | style="background:black; color:white" | 1 | style="background:white; color:black" | 0 | style="background:black; color:white" | 1 |}<br> ==='"`UNIQ--h-51--QINU`"'Table 10=== {| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" |+ '''Table 10. Higher Order Propositions (''n'' = 1)''' |- style="background:ghostwhite" | align="right" | \(x\):

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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