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

Revision as of 03:32, 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

\(\text{Table A1.}~~\text{Propositional Forms on Two Variables}\)
\(\mathcal{L}_1\) \(\text{Decimal}\)	\(\mathcal{L}_2\) \(\text{Binary}\)	\(\mathcal{L}_3\) \(\text{Vector}\)	\(\mathcal{L}_4\) \(\text{Cactus}\)	\(\mathcal{L}_5\) \(\text{English}\)	\(\mathcal{L}_6\) \(\text{Ordinary}\)
	\(x\colon\!\)	\(1~1~0~0\!\)
	\(y\colon\!\)	\(1~0~1~0\!\)
\(f_{0}\!\)	\(f_{0000}\!\)	\(0~0~0~0\!\)	\((~)\!\)	\(\text{false}\!\)	\(0\!\)
\(f_{1}\!\)	\(f_{0001}\!\)	\(0~0~0~1\!\)	\((x)(y)\!\)	\(\text{neither}~ x ~\text{nor}~ y\!\)	\(\lnot x \land \lnot y\!\)
\(f_{2}\!\)	\(f_{0010}\!\)	\(0~0~1~0\!\)	\((x)~y\!\)	\(y ~\text{without}~ x\!\)	\(\lnot x \land y\!\)
\(f_{3}\!\)	\(f_{0011}\!\)	\(0~0~1~1\!\)	\((x)\!\)	\(\text{not}~ x\!\)	\(\lnot x\!\)
\(f_{4}\!\)	\(f_{0100}\!\)	\(0~1~0~0\!\)	\(x~(y)\!\)	\(x ~\text{without}~ y\!\)	\(x \land \lnot y\!\)
\(f_{5}\!\)	\(f_{0101}\!\)	\(0~1~0~1\!\)	\((y)\!\)	\(\text{not}~ y\!\)	\(\lnot y\!\)
\(f_{6}\!\)	\(f_{0110}\!\)	\(0~1~1~0\!\)	\((x,~y)\!\)	\(x ~\text{not equal to}~ y\!\)	\(x \ne y\!\)
\(f_{7}\!\)	\(f_{0111}\!\)	\(0~1~1~1\!\)	\((x~y)\!\)	\(\text{not both}~ x ~\text{and}~ y\!\)	\(\lnot x \lor \lnot y\!\)
\(f_{8}\!\)	\(f_{1000}\!\)	\(1~0~0~0\!\)	\(x~y\!\)	\(x ~\text{and}~ y\!\)	\(x \land y\!\)
\(f_{9}\!\)	\(f_{1001}\!\)	\(1~0~0~1\!\)	\(((x,~y))\!\)	\(x ~\text{equal to}~ y\!\)	\(x = y\!\)
\(f_{10}\!\)	\(f_{1010}\!\)	\(1~0~1~0\!\)	\(y\!\)	\(y\!\)	\(y\!\)
\(f_{11}\!\)	\(f_{1011}\!\)	\(1~0~1~1\!\)	\((x~(y))\!\)	\(\text{not}~ x ~\text{without}~ y\!\)	\(x \Rightarrow y\!\)
\(f_{12}\!\)	\(f_{1100}\!\)	\(1~1~0~0\!\)	\(x\!\)	\(x\!\)	\(x\!\)
\(f_{13}\!\)	\(f_{1101}\!\)	\(1~1~0~1\!\)	\(((x)~y)\!\)	\(\text{not}~ y ~\text{without}~ x\!\)	\(x \Leftarrow y\!\)
\(f_{14}\!\)	\(f_{1110}\!\)	\(1~1~1~0\!\)	\(((x)(y))\!\)	\(x ~\text{or}~ y\!\)	\(x \lor y\!\)
\(f_{15}\!\)	\(f_{1111}\!\)	\(1~1~1~1\!\)	\(((~))\!\)	\(\text{true}\!\)	\(1\!\)

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

\(\mathcal{L}_1\)

\(\text{Decimal}\)

\(\mathcal{L}_2\)

\(\text{Binary}\)

\(\mathcal{L}_3\)

\(\text{Vector}\)

\(\mathcal{L}_4\)

\(\text{Cactus}\)

\(\mathcal{L}_5\)

\(\text{English}\)

\(\mathcal{L}_6\)

\(\text{Ordinary}\)

\(x\colon\!\)

\(1~1~0~0\!\)

\(y\colon\!\)

\(1~0~1~0\!\)

\(\begin{matrix} f_0 \'"`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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Revision as of 03:32, 30 May 2009

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-| <math>f_{1010}\!</math>
+{| align="center"
-| <math>1~0~1~0\!</math>
+|
-| <math>y\!</math>
+<math>\begin{matrix}
-| <math>y\!</math>
+~~x~~y~~
-| <math>y\!</math>
+\\[4pt]
-|-
+((x,~y))
-| <math>f_{11}\!</math>
+\\[4pt]
-| <math>f_{1011}\!</math>
+~~~~~y~~
-| <math>1~0~1~1\!</math>
+\\[4pt]
-| <math>(x~(y))\!</math>
+~(x~(y))
-| <math>\text{not}~ x ~\text{without}~ y\!</math>
+\\[4pt]
-| <math>x \Rightarrow y\!</math>
+~~x~~~~~
-|-
+\\[4pt]
-| <math>f_{12}\!</math>
+((x)~y)~
-| <math>f_{1100}\!</math>
+\\[4pt]
-| <math>1~1~0~0\!</math>
+((x)(y))
-| <math>x\!</math>
+\\[4pt]
-| <math>x\!</math>
+((~))
-| <math>x\!</math>
+\end{matrix}</math>
-|-
+|}
-| <math>f_{13}\!</math>
+|
-| <math>f_{1101}\!</math>
+{| align="center"
-| <math>1~1~0~1\!</math>
+|
-| <math>((x)~y)\!</math>
+<math>\begin{matrix}
-| <math>\text{not}~ y ~\text{without}~ x\!</math>
+x ~\text{and}~ y
-| <math>x \Leftarrow y\!</math>
+\\[4pt]
-|-
+x ~\text{equal to}~ y
-| <math>f_{14}\!</math>
+\\[4pt]
-| <math>f_{1110}\!</math>
+y
-| <math>1~1~1~0\!</math>
+\\[4pt]
-| <math>((x)(y))\!</math>
+\text{not}~ x ~\text{without}~ y
-| <math>x ~\text{or}~ y\!</math>
+\\[4pt]
-| <math>x \lor y\!</math>
+x
-|-
+\\[4pt]
-| <math>f_{15}\!</math>
+\text{not}~ y ~\text{without}~ x
-| <math>f_{1111}\!</math>
+\\[4pt]
-| <math>1~1~1~1\!</math>
+x ~\text{or}~ y
-| <math>((~))\!</math>
+\\[4pt]
-| <math>\text{true}\!</math>
+\text{true}
-| <math>1\!</math>
+\end{matrix}</math>
+|}
+|
+{| align="center"
+|
+<math>\begin{matrix}
+x \land y
+\\[4pt]
+x = y
+\\[4pt]
+y
+\\[4pt]
+x \Rightarrow y
+\\[4pt]
+x
+\\[4pt]
+x \Leftarrow y
+\\[4pt]
+x \lor y
+\\[4pt]
+\end{matrix}</math>
+|}
 |}