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

Revision as of 20:08, 29 May 2009

Differential Logic

Ascii Tables

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

Table A2.  Propositional Forms On Two Variables
o---------o---------o---------o----------o------------------o----------o
| L_1     | L_2     | L_3     | L_4      | L_5              | L_6      |
|         |         |         |          |                  |          |
| Decimal | Binary  | Vector  | Cactus   | English          | Ordinary |
o---------o---------o---------o----------o------------------o----------o
|         |       x : 1 1 0 0 |          |                  |          |
|         |       y : 1 0 1 0 |          |                  |          |
o---------o---------o---------o----------o------------------o----------o
|         |         |         |          |                  |          |
| f_0     | f_0000  | 0 0 0 0 |    ()    | false            |    0     |
|         |         |         |          |                  |          |
o---------o---------o---------o----------o------------------o----------o
|         |         |         |          |                  |          |
| f_1     | f_0001  | 0 0 0 1 |  (x)(y)  | neither x nor y  | ~x & ~y  |
|         |         |         |          |                  |          |
| f_2     | f_0010  | 0 0 1 0 |  (x) y   | y and not x      | ~x &  y  |
|         |         |         |          |                  |          |
| f_4     | f_0100  | 0 1 0 0 |   x (y)  | x and not y      |  x & ~y  |
|         |         |         |          |                  |          |
| f_8     | f_1000  | 1 0 0 0 |   x  y   | x and y          |  x &  y  |
|         |         |         |          |                  |          |
o---------o---------o---------o----------o------------------o----------o
|         |         |         |          |                  |          |
| f_3     | f_0011  | 0 0 1 1 |  (x)     | not x            | ~x       |
|         |         |         |          |                  |          |
| f_12    | f_1100  | 1 1 0 0 |   x      | x                |  x       |
|         |         |         |          |                  |          |
o---------o---------o---------o----------o------------------o----------o
|         |         |         |          |                  |          |
| f_6     | f_0110  | 0 1 1 0 |  (x, y)  | x not equal to y |  x +  y  |
|         |         |         |          |                  |          |
| f_9     | f_1001  | 1 0 0 1 | ((x, y)) | x equal to y     |  x =  y  |
|         |         |         |          |                  |          |
o---------o---------o---------o----------o------------------o----------o
|         |         |         |          |                  |          |
| f_5     | f_0101  | 0 1 0 1 |     (y)  | not y            |      ~y  |
|         |         |         |          |                  |          |
| f_10    | f_1010  | 1 0 1 0 |      y   | y                |       y  |
|         |         |         |          |                  |          |
o---------o---------o---------o----------o------------------o----------o
|         |         |         |          |                  |          |
| f_7     | f_0111  | 0 1 1 1 |  (x  y)  | not both x and y | ~x v ~y  |
|         |         |         |          |                  |          |
| f_11    | f_1011  | 1 0 1 1 |  (x (y)) | not x without y  |  x => y  |
|         |         |         |          |                  |          |
| f_13    | f_1101  | 1 1 0 1 | ((x) y)  | not y without x  |  x <= y  |
|         |         |         |          |                  |          |
| f_14    | f_1110  | 1 1 1 0 | ((x)(y)) | x or y           |  x v  y  |
|         |         |         |          |                  |          |
o---------o---------o---------o----------o------------------o----------o
|         |         |         |          |                  |          |
| f_15    | f_1111  | 1 1 1 1 |   (())   | true             |    1     |
|         |         |         |          |                  |          |
o---------o---------o---------o----------o------------------o----------o

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

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

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

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

Wiki Tables

New Versions

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

Table 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

Old Versions

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

Table 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

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

TeX Tables

\tableofcontents

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

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

\begin{quote}\begin{tabular}{|c|c|c|c|c|c|c|}
\multicolumn{7}{c}{\textbf{Table A1.  Propositional Forms on Two Variables}} \\
\hline
$\mathcal{L}_1$ &
$\mathcal{L}_2$ &&
$\mathcal{L}_3$ &
$\mathcal{L}_4$ &
$\mathcal{L}_5$ &
$\mathcal{L}_6$ \\
\hline
& & $x =$ & 1 1 0 0 & & & \\
& & $y =$ & 1 0 1 0 & & & \\
\hline
$f_{0}$     &
$f_{0000}$  &&
0 0 0 0     &
$(~)$       &
$\operatorname{false}$ &
$0$         \\
$f_{1}$     &
$f_{0001}$  &&
0 0 0 1     &
$(x)(y)$    &
$\operatorname{neither}\ x\ \operatorname{nor}\ y$ &
$\lnot x \land \lnot y$ \\
$f_{2}$     &
$f_{0010}$  &&
0 0 1 0     &
$(x)\ y$    &
$y\ \operatorname{without}\ x$ &
$\lnot x \land y$ \\
$f_{3}$     &
$f_{0011}$  &&
0 0 1 1     &
$(x)$       &
$\operatorname{not}\ x$ &
$\lnot x$   \\
$f_{4}$     &
$f_{0100}$  &&
0 1 0 0     &
$x\ (y)$    &
$x\ \operatorname{without}\ y$ &
$x \land \lnot y$ \\
$f_{5}$     &
$f_{0101}$  &&
0 1 0 1     &
$(y)$       &
$\operatorname{not}\ y$ &
$\lnot y$   \\
$f_{6}$     &
$f_{0110}$  &&
0 1 1 0     &
$(x,\ y)$   &
$x\ \operatorname{not~equal~to}\ y$ &
$x \ne y$   \\
$f_{7}$     &
$f_{0111}$  &&
0 1 1 1     &
$(x\ y)$    &
$\operatorname{not~both}\ x\ \operatorname{and}\ y$ &
$\lnot x \lor \lnot y$ \\
\hline
$f_{8}$     &
$f_{1000}$  &&
1 0 0 0     &
$x\ y$      &
$x\ \operatorname{and}\ y$ &
$x \land y$ \\
$f_{9}$     &
$f_{1001}$  &&
1 0 0 1     &
$((x,\ y))$ &
$x\ \operatorname{equal~to}\ y$ &
$x = y$     \\
$f_{10}$    &
$f_{1010}$  &&
1 0 1 0     &
$y$         &
$y$         &
$y$         \\
$f_{11}$    &
$f_{1011}$  &&
1 0 1 1     &
$(x\ (y))$  &
$\operatorname{not}\ x\ \operatorname{without}\ y$ &
$x \Rightarrow y$ \\
$f_{12}$    &
$f_{1100}$  &&
1 1 0 0     &
$x$         &
$x$         &
$x$         \\
$f_{13}$    &
$f_{1101}$  &&
1 1 0 1     &
$((x)\ y)$  &
$\operatorname{not}\ y\ \operatorname{without}\ x$ &
$x \Leftarrow y$ \\
$f_{14}$    &
$f_{1110}$  &&
1 1 1 0     &
$((x)(y))$  &
$x\ \operatorname{or}\ y$ &
$x \lor y$  \\
$f_{15}$    &
$f_{1111}$  &&
1 1 1 1     &
$((~))$     &
$\operatorname{true}$ &
$1$         \\
\hline
\end{tabular}\end{quote}

\subsection{Table A2.  Propositional Forms on Two Variables}

Table A2 lists the sixteen Boolean functions of two variables in a different order, grouping them by structural similarity into seven natural classes.

\begin{quote}\begin{tabular}{|c|c|c|c|c|c|c|}
\multicolumn{7}{c}{\textbf{Table A2.  Propositional Forms on Two Variables}} \\
\hline
$\mathcal{L}_1$ &
$\mathcal{L}_2$ &&
$\mathcal{L}_3$ &
$\mathcal{L}_4$ &
$\mathcal{L}_5$ &
$\mathcal{L}_6$ \\
\hline
& & $x =$ & 1 1 0 0 & & & \\
& & $y =$ & 1 0 1 0 & & & \\
\hline
$f_{0}$     &
$f_{0000}$  &&
0 0 0 0     &
$(~)$       &
$\operatorname{false}$ &
$0$         \\
\hline
$f_{1}$     &
$f_{0001}$  &&
0 0 0 1     &
$(x)(y)$    &
$\operatorname{neither}\ x\ \operatorname{nor}\ y$ &
$\lnot x \land \lnot y$ \\
$f_{2}$     &
$f_{0010}$  &&
0 0 1 0     &
$(x)\ y$    &
$y\ \operatorname{without}\ x$ &
$\lnot x \land y$ \\
$f_{4}$     &
$f_{0100}$  &&
0 1 0 0     &
$x\ (y)$    &
$x\ \operatorname{without}\ y$ &
$x \land \lnot y$ \\
$f_{8}$     &
$f_{1000}$  &&
1 0 0 0     &
$x\ y$      &
$x\ \operatorname{and}\ y$ &
$x \land y$ \\
\hline
$f_{3}$     &
$f_{0011}$  &&
0 0 1 1     &
$(x)$       &
$\operatorname{not}\ x$ &
$\lnot x$   \\
$f_{12}$    &
$f_{1100}$  &&
1 1 0 0     &
$x$         &
$x$         &
$x$         \\
\hline
$f_{6}$     &
$f_{0110}$  &&
0 1 1 0     &
$(x,\ y)$   &
$x\ \operatorname{not~equal~to}\ y$ &
$x \ne y$   \\
$f_{9}$     &
$f_{1001}$  &&
1 0 0 1     &
$((x,\ y))$ &
$x\ \operatorname{equal~to}\ y$ &
$x = y$     \\
\hline
$f_{5}$     &
$f_{0101}$  &&
0 1 0 1     &
$(y)$       &
$\operatorname{not}\ y$ &
$\lnot y$   \\
$f_{10}$    &
$f_{1010}$  &&
1 0 1 0     &
$y$         &
$y$         &
$y$         \\
\hline
$f_{7}$     &
$f_{0111}$  &&
0 1 1 1     &
$(x\ y)$    &
$\operatorname{not~both}\ x\ \operatorname{and}\ y$ &
$\lnot x \lor \lnot y$ \\
$f_{11}$    &
$f_{1011}$  &&
1 0 1 1     &
$(x\ (y))$  &
$\operatorname{not}\ x\ \operatorname{without}\ y$ &
$x \Rightarrow y$ \\
$f_{13}$    &
$f_{1101}$  &&
1 1 0 1     &
$((x)\ y)$  &
$\operatorname{not}\ y\ \operatorname{without}\ x$ &
$x \Leftarrow y$ \\
$f_{14}$    &
$f_{1110}$  &&
1 1 1 0     &
$((x)(y))$  &
$x\ \operatorname{or}\ y$ &
$x \lor y$  \\
\hline
$f_{15}$    &
$f_{1111}$  &&
1 1 1 1     &
$((~))$     &
$\operatorname{true}$ &
$1$         \\
\hline
\end{tabular}\end{quote}

\subsection{Table A3.  $\operatorname{E}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}

\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
\multicolumn{6}{c}{\textbf{Table A3.  $\operatorname{E}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}} \\
\hline
& &
$\operatorname{T}_{11}$ &
$\operatorname{T}_{10}$ &
$\operatorname{T}_{01}$ &
$\operatorname{T}_{00}$ \\
& $f$ &
$\operatorname{E}f|_{\operatorname{d}x\ \operatorname{d}y}$   &
$\operatorname{E}f|_{\operatorname{d}x (\operatorname{d}y)}$  &
$\operatorname{E}f|_{(\operatorname{d}x) \operatorname{d}y}$  &
$\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}$ \\
\hline
$f_{0}$  & $(~)$       & $(~)$       & $(~)$       & $(~)$       & $(~)$       \\
\hline
$f_{1}$  & $(x)(y)$    & $x\ y$      & $x\ (y)$    & $(x)\ y$    & $(x)(y)$    \\
$f_{2}$  & $(x)\ y$    & $x\ (y)$    & $x\ y$      & $(x)(y)$    & $(x)\ y$    \\
$f_{4}$  & $x\ (y)$    & $(x)\ y$    & $(x)(y)$    & $x\ y$      & $x\ (y)$    \\
$f_{8}$  & $x\ y$      & $(x)(y)$    & $(x)\ y$    & $x\ (y)$    & $x\ y$      \\
\hline
$f_{3}$  & $(x)$       & $x$         & $x$         & $(x)$       & $(x)$       \\
$f_{12}$ & $x$         & $(x)$       & $(x)$       & $x$         & $x$         \\
\hline
$f_{6}$  & $(x,\ y)$   & $(x,\ y)$   & $((x,\ y))$ & $((x,\ y))$ & $(x,\ y)$   \\
$f_{9}$  & $((x,\ y))$ & $((x,\ y))$ & $(x,\ y)$   & $(x,\ y)$   & $((x,\ y))$ \\
\hline
$f_{5}$  & $(y)$       & $y$         & $(y)$       & $y$         & $(y)$       \\
$f_{10}$ & $y$         & $(y)$       & $y$         & $(y)$       & $y$         \\
\hline
$f_{7}$  & $(x\ y)$    & $((x)(y))$  & $((x)\ y)$  & $(x\ (y))$  & $(x\ y)$    \\
$f_{11}$ & $(x\ (y))$  & $((x)\ y)$  & $((x)(y))$  & $(x\ y)$    & $(x\ (y))$  \\
$f_{13}$ & $((x)\ y)$  & $(x\ (y))$  & $(x\ y)$    & $((x)(y))$  & $((x)\ y)$  \\
$f_{14}$ & $((x)(y))$  & $(x\ y)$    & $(x\ (y))$  & $((x)\ y)$  & $((x)(y))$  \\
\hline
$f_{15}$ & $((~))$     & $((~))$     & $((~))$     & $((~))$     & $((~))$     \\
\hline
\multicolumn{2}{|c||}{\PMlinkname{Fixed Point}{FixedPoint} Total:} & 4 & 4 & 4 & 16 \\
\hline
\end{tabular}\end{quote}

\subsection{Table A4.  $\operatorname{D}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}

\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
\multicolumn{6}{c}{\textbf{Table A4.  $\operatorname{D}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}} \\
\hline
& $f$ &
$\operatorname{D}f|_{\operatorname{d}x\ \operatorname{d}y}$   &
$\operatorname{D}f|_{\operatorname{d}x (\operatorname{d}y)}$  &
$\operatorname{D}f|_{(\operatorname{d}x) \operatorname{d}y}$  &
$\operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)}$ \\
\hline
$f_{0}$  & $(~)$       & $(~)$       & $(~)$   & $(~)$   & $(~)$ \\
\hline
$f_{1}$  & $(x)(y)$    & $((x,\ y))$ & $(y)$   & $(x)$   & $(~)$ \\
$f_{2}$  & $(x)\ y$    & $(x,\ y)$   & $y$     & $(x)$   & $(~)$ \\
$f_{4}$  & $x\ (y)$    & $(x,\ y)$   & $(y)$   & $x$     & $(~)$ \\
$f_{8}$  & $x\ y$      & $((x,\ y))$ & $y$     & $x$     & $(~)$ \\
\hline
$f_{3}$  & $(x)$       & $((~))$     & $((~))$ & $(~)$   & $(~)$ \\
$f_{12}$ & $x$         & $((~))$     & $((~))$ & $(~)$   & $(~)$ \\
\hline
$f_{6}$  & $(x,\ y)$   & $(~)$       & $((~))$ & $((~))$ & $(~)$ \\
$f_{9}$  & $((x,\ y))$ & $(~)$       & $((~))$ & $((~))$ & $(~)$ \\
\hline
$f_{5}$  & $(y)$       & $((~))$     & $(~)$   & $((~))$ & $(~)$ \\
$f_{10}$ & $y$         & $((~))$     & $(~)$   & $((~))$ & $(~)$ \\
\hline
$f_{7}$  & $(x\ y)$    & $((x,\ y))$ & $y$     & $x$     & $(~)$ \\
$f_{11}$ & $(x\ (y))$  & $(x,\ y)$   & $(y)$   & $x$     & $(~)$ \\
$f_{13}$ & $((x)\ y)$  & $(x,\ y)$   & $y$     & $(x)$   & $(~)$ \\
$f_{14}$ & $((x)(y))$  & $((x,\ y))$ & $(y)$   & $(x)$   & $(~)$ \\
\hline
$f_{15}$ & $((~))$     & $(~)$       & $(~)$   & $(~)$   & $(~)$ \\
\hline
\end{tabular}\end{quote}

\subsection{Table A5.  $\operatorname{E}f$ Expanded Over Ordinary Features $\{ x, y \}$}

\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
\multicolumn{6}{c}{\textbf{Table A5.  $\operatorname{E}f$ Expanded Over Ordinary Features $\{ x, y \}$}} \\
\hline
& $f$ &
$\operatorname{E}f|_{x\ y}$   &
$\operatorname{E}f|_{x (y)}$  &
$\operatorname{E}f|_{(x) y}$  &
$\operatorname{E}f|_{(x)(y)}$ \\
\hline
$f_{0}$ &
$(~)$   &
$(~)$   &
$(~)$   &
$(~)$   &
$(~)$   \\
\hline
$f_{1}$  &
$(x)(y)$ &
$\operatorname{d}x\ \operatorname{d}y$   &
$\operatorname{d}x\ (\operatorname{d}y)$ &
$(\operatorname{d}x)\ \operatorname{d}y$ &
$(\operatorname{d}x)(\operatorname{d}y)$ \\
$f_{2}$  &
$(x)\ y$ &
$\operatorname{d}x\ (\operatorname{d}y)$ &
$\operatorname{d}x\ \operatorname{d}y$   &
$(\operatorname{d}x)(\operatorname{d}y)$ &
$(\operatorname{d}x)\ \operatorname{d}y$ \\
$f_{4}$  &
$x\ (y)$ &
$(\operatorname{d}x)\ \operatorname{d}y$ &
$(\operatorname{d}x)(\operatorname{d}y)$ &
$\operatorname{d}x\ \operatorname{d}y$   &
$\operatorname{d}x\ (\operatorname{d}y)$ \\
$f_{8}$ &
$x\ y$  &
$(\operatorname{d}x)(\operatorname{d}y)$ &
$(\operatorname{d}x)\ \operatorname{d}y$ &
$\operatorname{d}x\ (\operatorname{d}y)$ &
$\operatorname{d}x\ \operatorname{d}y$   \\
\hline
$f_{3}$ &
$(x)$   &
$\operatorname{d}x$   &
$\operatorname{d}x$   &
$(\operatorname{d}x)$ &
$(\operatorname{d}x)$ \\
$f_{12}$ &
$x$      &
$(\operatorname{d}x)$ &
$(\operatorname{d}x)$ &
$\operatorname{d}x$   &
$\operatorname{d}x$   \\
\hline
$f_{6}$   &
$(x,\ y)$ &
$(\operatorname{d}x,\ \operatorname{d}y)$   &
$((\operatorname{d}x,\ \operatorname{d}y))$ &
$((\operatorname{d}x,\ \operatorname{d}y))$ &
$(\operatorname{d}x,\ \operatorname{d}y)$   \\
$f_{9}$     &
$((x,\ y))$ &
$((\operatorname{d}x,\ \operatorname{d}y))$ &
$(\operatorname{d}x,\ \operatorname{d}y)$   &
$(\operatorname{d}x,\ \operatorname{d}y)$   &
$((\operatorname{d}x,\ \operatorname{d}y))$ \\
\hline
$f_{5}$ &
$(y)$   &
$\operatorname{d}y$   &
$(\operatorname{d}y)$ &
$\operatorname{d}y$   &
$(\operatorname{d}y)$ \\
$f_{10}$ &
$y$      &
$(\operatorname{d}y)$ &
$\operatorname{d}y$   &
$(\operatorname{d}y)$ &
$\operatorname{d}y$   \\
\hline
$f_{7}$  &
$(x\ y)$ &
$((\operatorname{d}x)(\operatorname{d}y))$ &
$((\operatorname{d}x)\ \operatorname{d}y)$ &
$(\operatorname{d}x\ (\operatorname{d}y))$ &
$(\operatorname{d}x\ \operatorname{d}y)$   \\
$f_{11}$   &
$(x\ (y))$ &
$((\operatorname{d}x)\ \operatorname{d}y)$ &
$((\operatorname{d}x)(\operatorname{d}y))$ &
$(\operatorname{d}x\ \operatorname{d}y)$   &
$(\operatorname{d}x\ (\operatorname{d}y))$ \\
$f_{13}$   &
$((x)\ y)$ &
$(\operatorname{d}x\ (\operatorname{d}y))$ &
$(\operatorname{d}x\ \operatorname{d}y)$   &
$((\operatorname{d}x)(\operatorname{d}y))$ &
$((\operatorname{d}x)\ \operatorname{d}y)$ \\
$f_{14}$   &
$((x)(y))$ &
$(\operatorname{d}x\ \operatorname{d}y)$   &
$(\operatorname{d}x\ (\operatorname{d}y))$ &
$((\operatorname{d}x)\ \operatorname{d}y)$ &
$((\operatorname{d}x)(\operatorname{d}y))$ \\
\hline
$f_{15}$ &
$((~))$  &
$((~))$  &
$((~))$  &
$((~))$  &
$((~))$  \\
\hline
\end{tabular}\end{quote}

\subsection{Table A6.  $\operatorname{D}f$ Expanded Over Ordinary Features $\{ x, y \}$}

\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
\multicolumn{6}{c}{\textbf{Table A6.  $\operatorname{D}f$ Expanded Over Ordinary Features $\{ x, y \}$}} \\
\hline
& $f$ &
$\operatorname{D}f|_{x\ y}$   &
$\operatorname{D}f|_{x (y)}$  &
$\operatorname{D}f|_{(x) y}$  &
$\operatorname{D}f|_{(x)(y)}$ \\
\hline
$f_{0}$ &
$(~)$   &
$(~)$   &
$(~)$   &
$(~)$   &
$(~)$   \\
\hline
$f_{1}$  &
$(x)(y)$ &
$\operatorname{d}x\ \operatorname{d}y$     &
$\operatorname{d}x\ (\operatorname{d}y)$   &
$(\operatorname{d}x)\ \operatorname{d}y$   &
$((\operatorname{d}x)(\operatorname{d}y))$ \\
$f_{2}$  &
$(x)\ y$ &
$\operatorname{d}x\ (\operatorname{d}y)$   &
$\operatorname{d}x\ \operatorname{d}y$     &
$((\operatorname{d}x)(\operatorname{d}y))$ &
$(\operatorname{d}x)\ \operatorname{d}y$   \\
$f_{4}$  &
$x\ (y)$ &
$(\operatorname{d}x)\ \operatorname{d}y$   &
$((\operatorname{d}x)(\operatorname{d}y))$ &
$\operatorname{d}x\ \operatorname{d}y$     &
$\operatorname{d}x\ (\operatorname{d}y)$   \\
$f_{8}$ &
$x\ y$  &
$((\operatorname{d}x)(\operatorname{d}y))$ &
$(\operatorname{d}x)\ \operatorname{d}y$   &
$\operatorname{d}x\ (\operatorname{d}y)$   &
$\operatorname{d}x\ \operatorname{d}y$     \\
\hline
$f_{3}$ &
$(x)$   &
$\operatorname{d}x$ &
$\operatorname{d}x$ &
$\operatorname{d}x$ &
$\operatorname{d}x$ \\
$f_{12}$ &
$x$      &
$\operatorname{d}x$ &
$\operatorname{d}x$ &
$\operatorname{d}x$ &
$\operatorname{d}x$ \\
\hline
$f_{6}$   &
$(x,\ y)$ &
$(\operatorname{d}x,\ \operatorname{d}y)$ &
$(\operatorname{d}x,\ \operatorname{d}y)$ &
$(\operatorname{d}x,\ \operatorname{d}y)$ &
$(\operatorname{d}x,\ \operatorname{d}y)$ \\
$f_{9}$     &
$((x,\ y))$ &
$(\operatorname{d}x,\ \operatorname{d}y)$ &
$(\operatorname{d}x,\ \operatorname{d}y)$ &
$(\operatorname{d}x,\ \operatorname{d}y)$ &
$(\operatorname{d}x,\ \operatorname{d}y)$ \\
\hline
$f_{5}$ &
$(y)$   &
$\operatorname{d}y$ &
$\operatorname{d}y$ &
$\operatorname{d}y$ &
$\operatorname{d}y$ \\
$f_{10}$ &
$y$      &
$\operatorname{d}y$ &
$\operatorname{d}y$ &
$\operatorname{d}y$ &
$\operatorname{d}y$ \\
\hline
$f_{7}$  &
$(x\ y)$ &
$((\operatorname{d}x)(\operatorname{d}y))$ &
$(\operatorname{d}x)\ \operatorname{d}y$   &
$\operatorname{d}x\ (\operatorname{d}y)$   &
$\operatorname{d}x\ \operatorname{d}y$     \\
$f_{11}$   &
$(x\ (y))$ &
$(\operatorname{d}x)\ \operatorname{d}y$   &
$((\operatorname{d}x)(\operatorname{d}y))$ &
$\operatorname{d}x\ \operatorname{d}y$     &
$\operatorname{d}x\ (\operatorname{d}y)$   \\
$f_{13}$   &
$((x)\ y)$ &
$\operatorname{d}x\ (\operatorname{d}y)$   &
$\operatorname{d}x\ \operatorname{d}y$     &
$((\operatorname{d}x)(\operatorname{d}y))$ &
$(\operatorname{d}x)\ \operatorname{d}y$   \\
$f_{14}$   &
$((x)(y))$ &
$\operatorname{d}x\ \operatorname{d}y$     &
$\operatorname{d}x\ (\operatorname{d}y)$   &
$(\operatorname{d}x)\ \operatorname{d}y$   &
$((\operatorname{d}x)(\operatorname{d}y))$ \\
\hline
$f_{15}$ &
$((~))$  &
$(~)$    &
$(~)$    &
$(~)$    &
$(~)$    \\
\hline
\end{tabular}\end{quote}

Inquiry Driven Systems

Table 1. Sign Relation of Interpreter A

Table 1.  Sign Relation of Interpreter A
o---------------o---------------o---------------o
| Object        | Sign          | Interpretant  |
o---------------o---------------o---------------o
| A             | "A"           | "A"           |
| A             | "A"           | "i"           |
| A             | "i"           | "A"           |
| A             | "i"           | "i"           |
| B             | "B"           | "B"           |
| B             | "B"           | "u"           |
| B             | "u"           | "B"           |
| B             | "u"           | "u"           |
o---------------o---------------o---------------o

Table 1. Sign Relation of Interpreter A
Object	Sign	Interpretant
A	"A"	"A"
A	"A"	"i"
A	"i"	"A"
A	"i"	"i"
B	"B"	"B"
B	"B"	"u"
B	"u"	"B"
B	"u"	"u"

Table 2. Sign Relation of Interpreter B

Table 2.  Sign Relation of Interpreter B
o---------------o---------------o---------------o
| Object        | Sign          | Interpretant  |
o---------------o---------------o---------------o
| A             | "A"           | "A"           |
| A             | "A"           | "u"           |
| A             | "u"           | "A"           |
| A             | "u"           | "u"           |
| B             | "B"           | "B"           |
| B             | "B"           | "i"           |
| B             | "i"           | "B"           |
| B             | "i"           | "i"           |
o---------------o---------------o---------------o

Table 2. Sign Relation of Interpreter B
Object	Sign	Interpretant
A	"A"	"A"
A	"A"	"u"
A	"u"	"A"
A	"u"	"u"
B	"B"	"B"
B	"B"	"i"
B	"i"	"B"
B	"i"	"i"

Table 3. Semiotic Partition of Interpreter A

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

Table 3. Semiotic Partition of Interpreter A

"A"

"i"

"u"

"B"

Table 4. Semiotic Partition of Interpreter B

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

Table 4. Semiotic Partition of Interpreter B

"A"

"u"

"i"

"B"

Table 5. Alignments of Capacities

Table 5.  Alignments of Capacities
o-------------------o-----------------------------o
|      Formal       |          Formative          |
o-------------------o-----------------------------o
|     Objective     |        Instrumental         |
|      Passive      |           Active            |
o-------------------o--------------o--------------o
|     Afforded      |  Possessed   |  Exercised   |
o-------------------o--------------o--------------o

Table 6. Alignments of Capacities in Aristotle

Table 6.  Alignments of Capacities in Aristotle
o-------------------o-----------------------------o
|      Matter       |            Form             |
o-------------------o-----------------------------o
|   Potentiality    |          Actuality          |
|    Receptivity    |  Possession  |   Exercise   |
|       Life        |    Sleep     |    Waking    |
|        Wax        |         Impression          |
|        Axe        |    Edge      |   Cutting    |
|        Eye        |   Vision     |    Seeing    |
|       Body        |            Soul             |
o-------------------o-----------------------------o
|       Ship?       |           Sailor?           |
o-------------------o-----------------------------o

Table 7. Synthesis of Alignments

Table 7.  Synthesis of Alignments
o-------------------o-----------------------------o
|      Formal       |          Formative          |
o-------------------o-----------------------------o
|     Objective     |        Instrumental         |
|      Passive      |           Active            |
|     Afforded      |  Possessed   |  Exercised   |
|      To Hold      |   To Have    |    To Use    |
|    Receptivity    |  Possession  |   Exercise   |
|   Potentiality    |          Actuality          |
|      Matter       |            Form             |
o-------------------o-----------------------------o

Table 8. Boolean Product

Table 8.  Boolean Product
o---------o---------o---------o
|   %*%   %   %0%   |   %1%   |
o=========o=========o=========o
|   %0%   %   %0%   |   %0%   |
o---------o---------o---------o
|   %1%   %   %0%   |   %1%   |
o---------o---------o---------o

Table 9. Boolean Sum

Table 9.  Boolean Sum
o---------o---------o---------o
|   %+%   %   %0%   |   %1%   |
o=========o=========o=========o
|   %0%   %   %0%   |   %1%   |
o---------o---------o---------o
|   %1%   %   %1%   |   %0%   |
o---------o---------o---------o

Logical Tables

Table Templates

Table 1. Two Variable Template

u :

v :

1 1 0 0

1 0 1 0

f

f₀

f₁

f₂

f₃

f₄

f₅

f₆

f₇

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

f₀

f₁

f₂

f₃

f₄

f₅

f₆

f₇

f₀

f₁

f₂

f₃

f₄

f₅

f₆

f₇

f₀

f₁

f₂

f₃

f₄

f₅

f₆

f₇

f₈

f₉

f₁₀

f₁₁

f₁₂

f₁₃

f₁₄

f₁₅

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

f₈

f₉

f₁₀

f₁₁

f₁₂

f₁₃

f₁₄

f₁₅

f₈

f₉

f₁₀

f₁₁

f₁₂

f₁₃

f₁₄

f₁₅

f₈

f₉

f₁₀

f₁₁

f₁₂

f₁₃

f₁₄

f₁₅

Table 2. Two Variable Template

u :

v :

1100

1010

f

f₀

f₁

f₂

f₃

f₄

f₅

f₆

f₇

0000
0001
0010
0011
0100
0101
0110
0111

()

(u)(v)

(u) v

(u)

u (v)

(v)

(u, v)

(u v)

f₀

f₁

f₂

f₃

f₄

f₅

f₆

f₇

f₀

f₁

f₂

f₃

f₄

f₅

f₆

f₇

f₈

f₉

f₁₀

f₁₁

f₁₂

f₁₃

f₁₄

f₁₅

1000
1001
1010
1011
1100
1101
1110
1111

u v

((u, v))

v

(u (v))

u

((u) v)

((u)(v))

(())

f₈

f₉

f₁₀

f₁₁

f₁₂

f₁₃

f₁₄

f₁₅

f₈

f₉

f₁₀

f₁₁

f₁₂

f₁₃

f₁₄

f₁₅

Higher Order Propositions

**Table 7. Higher Order Propositions (n = 1)**
\ x	1 0	F	m	m	m	m	m	m	m	m	m	m	m	m	m	m	m	m
F \			00	01	02	03	04	05	06	07	08	09	10	11	12	13	14	15
F₀	0 0	0	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
F₁	0 1	(x)	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1
F₂	1 0	x	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1
F₃	1 1	1	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1

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

**Table 9. Higher Order Propositions (n = 2)**
x :	1100	f	m	m	m	m	m	m	m	m	m	m	m	m	m	m	m	m	m	m	m	m	m	m	m	m
y :	1010		0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23
f₀	0000	( )	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
f₁	0001	(x)(y)			1	1	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1
f₂	0010	(x) y					1	1	1	1	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1
f₃	0011	(x)									1	1	1	1	1	1	1	1	0	0	0	0	0	0	0	0
f₄	0100	x (y)																	1	1	1	1	1	1	1	1
f₅	0101	(y)
f₆	0110	(x, y)
f₇	0111	(x y)
f₈	1000	x y
f₉	1001	((x, y))
f₁₀	1010	y
f₁₁	1011	(x (y))
f₁₂	1100	x
f₁₃	1101	((x) y)
f₁₄	1110	((x)(y))
f₁₅	1111	(( ))

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

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

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

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

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

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

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

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

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

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

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

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

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

Zeroth Order Logic

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

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

Template Draft

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

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

The truth table of NOT p (also written as ~p or ¬p) is as follows:

**Logical Negation**
p	¬p
F	T
T	F

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

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

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

Within a system of classical logic, double negation, that is, the negation of the negation of a proposition p, is logically equivalent to the initial proposition p. Expressed in symbolic terms, ¬(¬p) ⇔ p.

Within a system of intuitionistic logic, however, ¬¬p is a weaker statement than p. On the other hand, the logical equivalence ¬¬¬p ⇔ ¬p remains valid.

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

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

Logical conjunction

Logical conjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both of its operands are true.

The truth table of p AND q (also written as p ∧ q, p & q, or p\(\cdot\)q) is as follows:

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

Logical disjunction

Logical disjunction is an operation on two logical values, typically the values of two propositions, that produces a value of false if and only if both of its operands are false.

The truth table of p OR q (also written as p ∨ q) is as follows:

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

Logical equality

Logical equality is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both operands are false or both operands are true.

The truth table of p EQ q (also written as p = q, p ↔ q, or p ≡ q) is as follows:

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

Exclusive disjunction

Exclusive disjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true just in case exactly one of its operands is true.

The truth table of p XOR q (also written as p + q, p ⊕ q, or p ≠ q) is as follows:

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

The following equivalents can then be deduced:

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

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

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

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

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

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

Logical implication

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

The truth table associated with the material conditional if p then q (symbolized as p → q) and the logical implication p implies q (symbolized as p ⇒ q) is as follows:

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

Logical NAND

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

The truth table of p NAND q (also written as p | q or p ↑ q) is as follows:

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

Logical NNOR

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

The truth table of p NNOR q (also written as p ⊥ q or p ↓ q) is as follows:

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

Relational Tables

Sign Relations

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

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

L_A = Sign Relation of Interpreter A
Object	Sign	Interpretant
A	"A"	"A"
A	"A"	"i"
A	"i"	"A"
A	"i"	"i"
B	"B"	"B"
B	"B"	"u"
B	"u"	"B"
B	"u"	"u"

L_B = Sign Relation of Interpreter B
Object	Sign	Interpretant
A	"A"	"A"
A	"A"	"u"
A	"u"	"A"
A	"u"	"u"
B	"B"	"B"
B	"B"	"i"
B	"i"	"B"
B	"i"	"i"

Triadic Relations

Algebraic Examples

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

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

Semiotic Examples

L_A = Sign Relation of Interpreter A
Object	Sign	Interpretant
A	"A"	"A"
A	"A"	"i"
A	"i"	"A"
A	"i"	"i"
B	"B"	"B"
B	"B"	"u"
B	"u"	"B"
B	"u"	"u"

L_B = Sign Relation of Interpreter B
Object	Sign	Interpretant
A	"A"	"A"
A	"A"	"u"
A	"u"	"A"
A	"u"	"u"
B	"B"	"B"
B	"B"	"i"
B	"i"	"B"
B	"i"	"i"

Dyadic Projections

L_OS	=	proj_OS(L)	=	{ (o, s) ∈ O × S : (o, s, i) ∈ L for some i ∈ I }
L_SO	=	proj_SO(L)	=	{ (s, o) ∈ S × O : (o, s, i) ∈ L for some i ∈ I }
L_IS	=	proj_IS(L)	=	{ (i, s) ∈ I × S : (o, s, i) ∈ L for some o ∈ O }
L_SI	=	proj_SI(L)	=	{ (s, i) ∈ S × I : (o, s, i) ∈ L for some o ∈ O }
L_OI	=	proj_OI(L)	=	{ (o, i) ∈ O × I : (o, s, i) ∈ L for some s ∈ S }
L_IO	=	proj_IO(L)	=	{ (i, o) ∈ I × O : (o, s, i) ∈ L for some s ∈ S }

Method 1 : Subtitles as Captions

*proj*_OS(L_A)
Object	Sign
A	"A"
A	"i"
B	"B"
B	"u"

*proj*_OS(L_B)
Object	Sign
A	"A"
A	"u"
B	"B"
B	"i"

*proj*_SI(L_A)
Sign	Interpretant
"A"	"A"
"A"	"i"
"i"	"A"
"i"	"i"
"B"	"B"
"B"	"u"
"u"	"B"
"u"	"u"

*proj*_SI(L_B)
Sign	Interpretant
"A"	"A"
"A"	"u"
"u"	"A"
"u"	"u"
"B"	"B"
"B"	"i"
"i"	"B"
"i"	"i"

*proj*_OI(L_A)
Object	Interpretant
A	"A"
A	"i"
B	"B"
B	"u"

*proj*_OI(L_B)
Object	Interpretant
A	"A"
A	"u"
B	"B"
B	"i"

Method 2 : Subtitles as Top Rows

proj_OS(L_A)

Object	Sign
A	"A"
A	"i"
B	"B"
B	"u"

proj_OS(L_B)

Object	Sign
A	"A"
A	"u"
B	"B"
B	"i"

proj_SI(L_A)

Sign	Interpretant
"A"	"A"
"A"	"i"
"i"	"A"
"i"	"i"
"B"	"B"
"B"	"u"
"u"	"B"
"u"	"u"

proj_SI(L_B)

Sign	Interpretant
"A"	"A"
"A"	"u"
"u"	"A"
"u"	"u"
"B"	"B"
"B"	"i"
"i"	"B"
"i"	"i"

proj_OI(L_A)

Object	Interpretant
A	"A"
A	"i"
B	"B"
B	"u"

proj_OI(L_B)

Object	Interpretant
A	"A"
A	"u"
B	"B"
B	"i"

Relation Reduction

Method 1 : Subtitles as Captions

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

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

proj_XY(L₀)
X	Y
0	0
0	1
1	0
1	1

proj_XZ(L₀)
X	Z
0	0
0	1
1	1
1	0

proj_YZ(L₀)
Y	Z
0	0
1	1
0	1
1	0

proj_XY(L₁)
X	Y
0	0
0	1
1	0
1	1

proj_XZ(L₁)
X	Z
0	1
0	0
1	0
1	1

proj_YZ(L₁)
Y	Z
0	1
1	0
0	0
1	1

proj_XY(L₀) = proj_XY(L₁)

proj_XZ(L₀) = proj_XZ(L₁)

proj_YZ(L₀) = proj_YZ(L₁)

L_A = Sign Relation of Interpreter A
Object	Sign	Interpretant
A	"A"	"A"
A	"A"	"i"
A	"i"	"A"
A	"i"	"i"
B	"B"	"B"
B	"B"	"u"
B	"u"	"B"
B	"u"	"u"

L_B = Sign Relation of Interpreter B
Object	Sign	Interpretant
A	"A"	"A"
A	"A"	"u"
A	"u"	"A"
A	"u"	"u"
B	"B"	"B"
B	"B"	"i"
B	"i"	"B"
B	"i"	"i"

proj_XY(L_A)
Object	Sign
A	"A"
A	"i"
B	"B"
B	"u"

proj_XZ(L_A)
Object	Interpretant
A	"A"
A	"i"
B	"B"
B	"u"

proj_YZ(L_A)
Sign	Interpretant
"A"	"A"
"A"	"i"
"i"	"A"
"i"	"i"
"B"	"B"
"B"	"u"
"u"	"B"
"u"	"u"

proj_XY(L_B)
Object	Sign
A	"A"
A	"u"
B	"B"
B	"i"

proj_XZ(L_B)
Object	Interpretant
A	"A"
A	"u"
B	"B"
B	"i"

proj_YZ(L_B)
Sign	Interpretant
"A"	"A"
"A"	"u"
"u"	"A"
"u"	"u"
"B"	"B"
"B"	"i"
"i"	"B"
"i"	"i"

proj_XY(L_A) ≠ proj_XY(L_B)

proj_XZ(L_A) ≠ proj_XZ(L_B)

proj_YZ(L_A) ≠ proj_YZ(L_B)

Method 2 : Subtitles as Top Rows

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

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

proj_XY(L₀)

X	Y
0	0
0	1
1	0
1	1

proj_XZ(L₀)

X	Z
0	0
0	1
1	1
1	0

proj_YZ(L₀)

Y	Z
0	0
1	1
0	1
1	0

proj_XY(L₁)

X	Y
0	0
0	1
1	0
1	1

proj_XZ(L₁)

X	Z
0	1
0	0
1	0
1	1

proj_YZ(L₁)

Y	Z
0	1
1	0
0	0
1	1

proj_XY(L₀) = proj_XY(L₁)

proj_XZ(L₀) = proj_XZ(L₁)

proj_YZ(L₀) = proj_YZ(L₁)

L_A = Sign Relation of Interpreter A
Object	Sign	Interpretant
A	"A"	"A"
A	"A"	"i"
A	"i"	"A"
A	"i"	"i"
B	"B"	"B"
B	"B"	"u"
B	"u"	"B"
B	"u"	"u"

L_B = Sign Relation of Interpreter B
Object	Sign	Interpretant
A	"A"	"A"
A	"A"	"u"
A	"u"	"A"
A	"u"	"u"
B	"B"	"B"
B	"B"	"i"
B	"i"	"B"
B	"i"	"i"

proj_XY(L_A)

Object	Sign
A	"A"
A	"i"
B	"B"
B	"u"

proj_XZ(L_A)

Object	Interpretant
A	"A"
A	"i"
B	"B"
B	"u"

proj_YZ(L_A)

Sign	Interpretant
"A"	"A"
"A"	"i"
"i"	"A"
"i"	"i"
"B"	"B"
"B"	"u"
"u"	"B"
"u"	"u"

proj_XY(L_B)

Object	Sign
A	"A"
A	"u"
B	"B"
B	"i"

proj_XZ(L_B)

Object	Interpretant
A	"A"
A	"u"
B	"B"
B	"i"

proj_YZ(L_B)

Sign	Interpretant
"A"	"A"
"A"	"u"
"u"	"A"
"u"	"u"
"B"	"B"
"B"	"i"
"i"	"B"
"i"	"i"

proj_XY(L_A) ≠ proj_XY(L_B)

proj_XZ(L_A) ≠ proj_XZ(L_B)

proj_YZ(L_A) ≠ proj_YZ(L_B)

Formatted Text Display

So in a triadic fact, say, the example

A gives B to C

we make no distinction in the ordinary logic of relations between the subject nominative, the direct object, and the indirect object. We say that the proposition has three logical subjects. We regard it as a mere affair of English grammar that there are six ways of expressing this:

A gives B to C	A benefits C with B
B enriches C at expense of A	C receives B from A
C thanks A for B	B leaves A for C

These six sentences express one and the same indivisible phenomenon. (C.S. Peirce, "The Categories Defended", MS 308 (1903), EP 2, 170-171).

Work Area

Binary Operations
x₀	x₁	²f₀	²f₁	²f₂	²f₃	²f₄	²f₅	²f₆	²f₇	²f₈	²f₉	²f₁₀	²f₁₁	²f₁₂	²f₁₃	²f₁₄	²f₁₅
0	0	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
1	0	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1
0	1	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1
1	1	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1

Draft 1

TRUTH TABLES FOR THE BOOLEAN OPERATIONS OF ARITY UP TO 2

Constants
	⁰f₀	⁰f₁
	0	1

Unary Operations
x₀	¹f₀	¹f₁	¹f₂	¹f₃
0	0	1	0	1
1	0	0	1	1

Binary Operations
x₀	x₁	²f₀	²f₁	²f₂	²f₃	²f₄	²f₅	²f₆	²f₇	²f₈	²f₉	²f₁₀	²f₁₁	²f₁₂	²f₁₃	²f₁₄	²f₁₅
0	0	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
1	0	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1
0	1	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1
1	1	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1

Draft 2

TRUTH TABLES FOR THE BOOLEAN OPERATIONS OF ARITY UP TO 2

Constants
	⁰f₀	⁰f₁
	0	1

Unary Operations
x₀	¹f₀	¹f₁	¹f₂	¹f₃
0	0	1	0	1
1	0	0	1	1

Binary Operations
x₀	x₁	²f₀	²f₁	²f₂	²f₃	²f₄	²f₅	²f₆	²f₇	²f₈	²f₉	²f₁₀	²f₁₁	²f₁₂	²f₁₃	²f₁₄	²f₁₅
0	0	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
1	0	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1
0	1	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1
1	1	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1

Inquiry and Analogy

Test Patterns

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

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

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

Table 10

**Table 10. Higher Order Propositions (n = 1)**
\(x\):	1 0	\(f\)	\(m_0\)	\(m_1\)	\(m_2\)	\(m_3\)	\(m_4\)	\(m_5\)	\(m_6\)	\(m_7\)	\(m_8\)	\(m_9\)	\(m_{10}\)	\(m_{11}\)	\(m_{12}\)	\(m_{13}\)	\(m_{14}\)	\(m_{15}\)
\(f_0\)	0 0	\(0\!\)	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
\(f_1\)	0 1	\((x)\!\)	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1
\(f_2\)	1 0	\(x\!\)	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1
\(f_3\)	1 1	\(1\!\)	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1

**Table 10. Higher Order Propositions (n = 1)**
\(x:\)	1 0	\(f\!\)	\(m_0\)	\(m_1\)	\(m_2\)	\(m_3\)	\(m_4\)	\(m_5\)	\(m_6\)	\(m_7\)	\(m_8\)	\(m_9\)	\(m_{10}\)	\(m_{11}\)	\(m_{12}\)	\(m_{13}\)	\(m_{14}\)	\(m_{15}\)
\(f_0\)	0 0	\(0\!\)	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
\(f_1\)	0 1	\((x)\!\)	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1
\(f_2\)	1 0	\(x\!\)	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1
\(f_3\)	1 1	\(1\!\)	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1

Table 11

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

Table 12

**Table 12. Higher Order Propositions (n = 2)**
\(x:\) \(y:\)	1100 1010	\(f\!\)	\(m_0\)	\(m_1\)	\(m_2\)	\(m_3\)	\(m_4\)	\(m_5\)	\(m_6\)	\(m_7\)	\(m_8\)	\(m_9\)	\(m_{10}\)	\(m_{11}\)	\(m_{12}\)	\(m_{13}\)	\(m_{14}\)	\(m_{15}\)	\(m_{16}\)	\(m_{17}\)	\(m_{18}\)	\(m_{19}\)	\(m_{20}\)	\(m_{21}\)	\(m_{22}\)	\(m_{23}\)
\(f_0\)	0000	\((~)\)	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
\(f_1\)	0001	\((x)(y)\!\)			1	1	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1
\(f_2\)	0010	\((x) y\!\)					1	1	1	1	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1
\(f_3\)	0011	\((x)\!\)									1	1	1	1	1	1	1	1	0	0	0	0	0	0	0	0
\(f_4\)	0100	\(x (y)\!\)																	1	1	1	1	1	1	1	1
\(f_5\)	0101	\((y)\!\)
\(f_6\)	0110	\((x, y)\!\)
\(f_7\)	0111	\((x y)\!\)
\(f_8\)	1000	\(x y\!\)
\(f_9\)	1001	\(((x, y))\!\)
\(f_{10}\)	1010	\(y\!\)
\(f_{11}\)	1011	\((x (y))\!\)
\(f_{12}\)	1100	\(x\!\)
\(f_{13}\)	1101	\(((x) y)\!\)
\(f_{14}\)	1110	\(((x)(y))\!\)
\(f_{15}\)	1111	\(((~))\!\)

**Table 12. Higher Order Propositions (n = 2)**
\(u:\) \(v:\)	1100 1010	\(f\!\)	\(m_0\)	\(m_1\)	\(m_2\)	\(m_3\)	\(m_4\)	\(m_5\)	\(m_6\)	\(m_7\)	\(m_8\)	\(m_9\)	\(m_{10}\)	\(m_{11}\)	\(m_{12}\)	\(m_{13}\)	\(m_{14}\)	\(m_{15}\)	\(m_{16}\)	\(m_{17}\)	\(m_{18}\)	\(m_{19}\)	\(m_{20}\)	\(m_{21}\)	\(m_{22}\)	\(m_{23}\)
\(f_0\)	0000	\((~)\)	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
\(f_1\)	0001	\((u)(v)\!\)	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1
\(f_2\)	0010	\((u) v\!\)	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1
\(f_3\)	0011	\((u)\!\)	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	0	0	0	0	0	0	0	0
\(f_4\)	0100	\(u (v)\!\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1
\(f_5\)	0101	\((v)\!\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(f_6\)	0110	\((u, v)\!\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(f_7\)	0111	\((u v)\!\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(f_8\)	1000	\(u v\!\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(f_9\)	1001	\(((u, v))\!\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(f_{10}\)	1010	\(v\!\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(f_{11}\)	1011	\((u (v))\!\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(f_{12}\)	1100	\(u\!\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(f_{13}\)	1101	\(((u) v)\!\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(f_{14}\)	1110	\(((u)(v))\!\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(f_{15}\)	1111	\(((~))\!\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0

Table 13

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

Table 14

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

Figure 15

Table 16

**Table 16. Syllogistic Premisses as Higher Order Indicator Functions**
\(\begin{array}{clcl} \mathrm{A} & \mathrm{Universal~Affirmative} & \mathrm{All}\ u\ \mathrm{is}\ v & \mathrm{Indicator~of}\ u (v) = 0 \\ \mathrm{E} & \mathrm{Universal~Negative} & \mathrm{All}\ u\ \mathrm{is}\ (v) & \mathrm{Indicator~of}\ u \cdot v = 0 \\ \mathrm{I} & \mathrm{Particular~Affirmative} & \mathrm{Some}\ u\ \mathrm{is}\ v & \mathrm{Indicator~of}\ u \cdot v = 1 \\ \mathrm{O} & \mathrm{Particular~Negative} & \mathrm{Some}\ u\ \mathrm{is}\ (v) & \mathrm{Indicator~of}\ u (v) = 1 \\ \end{array}\)

Table 17

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

Table 18

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

Table 19

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

@@ Line 336: / Line 336: @@
 ====New Versions====
+<br>
+{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|+ '''Table 1.  Propositional Forms on Two Variables'''
+|- style="background:paleturquoise"
+! style="width:15%" | L<sub>1</sub>
+! style="width:15%" | L<sub>2</sub>
+! style="width:15%" | L<sub>3</sub>
+! style="width:15%" | L<sub>4</sub>
+! style="width:25%" | L<sub>5</sub>
+! style="width:15%" | L<sub>6</sub>
+|- style="background:paleturquoise"
+| &nbsp;
+| align="right" | x :
+| 1 1 0 0
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|- style="background:paleturquoise"
+| &nbsp;
+| align="right" | y :
+| 1 0 1 0
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|-
+| f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || (&nbsp;) || false || 0
+|-
+| f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || &not;x &and; &not;y
+|-
+| f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || &not;x &and; y
+|-
+| f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || &not;x
+|-
+| f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x &and; &not;y
+|-
+| f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || &not;y
+|-
+| f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x &ne; y
+|-
+| f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x&nbsp;y) || not both x and y || &not;x &or; &not;y
+|-
+| f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x&nbsp;y || x and y || x &and; y
+|-
+| f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y
+|-
+| f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y
+|-
+| f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x &rarr; y
+|-
+| f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x
+|-
+| f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x &larr; y
+|-
+| f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y  || x &or; y
+|-
+| f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || ((&nbsp;)) || true || 1
+|}
+<br>
+{| align="center" border="1" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"
+|+ '''Table 14.  Differential Propositions'''
+|- style="background:ghostwhite"
+| &nbsp;
+| align="right" | A :
+| 1 1 0 0
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|- style="background:ghostwhite"
+| &nbsp;
+| align="right" | dA :
+| 1 0 1 0
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|-
+| f<sub>0</sub>
+| g<sub>0</sub>
+| 0 0 0 0
+| (&nbsp;)
+| False
+| 0
+|-
+|
+{|
+|
+&nbsp;<br>
+&nbsp;<br>
+&nbsp;<br>
+&nbsp;
+|}
+|
+{|
+|
+g<sub>1</sub><br>
+g<sub>2</sub><br>
+g<sub>4</sub><br>
+g<sub>8</sub>
+|}
+|
+{|
+|
+0 0 1<br>
+0 1 0<br>
+1 0 0<br>
+0 0 0
+|}
+|
+{|
+|
+(A)(dA)<br>
+(A) dA <br>
+A (dA)<br>
+A dA
+|}
+|
+{|
+|
+Neither A nor dA<br>
+Not A but dA<br>
+A but not dA<br>
+A and dA
+|}
+|
+{|
+|
+&not;A &and; &not;dA<br>
+&not;A &and; dA<br>
+A &and; &not;dA<br>
+A &and; dA
+|}
+|-
+|
+{|
+|
+f<sub>1</sub><br>
+f<sub>2</sub>
+|}
+|
+{|
+|
+g<sub>3</sub><br>
+g<sub>12</sub>
+|}
+|
+{|
+|
+0 1 1<br>
+1 0 0
+|}
+|
+{|
+|
+(A)<br>
+A
+|}
+|
+{|
+|
+Not A<br>
+A
+|}
+|
+{|
+|
+&not;A<br>
+A
+|}
+|-
+|
+{|
+|
+&nbsp;<br>
+&nbsp;
+|}
+|
+{|
+|
+g<sub>6</sub><br>
+g<sub>9</sub>
+|}
+|
+{|
+|
+1 1 0<br>
+0 0 1
+|}
+|
+{|
+|
+(A, dA)<br>
+((A, dA))
+|}
+|
+{|
+|
+A not equal to dA<br>
+A equal to dA
+|}
+|
+{|
+|
+A &ne; dA<br>
+A = dA
+|}
+|-
+|
+{|
+|
+&nbsp;<br>
+&nbsp;
+|}
+|
+{|
+|
+g<sub>5</sub><br>
+g<sub>10</sub>
+|}
+|
+{|
+|
+1 0 1<br>
+0 1 0
+|}
+|
+{|
+|
+(dA)<br>
+dA
+|}
+|
+{|
+|
+Not dA<br>
+dA
+|}
+|
+{|
+|
+&not;dA<br>
+dA
+|}
+|-
+|
+{|
+|
+&nbsp;<br>
+&nbsp;<br>
+&nbsp;<br>
+&nbsp;
+|}
+|
+{|
+|
+g<sub>7</sub><br>
+g<sub>11</sub><br>
+g<sub>13</sub><br>
+g<sub>14</sub>
+|}
+|
+{|
+|
+1 1 1<br>
+0 1 1<br>
+1 0 1<br>
+1 1 0
+|}
+|
+{|
+|
+(A dA)<br>
+(A (dA))<br>
+((A) dA)<br>
+((A)(dA))
+|}
+|
+{|
+|
+Not both A and dA<br>
+Not A without dA<br>
+Not dA without A<br>
+A or dA
+|}
+|
+{|
+|
+&not;A &or; &not;dA<br>
+A &rarr; dA<br>
+A &larr; dA<br>
+A &or; dA
+|}
+|-
+| f<sub>3</sub>
+| g<sub>15</sub>
+| 1 1 1 1
+| ((&nbsp;))
+| True
+| 1
+|}
+<br>
 ====Old Versions====

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