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
o----------o----------o----------o----------o----------o
| % | | | |
| · % T_00 | T_01 | T_10 | T_11 |
| % | | | |
o==========o==========o==========o==========o==========o
| % | | | |
| T_00 % T_00 | T_01 | T_10 | T_11 |
| % | | | |
o----------o----------o----------o----------o----------o
| % | | | |
| T_01 % T_01 | T_00 | T_11 | T_10 |
| % | | | |
o----------o----------o----------o----------o----------o
| % | | | |
| T_10 % T_10 | T_11 | T_00 | T_01 |
| % | | | |
o----------o----------o----------o----------o----------o
| % | | | |
| T_11 % T_11 | T_10 | T_01 | T_00 |
| % | | | |
o----------o----------o----------o----------o----------o
o---------o---------o---------o---------o---------o
| % | | | |
| · % e | f | g | h |
| % | | | |
o=========o=========o=========o=========o=========o
| % | | | |
| e % e | f | g | h |
| % | | | |
o---------o---------o---------o---------o---------o
| % | | | |
| f % f | e | h | g |
| % | | | |
o---------o---------o---------o---------o---------o
| % | | | |
| g % g | h | e | f |
| % | | | |
o---------o---------o---------o---------o---------o
| % | | | |
| h % h | g | f | e |
| % | | | |
o---------o---------o---------o---------o---------o
Wiki Tables : New Versions
Propositional Forms on Two Variables
Table A1. Propositional Forms on Two Variables
| L1
|
L2
|
L3
|
L4
|
L5
|
L6
|
|
|
x :
|
1 1 0 0
|
|
|
|
|
|
y :
|
1 0 1 0
|
|
|
|
| f0
|
f0000
|
0 0 0 0
|
( )
|
false
|
0
|
| f1
|
f0001
|
0 0 0 1
|
(x)(y)
|
neither x nor y
|
¬x ∧ ¬y
|
| f2
|
f0010
|
0 0 1 0
|
(x) y
|
y and not x
|
¬x ∧ y
|
| f3
|
f0011
|
0 0 1 1
|
(x)
|
not x
|
¬x
|
| f4
|
f0100
|
0 1 0 0
|
x (y)
|
x and not y
|
x ∧ ¬y
|
| f5
|
f0101
|
0 1 0 1
|
(y)
|
not y
|
¬y
|
| f6
|
f0110
|
0 1 1 0
|
(x, y)
|
x not equal to y
|
x ≠ y
|
| f7
|
f0111
|
0 1 1 1
|
(x y)
|
not both x and y
|
¬x ∨ ¬y
|
| f8
|
f1000
|
1 0 0 0
|
x y
|
x and y
|
x ∧ y
|
| f9
|
f1001
|
1 0 0 1
|
((x, y))
|
x equal to y
|
x = y
|
| f10
|
f1010
|
1 0 1 0
|
y
|
y
|
y
|
| f11
|
f1011
|
1 0 1 1
|
(x (y))
|
not x without y
|
x ⇒ y
|
| f12
|
f1100
|
1 1 0 0
|
x
|
x
|
x
|
| f13
|
f1101
|
1 1 0 1
|
((x) y)
|
not y without x
|
x ⇐ y
|
| f14
|
f1110
|
1 1 1 0
|
((x)(y))
|
x or y
|
x ∨ y
|
| f15
|
f1111
|
1 1 1 1
|
(( ))
|
true |
1
|
Table A2. Propositional Forms on Two Variables
| L1
|
L2
|
L3
|
L4
|
L5
|
L6
|
|
|
x :
|
1 1 0 0
|
|
|
|
|
|
y :
|
1 0 1 0
|
|
|
|
| f0
|
f0000
|
0 0 0 0
|
( )
|
false
|
0
|
|
|
|
|
0 0 0 1
0 0 1 0
0 1 0 0
1 0 0 0
|
|
|
|
neither x nor y
not x but y
x but not y
x and y
|
|
|
¬x ∧ ¬y
¬x ∧ y
x ∧ ¬y
x ∧ y
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
x not equal to y
x equal to y
|
|
|
|
|
|
|
|
|
|
|
|
|
|
0 1 1 1
1 0 1 1
1 1 0 1
1 1 1 0
|
|
|
(x y)
(x (y))
((x) y)
((x)(y))
|
|
|
not both x and y
not x without y
not y without x
x or y
|
|
|
¬x ∨ ¬y
x ⇒ y
x ⇐ y
x ∨ y
|
|
| f15
|
f1111
|
1 1 1 1
|
(( ))
|
true
|
1
|
Differential Propositions
Table 14. Differential Propositions
|
|
A :
|
1 1 0 0
|
|
|
|
|
|
dA :
|
1 0 1 0
|
|
|
|
| f0
|
g0
|
0 0 0 0
|
( )
|
False
|
0
|
|
|
|
|
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
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
A not equal to dA
A equal to dA
|
|
|
|
|
|
|
|
|
|
|
|
|
|
0 1 1 1
1 0 1 1
1 1 0 1
1 1 1 0
|
|
|
(A dA)
(A (dA))
((A) dA)
((A)(dA))
|
|
|
Not both A and dA
Not A without dA
Not dA without A
A or dA
|
|
|
¬A ∨ ¬dA
A ⇒ dA
A ⇐ dA
A ∨ dA
|
|
| f3
|
g15
|
1 1 1 1
|
(( ))
|
True
|
1
|
Wiki Tables : Old Versions
Propositional Forms on Two Variables
Table 1. Propositional Forms on Two Variables
| L1
|
L2
|
L3
|
L4
|
L5
|
L6
|
|
|
x :
|
1 1 0 0
|
|
|
|
|
|
y :
|
1 0 1 0
|
|
|
|
| f0 |
f0000 |
0 0 0 0 |
( ) |
false |
0
|
| f1 |
f0001 |
0 0 0 1 |
(x)(y) |
neither x nor y |
¬x ∧ ¬y
|
| f2 |
f0010 |
0 0 1 0 |
(x) y |
y and not x |
¬x ∧ y
|
| f3 |
f0011 |
0 0 1 1 |
(x) |
not x |
¬x
|
| f4 |
f0100 |
0 1 0 0 |
x (y) |
x and not y |
x ∧ ¬y
|
| f5 |
f0101 |
0 1 0 1 |
(y) |
not y |
¬y
|
| f6 |
f0110 |
0 1 1 0 |
(x, y) |
x not equal to y |
x ≠ y
|
| f7 |
f0111 |
0 1 1 1 |
(x y) |
not both x and y |
¬x ∨ ¬y
|
| f8 |
f1000 |
1 0 0 0 |
x y |
x and y |
x ∧ y
|
| f9 |
f1001 |
1 0 0 1 |
((x, y)) |
x equal to y |
x = y
|
| f10 |
f1010 |
1 0 1 0 |
y |
y |
y
|
| f11 |
f1011 |
1 0 1 1 |
(x (y)) |
not x without y |
x → y
|
| f12 |
f1100 |
1 1 0 0 |
x |
x |
x
|
| f13 |
f1101 |
1 1 0 1 |
((x) y) |
not y without x |
x ← y
|
| f14 |
f1110 |
1 1 1 0 |
((x)(y)) |
x or y |
x ∨ y
|
| f15 |
f1111 |
1 1 1 1 |
(( )) |
true |
1
|
Differential Propositions
Table 14. Differential Propositions
|
|
A :
|
1 1 0 0
|
|
|
|
|
|
dA :
|
1 0 1 0
|
|
|
|
| f0
|
g0
|
0 0 0 0
|
( )
|
False
|
0
|
|
|
|
|
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
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
A not equal to dA
A equal to dA
|
|
|
|
|
|
|
|
|
|
|
|
|
|
0 1 1 1
1 0 1 1
1 1 0 1
1 1 1 0
|
|
|
(A dA)
(A (dA))
((A) dA)
((A)(dA))
|
|
|
Not both A and dA
Not A without dA
Not dA without A
A or dA
|
|
|
¬A ∨ ¬dA
A → dA
A ← dA
A ∨ dA
|
|
| f3
|
g15
|
1 1 1 1
|
(( ))
|
True
|
1
|
Wiki TeX Tables
\(\text{Table A1.}~~\text{Propositional Forms on Two Variables}\)
|
\(\mathcal{L}_1\)
\(\text{Decimal}\)
|
\(\mathcal{L}_2\)
\(\text{Binary}\)
|
\(\mathcal{L}_3\)
\(\text{Vector}\)
|
\(\mathcal{L}_4\)
\(\text{Cactus}\)
|
\(\mathcal{L}_5\)
\(\text{English}\)
|
\(\mathcal{L}_6\)
\(\text{Ordinary}\)
|
|
|
\(x\colon\!\)
|
\(1~1~0~0\!\)
|
|
|
|
|
|
\(y\colon\!\)
|
\(1~0~1~0\!\)
|
|
|
|
| \(f_{0}\!\)
|
\(f_{0000}\!\)
|
\(0~0~0~0\!\)
|
\((~)\!\)
|
\(\text{false}\!\)
|
\(0\!\)
|
| \(f_{1}\!\)
|
\(f_{0001}\!\)
|
\(0~0~0~1\!\)
|
\((x)(y)\!\)
|
\(\text{neither}~ x ~\text{nor}~ y\!\)
|
\(\lnot x \land \lnot y\!\)
|
| \(f_{2}\!\)
|
\(f_{0010}\!\)
|
\(0~0~1~0\!\)
|
\((x)~y\!\)
|
\(y ~\text{without}~ x\!\)
|
\(\lnot x \land y\!\)
|
| \(f_{3}\!\)
|
\(f_{0011}\!\)
|
\(0~0~1~1\!\)
|
\((x)\!\)
|
\(\text{not}~ x\!\)
|
\(\lnot x\!\)
|
| \(f_{4}\!\)
|
\(f_{0100}\!\)
|
\(0~1~0~0\!\)
|
\(x~(y)\!\)
|
\(x ~\text{without}~ y\!\)
|
\(x \land \lnot y\!\)
|
| \(f_{5}\!\)
|
\(f_{0101}\!\)
|
\(0~1~0~1\!\)
|
\((y)\!\)
|
\(\text{not}~ y\!\)
|
\(\lnot y\!\)
|
| \(f_{6}\!\)
|
\(f_{0110}\!\)
|
\(0~1~1~0\!\)
|
\((x,~y)\!\)
|
\(x ~\text{not equal to}~ y\!\)
|
\(x \ne y\!\)
|
| \(f_{7}\!\)
|
\(f_{0111}\!\)
|
\(0~1~1~1\!\)
|
\((x~y)\!\)
|
\(\text{not both}~ x ~\text{and}~ y\!\)
|
\(\lnot x \lor \lnot y\!\)
|
| \(f_{8}\!\)
|
\(f_{1000}\!\)
|
\(1~0~0~0\!\)
|
\(x~y\!\)
|
\(x ~\text{and}~ y\!\)
|
\(x \land y\!\)
|
| \(f_{9}\!\)
|
\(f_{1001}\!\)
|
\(1~0~0~1\!\)
|
\(((x,~y))\!\)
|
\(x ~\text{equal to}~ y\!\)
|
\(x = y\!\)
|
| \(f_{10}\!\)
|
\(f_{1010}\!\)
|
\(1~0~1~0\!\)
|
\(y\!\)
|
\(y\!\)
|
\(y\!\)
|
| \(f_{11}\!\)
|
\(f_{1011}\!\)
|
\(1~0~1~1\!\)
|
\((x~(y))\!\)
|
\(\text{not}~ x ~\text{without}~ y\!\)
|
\(x \Rightarrow y\!\)
|
| \(f_{12}\!\)
|
\(f_{1100}\!\)
|
\(1~1~0~0\!\)
|
\(x\!\)
|
\(x\!\)
|
\(x\!\)
|
| \(f_{13}\!\)
|
\(f_{1101}\!\)
|
\(1~1~0~1\!\)
|
\(((x)~y)\!\)
|
\(\text{not}~ y ~\text{without}~ x\!\)
|
\(x \Leftarrow y\!\)
|
| \(f_{14}\!\)
|
\(f_{1110}\!\)
|
\(1~1~1~0\!\)
|
\(((x)(y))\!\)
|
\(x ~\text{or}~ y\!\)
|
\(x \lor y\!\)
|
| \(f_{15}\!\)
|
\(f_{1111}\!\)
|
\(1~1~1~1\!\)
|
\(((~))\!\)
|
\(\text{true}\!\)
|
\(1\!\)
|
\(\text{Table A1.}~~\text{Propositional Forms on Two Variables}\)
|
\(\mathcal{L}_1\)
\(\text{Decimal}\)
|
\(\mathcal{L}_2\)
\(\text{Binary}\)
|
\(\mathcal{L}_3\)
\(\text{Vector}\)
|
\(\mathcal{L}_4\)
\(\text{Cactus}\)
|
\(\mathcal{L}_5\)
\(\text{English}\)
|
\(\mathcal{L}_6\)
\(\text{Ordinary}\)
|
|
|
\(x\colon\!\)
|
\(1~1~0~0\!\)
|
|
|
|
|
|
\(y\colon\!\)
|
\(1~0~1~0\!\)
|
|
|
|
|
\(\begin{matrix}
f_0
\'"`UNIQ-MathJax1-QINU`"'
'''Generalized''' or '''n-ary''' XOR is true when the number of 1-bits is odd.
'"`UNIQ--pre-00000012-QINU`"'
'"`UNIQ--pre-00000013-QINU`"'
'"`UNIQ--pre-00000014-QINU`"'
'"`UNIQ-MathJax2-QINU`"'
===='"`UNIQ--h-32--QINU`"'[[Logical implication]]====
The '''material conditional''' and '''logical implication''' are both associated with an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if the first operand is true and the second operand is false.
The [[truth table]] associated with the material conditional '''if p then q''' (symbolized as '''p → q''') and the logical implication '''p implies q''' (symbolized as '''p ⇒ q''') is as follows:
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%"
|+ '''Logical Implication'''
|- style="background:aliceblue"
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | p ⇒ q
|-
| F || F || T
|-
| F || T || T
|-
| T || F || F
|-
| T || T || T
|}
<br>
===='"`UNIQ--h-33--QINU`"'[[Logical NAND]]====
The '''NAND operation''' is a [[logical operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if both of its operands are true. In other words, it produces a value of ''true'' if and only if at least one of its operands is false.
The [[truth table]] of '''p NAND q''' (also written as '''p | q''' or '''p ↑ q''') is as follows:
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%"
|+ '''Logical NAND'''
|- style="background:aliceblue"
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | p ↑ q
|-
| F || F || T
|-
| F || T || T
|-
| T || F || T
|-
| T || T || F
|}
<br>
===='"`UNIQ--h-34--QINU`"'[[Logical NNOR]]====
The '''NNOR operation''' is a [[logical operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' if and only if both of its operands are false. In other words, it produces a value of ''false'' if and only if at least one of its operands is true.
The [[truth table]] of '''p NNOR q''' (also written as '''p ⊥ q''' or '''p ↓ q''') is as follows:
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%"
|+ '''Logical NOR'''
|- style="background:aliceblue"
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | p ↓ q
|-
| F || F || T
|-
| F || T || F
|-
| T || F || F
|-
| T || T || F
|}
<br>
=='"`UNIQ--h-35--QINU`"'Relational Tables==
==='"`UNIQ--h-36--QINU`"'Sign Relations===
{| cellpadding="4"
| width="20px" |
| align="center" | '''O''' || = || Object Domain
|-
| width="20px" |
| align="center" | '''S''' || = || Sign Domain
|-
| width="20px" |
| align="center" | '''I''' || = || Interpretant Domain
|}
<br>
{| cellpadding="4"
| width="20px" |
| align="center" | '''O'''
| =
| {Ann, Bob}
| =
| {A, B}
|-
| width="20px" |
| align="center" | '''S'''
| =
| {"Ann", "Bob", "I", "You"}
| =
| {"A", "B", "i", "u"}
|-
| width="20px" |
| align="center" | '''I'''
| =
| {"Ann", "Bob", "I", "You"}
| =
| {"A", "B", "i", "u"}
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
|+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A
|- style="background:paleturquoise"
! style="width:20%" | Object
! style="width:20%" | Sign
! style="width:20%" | Interpretant
|-
| '''A''' || '''"A"''' || '''"A"'''
|-
| '''A''' || '''"A"''' || '''"i"'''
|-
| '''A''' || '''"i"''' || '''"A"'''
|-
| '''A''' || '''"i"''' || '''"i"'''
|-
| '''B''' || '''"B"''' || '''"B"'''
|-
| '''B''' || '''"B"''' || '''"u"'''
|-
| '''B''' || '''"u"''' || '''"B"'''
|-
| '''B''' || '''"u"''' || '''"u"'''
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
|+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B
|- style="background:paleturquoise"
! style="width:20%" | Object
! style="width:20%" | Sign
! style="width:20%" | Interpretant
|-
| '''A''' || '''"A"''' || '''"A"'''
|-
| '''A''' || '''"A"''' || '''"u"'''
|-
| '''A''' || '''"u"''' || '''"A"'''
|-
| '''A''' || '''"u"''' || '''"u"'''
|-
| '''B''' || '''"B"''' || '''"B"'''
|-
| '''B''' || '''"B"''' || '''"i"'''
|-
| '''B''' || '''"i"''' || '''"B"'''
|-
| '''B''' || '''"i"''' || '''"i"'''
|}
<br>
==='"`UNIQ--h-37--QINU`"'Triadic Relations===
===='"`UNIQ--h-38--QINU`"'Algebraic Examples====
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
|+ '''L'''<sub>0</sub> = {(''x'', ''y'', ''z'') ∈ '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 0}
|- style="background:paleturquoise"
! X !! Y !! Z
|-
| '''0''' || '''0''' || '''0'''
|-
| '''0''' || '''1''' || '''1'''
|-
| '''1''' || '''0''' || '''1'''
|-
| '''1''' || '''1''' || '''0'''
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
|+ '''L'''<sub>1</sub> = {(''x'', ''y'', ''z'') ∈ '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 1}
|- style="background:paleturquoise"
! X !! Y !! Z
|-
| '''0''' || '''0''' || '''1'''
|-
| '''0''' || '''1''' || '''0'''
|-
| '''1''' || '''0''' || '''0'''
|-
| '''1''' || '''1''' || '''1'''
|}
<br>
===='"`UNIQ--h-39--QINU`"'Semiotic Examples====
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
|+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A
|- style="background:paleturquoise"
! style="width:20%" | Object
! style="width:20%" | Sign
! style="width:20%" | Interpretant
|-
| '''A''' || '''"A"''' || '''"A"'''
|-
| '''A''' || '''"A"''' || '''"i"'''
|-
| '''A''' || '''"i"''' || '''"A"'''
|-
| '''A''' || '''"i"''' || '''"i"'''
|-
| '''B''' || '''"B"''' || '''"B"'''
|-
| '''B''' || '''"B"''' || '''"u"'''
|-
| '''B''' || '''"u"''' || '''"B"'''
|-
| '''B''' || '''"u"''' || '''"u"'''
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
|+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B
|- style="background:paleturquoise"
! style="width:20%" | Object
! style="width:20%" | Sign
! style="width:20%" | Interpretant
|-
| '''A''' || '''"A"''' || '''"A"'''
|-
| '''A''' || '''"A"''' || '''"u"'''
|-
| '''A''' || '''"u"''' || '''"A"'''
|-
| '''A''' || '''"u"''' || '''"u"'''
|-
| '''B''' || '''"B"''' || '''"B"'''
|-
| '''B''' || '''"B"''' || '''"i"'''
|-
| '''B''' || '''"i"''' || '''"B"'''
|-
| '''B''' || '''"i"''' || '''"i"'''
|}
<br>
==='"`UNIQ--h-40--QINU`"'Dyadic Projections===
{| cellpadding="4"
| width="20px" |
| '''L'''<sub>OS</sub>
| =
| ''proj''<sub>OS</sub>('''L''')
| =
| { (''o'', ''s'') ∈ '''O''' × '''S''' : (''o'', ''s'', ''i'') ∈ '''L''' for some ''i'' ∈ '''I''' }
|-
| width="20px" |
| '''L'''<sub>SO</sub>
| =
| ''proj''<sub>SO</sub>('''L''')
| =
| { (''s'', ''o'') ∈ '''S''' × '''O''' : (''o'', ''s'', ''i'') ∈ '''L''' for some ''i'' ∈ '''I''' }
|-
| width="20px" |
| '''L'''<sub>IS</sub>
| =
| ''proj''<sub>IS</sub>('''L''')
| =
| { (''i'', ''s'') ∈ '''I''' × '''S''' : (''o'', ''s'', ''i'') ∈ '''L''' for some ''o'' ∈ '''O''' }
|-
| width="20px" |
| '''L'''<sub>SI</sub>
| =
| ''proj''<sub>SI</sub>('''L''')
| =
| { (''s'', ''i'') ∈ '''S''' × '''I''' : (''o'', ''s'', ''i'') ∈ '''L''' for some ''o'' ∈ '''O''' }
|-
| width="20px" |
| '''L'''<sub>OI</sub>
| =
| ''proj''<sub>OI</sub>('''L''')
| =
| { (''o'', ''i'') ∈ '''O''' × '''I''' : (''o'', ''s'', ''i'') ∈ '''L''' for some ''s'' ∈ '''S''' }
|-
| width="20px" |
| '''L'''<sub>IO</sub>
| =
| ''proj''<sub>IO</sub>('''L''')
| =
| { (''i'', ''o'') ∈ '''I''' × '''O''' : (''o'', ''s'', ''i'') ∈ '''L''' for some ''s'' ∈ '''S''' }
|}
<br>
===='"`UNIQ--h-41--QINU`"'Method 1 : Subtitles as Captions====
{| align="center" style="width:90%"
|
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ ''proj''<sub>OS</sub>('''L'''<sub>A</sub>)
|- style="background:paleturquoise"
! style="width:50%" | Object
! style="width:50%" | Sign
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"i"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"u"'''
|}
|
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ ''proj''<sub>OS</sub>('''L'''<sub>B</sub>)
|- style="background:paleturquoise"
! style="width:50%" | Object
! style="width:50%" | Sign
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"u"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"i"'''
|}
|}
<br>
{| align="center" style="width:90%"
|
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ ''proj''<sub>SI</sub>('''L'''<sub>A</sub>)
|- style="background:paleturquoise"
! style="width:50%" | Sign
! style="width:50%" | Interpretant
|-
| '''"A"''' || '''"A"'''
|-
| '''"A"''' || '''"i"'''
|-
| '''"i"''' || '''"A"'''
|-
| '''"i"''' || '''"i"'''
|-
| '''"B"''' || '''"B"'''
|-
| '''"B"''' || '''"u"'''
|-
| '''"u"''' || '''"B"'''
|-
| '''"u"''' || '''"u"'''
|}
|
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ ''proj''<sub>SI</sub>('''L'''<sub>B</sub>)
|- style="background:paleturquoise"
! style="width:50%" | Sign
! style="width:50%" | Interpretant
|-
| '''"A"''' || '''"A"'''
|-
| '''"A"''' || '''"u"'''
|-
| '''"u"''' || '''"A"'''
|-
| '''"u"''' || '''"u"'''
|-
| '''"B"''' || '''"B"'''
|-
| '''"B"''' || '''"i"'''
|-
| '''"i"''' || '''"B"'''
|-
| '''"i"''' || '''"i"'''
|}
|}
<br>
{| align="center" style="width:90%"
|
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ ''proj''<sub>OI</sub>('''L'''<sub>A</sub>)
|- style="background:paleturquoise"
! style="width:50%" | Object
! style="width:50%" | Interpretant
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"i"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"u"'''
|}
|
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ ''proj''<sub>OI</sub>('''L'''<sub>B</sub>)
|- style="background:paleturquoise"
! style="width:50%" | Object
! style="width:50%" | Interpretant
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"u"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"i"'''
|}
|}
<br>
===='"`UNIQ--h-42--QINU`"'Method 2 : Subtitles as Top Rows====
{| align="center" style="width:90%"
| align="center" style="width:45%" | ''proj''<sub>OS</sub>('''L'''<sub>A</sub>)
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! style="width:50%" | Object
! style="width:50%" | Sign
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"i"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"u"'''
|}
| align="center" style="width:45%" | ''proj''<sub>OS</sub>('''L'''<sub>B</sub>)
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! style="width:50%" | Object
! style="width:50%" | Sign
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"u"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"i"'''
|}
|}
<br>
{| align="center" style="width:90%"
| align="center" style="width:45%" | ''proj''<sub>SI</sub>('''L'''<sub>A</sub>)
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! style="width:50%" | Sign
! style="width:50%" | Interpretant
|-
| '''"A"''' || '''"A"'''
|-
| '''"A"''' || '''"i"'''
|-
| '''"i"''' || '''"A"'''
|-
| '''"i"''' || '''"i"'''
|-
| '''"B"''' || '''"B"'''
|-
| '''"B"''' || '''"u"'''
|-
| '''"u"''' || '''"B"'''
|-
| '''"u"''' || '''"u"'''
|}
| align="center" style="width:45%" | ''proj''<sub>SI</sub>('''L'''<sub>B</sub>)
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! style="width:50%" | Sign
! style="width:50%" | Interpretant
|-
| '''"A"''' || '''"A"'''
|-
| '''"A"''' || '''"u"'''
|-
| '''"u"''' || '''"A"'''
|-
| '''"u"''' || '''"u"'''
|-
| '''"B"''' || '''"B"'''
|-
| '''"B"''' || '''"i"'''
|-
| '''"i"''' || '''"B"'''
|-
| '''"i"''' || '''"i"'''
|}
|}
<br>
{| align="center" style="width:90%"
| align="center" style="width:45%" | ''proj''<sub>OI</sub>('''L'''<sub>A</sub>)
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! style="width:50%" | Object
! style="width:50%" | Interpretant
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"i"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"u"'''
|}
| align="center" style="width:45%" | ''proj''<sub>OI</sub>('''L'''<sub>B</sub>)
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! style="width:50%" | Object
! style="width:50%" | Interpretant
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"u"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"i"'''
|}
|}
<br>
==='"`UNIQ--h-43--QINU`"'Relation Reduction===
===='"`UNIQ--h-44--QINU`"'Method 1 : Subtitles as Captions====
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
|+ '''L'''<sub>0</sub> = {(''x'', ''y'', ''z'') ∈ '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 0}
|- style="background:paleturquoise"
! X !! Y !! Z
|-
| '''0''' || '''0''' || '''0'''
|-
| '''0''' || '''1''' || '''1'''
|-
| '''1''' || '''0''' || '''1'''
|-
| '''1''' || '''1''' || '''0'''
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
|+ '''L'''<sub>1</sub> = {(''x'', ''y'', ''z'') ∈ '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 1}
|- style="background:paleturquoise"
! X !! Y !! Z
|-
| '''0''' || '''0''' || '''1'''
|-
| '''0''' || '''1''' || '''0'''
|-
| '''1''' || '''0''' || '''0'''
|-
| '''1''' || '''1''' || '''1'''
|}
<br>
{| align="center" style="width:90%"
|
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ proj<sub>''XY''</sub>('''L'''<sub>0</sub>)
|- style="background:paleturquoise"
! X !! Y
|-
| '''0''' || '''0'''
|-
| '''0''' || '''1'''
|-
| '''1''' || '''0'''
|-
| '''1''' || '''1'''
|}
|
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ proj<sub>''XZ''</sub>('''L'''<sub>0</sub>)
|- style="background:paleturquoise"
! X !! Z
|-
| '''0''' || '''0'''
|-
| '''0''' || '''1'''
|-
| '''1''' || '''1'''
|-
| '''1''' || '''0'''
|}
|
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ proj<sub>''YZ''</sub>('''L'''<sub>0</sub>)
|- style="background:paleturquoise"
! Y !! Z
|-
| '''0''' || '''0'''
|-
| '''1''' || '''1'''
|-
| '''0''' || '''1'''
|-
| '''1''' || '''0'''
|}
|}
<br>
{| align="center" style="width:90%"
|
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ proj<sub>''XY''</sub>('''L'''<sub>1</sub>)
|- style="background:paleturquoise"
! X !! Y
|-
| '''0''' || '''0'''
|-
| '''0''' || '''1'''
|-
| '''1''' || '''0'''
|-
| '''1''' || '''1'''
|}
|
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ proj<sub>''XZ''</sub>('''L'''<sub>1</sub>)
|- style="background:paleturquoise"
! X !! Z
|-
| '''0''' || '''1'''
|-
| '''0''' || '''0'''
|-
| '''1''' || '''0'''
|-
| '''1''' || '''1'''
|}
|
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ proj<sub>''YZ''</sub>('''L'''<sub>1</sub>)
|- style="background:paleturquoise"
! Y !! Z
|-
| '''0''' || '''1'''
|-
| '''1''' || '''0'''
|-
| '''0''' || '''0'''
|-
| '''1''' || '''1'''
|}
|}
<br>
{| align="center" cellpadding="4" style="text-align:center; width:90%"
| proj<sub>''XY''</sub>('''L'''<sub>0</sub>) = proj<sub>''XY''</sub>('''L'''<sub>1</sub>)
| proj<sub>''XZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''XZ''</sub>('''L'''<sub>1</sub>)
| proj<sub>''YZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''YZ''</sub>('''L'''<sub>1</sub>)
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
|+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A
|- style="background:paleturquoise"
! style="width:20%" | Object
! style="width:20%" | Sign
! style="width:20%" | Interpretant
|-
| '''A''' || '''"A"''' || '''"A"'''
|-
| '''A''' || '''"A"''' || '''"i"'''
|-
| '''A''' || '''"i"''' || '''"A"'''
|-
| '''A''' || '''"i"''' || '''"i"'''
|-
| '''B''' || '''"B"''' || '''"B"'''
|-
| '''B''' || '''"B"''' || '''"u"'''
|-
| '''B''' || '''"u"''' || '''"B"'''
|-
| '''B''' || '''"u"''' || '''"u"'''
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
|+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B
|- style="background:paleturquoise"
! style="width:20%" | Object
! style="width:20%" | Sign
! style="width:20%" | Interpretant
|-
| '''A''' || '''"A"''' || '''"A"'''
|-
| '''A''' || '''"A"''' || '''"u"'''
|-
| '''A''' || '''"u"''' || '''"A"'''
|-
| '''A''' || '''"u"''' || '''"u"'''
|-
| '''B''' || '''"B"''' || '''"B"'''
|-
| '''B''' || '''"B"''' || '''"i"'''
|-
| '''B''' || '''"i"''' || '''"B"'''
|-
| '''B''' || '''"i"''' || '''"i"'''
|}
<br>
{| align="center" style="width:90%"
|
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ proj<sub>''XY''</sub>('''L'''<sub>A</sub>)
|- style="background:paleturquoise"
! style="width:50%" | Object
! style="width:50%" | Sign
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"i"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"u"'''
|}
|
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ proj<sub>''XZ''</sub>('''L'''<sub>A</sub>)
|- style="background:paleturquoise"
! style="width:50%" | Object
! style="width:50%" | Interpretant
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"i"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"u"'''
|}
|
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ proj<sub>''YZ''</sub>('''L'''<sub>A</sub>)
|- style="background:paleturquoise"
! style="width:50%" | Sign
! style="width:50%" | Interpretant
|-
| '''"A"''' || '''"A"'''
|-
| '''"A"''' || '''"i"'''
|-
| '''"i"''' || '''"A"'''
|-
| '''"i"''' || '''"i"'''
|-
| '''"B"''' || '''"B"'''
|-
| '''"B"''' || '''"u"'''
|-
| '''"u"''' || '''"B"'''
|-
| '''"u"''' || '''"u"'''
|}
|}
<br>
{| align="center" style="width:90%"
|
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ proj<sub>''XY''</sub>('''L'''<sub>B</sub>)
|- style="background:paleturquoise"
! style="width:50%" | Object
! style="width:50%" | Sign
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"u"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"i"'''
|}
|
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ proj<sub>''XZ''</sub>('''L'''<sub>B</sub>)
|- style="background:paleturquoise"
! style="width:50%" | Object
! style="width:50%" | Interpretant
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"u"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"i"'''
|}
|
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ proj<sub>''YZ''</sub>('''L'''<sub>B</sub>)
|- style="background:paleturquoise"
! style="width:50%" | Sign
! style="width:50%" | Interpretant
|-
| '''"A"''' || '''"A"'''
|-
| '''"A"''' || '''"u"'''
|-
| '''"u"''' || '''"A"'''
|-
| '''"u"''' || '''"u"'''
|-
| '''"B"''' || '''"B"'''
|-
| '''"B"''' || '''"i"'''
|-
| '''"i"''' || '''"B"'''
|-
| '''"i"''' || '''"i"'''
|}
|}
<br>
{| align="center" cellpadding="4" style="text-align:center; width:90%"
| proj<sub>''XY''</sub>('''L'''<sub>A</sub>) ≠ proj<sub>''XY''</sub>('''L'''<sub>B</sub>)
| proj<sub>''XZ''</sub>('''L'''<sub>A</sub>) ≠ proj<sub>''XZ''</sub>('''L'''<sub>B</sub>)
| proj<sub>''YZ''</sub>('''L'''<sub>A</sub>) ≠ proj<sub>''YZ''</sub>('''L'''<sub>B</sub>)
|}
<br>
===='"`UNIQ--h-45--QINU`"'Method 2 : Subtitles as Top Rows====
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
|+ '''L'''<sub>0</sub> = {(''x'', ''y'', ''z'') ∈ '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 0}
|- style="background:paleturquoise"
! X !! Y !! Z
|-
| '''0''' || '''0''' || '''0'''
|-
| '''0''' || '''1''' || '''1'''
|-
| '''1''' || '''0''' || '''1'''
|-
| '''1''' || '''1''' || '''0'''
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
|+ '''L'''<sub>1</sub> = {(''x'', ''y'', ''z'') ∈ '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 1}
|- style="background:paleturquoise"
! X !! Y !! Z
|-
| '''0''' || '''0''' || '''1'''
|-
| '''0''' || '''1''' || '''0'''
|-
| '''1''' || '''0''' || '''0'''
|-
| '''1''' || '''1''' || '''1'''
|}
<br>
{| align="center" style="width:90%"
| align="center" | proj<sub>''XY''</sub>('''L'''<sub>0</sub>)
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! X !! Y
|-
| '''0''' || '''0'''
|-
| '''0''' || '''1'''
|-
| '''1''' || '''0'''
|-
| '''1''' || '''1'''
|}
| align="center" | proj<sub>''XZ''</sub>('''L'''<sub>0</sub>)
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! X !! Z
|-
| '''0''' || '''0'''
|-
| '''0''' || '''1'''
|-
| '''1''' || '''1'''
|-
| '''1''' || '''0'''
|}
| align="center" | proj<sub>''YZ''</sub>('''L'''<sub>0</sub>)
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! Y !! Z
|-
| '''0''' || '''0'''
|-
| '''1''' || '''1'''
|-
| '''0''' || '''1'''
|-
| '''1''' || '''0'''
|}
|}
<br>
{| align="center" style="width:90%"
| align="center" | proj<sub>''XY''</sub>('''L'''<sub>1</sub>)
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! X !! Y
|-
| '''0''' || '''0'''
|-
| '''0''' || '''1'''
|-
| '''1''' || '''0'''
|-
| '''1''' || '''1'''
|}
| align="center" | proj<sub>''XZ''</sub>('''L'''<sub>1</sub>)
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! X !! Z
|-
| '''0''' || '''1'''
|-
| '''0''' || '''0'''
|-
| '''1''' || '''0'''
|-
| '''1''' || '''1'''
|}
| align="center" | proj<sub>''YZ''</sub>('''L'''<sub>1</sub>)
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! Y !! Z
|-
| '''0''' || '''1'''
|-
| '''1''' || '''0'''
|-
| '''0''' || '''0'''
|-
| '''1''' || '''1'''
|}
|}
<br>
{| align="center" cellpadding="4" style="text-align:center; width:90%"
| proj<sub>''XY''</sub>('''L'''<sub>0</sub>) = proj<sub>''XY''</sub>('''L'''<sub>1</sub>)
| proj<sub>''XZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''XZ''</sub>('''L'''<sub>1</sub>)
| proj<sub>''YZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''YZ''</sub>('''L'''<sub>1</sub>)
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
|+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A
|- style="background:paleturquoise"
! style="width:20%" | Object
! style="width:20%" | Sign
! style="width:20%" | Interpretant
|-
| '''A''' || '''"A"''' || '''"A"'''
|-
| '''A''' || '''"A"''' || '''"i"'''
|-
| '''A''' || '''"i"''' || '''"A"'''
|-
| '''A''' || '''"i"''' || '''"i"'''
|-
| '''B''' || '''"B"''' || '''"B"'''
|-
| '''B''' || '''"B"''' || '''"u"'''
|-
| '''B''' || '''"u"''' || '''"B"'''
|-
| '''B''' || '''"u"''' || '''"u"'''
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
|+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B
|- style="background:paleturquoise"
! style="width:20%" | Object
! style="width:20%" | Sign
! style="width:20%" | Interpretant
|-
| '''A''' || '''"A"''' || '''"A"'''
|-
| '''A''' || '''"A"''' || '''"u"'''
|-
| '''A''' || '''"u"''' || '''"A"'''
|-
| '''A''' || '''"u"''' || '''"u"'''
|-
| '''B''' || '''"B"''' || '''"B"'''
|-
| '''B''' || '''"B"''' || '''"i"'''
|-
| '''B''' || '''"i"''' || '''"B"'''
|-
| '''B''' || '''"i"''' || '''"i"'''
|}
<br>
{| align="center" style="width:90%"
| align="center" style="width:30%" | proj<sub>''XY''</sub>('''L'''<sub>A</sub>)
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! style="width:50%" | Object
! style="width:50%" | Sign
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"i"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"u"'''
|}
| align="center" style="width:30%" | proj<sub>''XZ''</sub>('''L'''<sub>A</sub>)
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! style="width:50%" | Object
! style="width:50%" | Interpretant
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"i"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"u"'''
|}
| align="center" style="width:30%" | proj<sub>''YZ''</sub>('''L'''<sub>A</sub>)
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! style="width:50%" | Sign
! style="width:50%" | Interpretant
|-
| '''"A"''' || '''"A"'''
|-
| '''"A"''' || '''"i"'''
|-
| '''"i"''' || '''"A"'''
|-
| '''"i"''' || '''"i"'''
|-
| '''"B"''' || '''"B"'''
|-
| '''"B"''' || '''"u"'''
|-
| '''"u"''' || '''"B"'''
|-
| '''"u"''' || '''"u"'''
|}
|}
<br>
{| align="center" style="width:90%"
| align="center" style="width:30%" | proj<sub>''XY''</sub>('''L'''<sub>B</sub>)
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! style="width:50%" | Object
! style="width:50%" | Sign
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"u"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"i"'''
|}
| align="center" style="width:30%" | proj<sub>''XZ''</sub>('''L'''<sub>B</sub>)
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! style="width:50%" | Object
! style="width:50%" | Interpretant
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"u"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"i"'''
|}
| align="center" style="width:30%" | proj<sub>''YZ''</sub>('''L'''<sub>B</sub>)
{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|- style="background:paleturquoise"
! style="width:50%" | Sign
! style="width:50%" | Interpretant
|-
| '''"A"''' || '''"A"'''
|-
| '''"A"''' || '''"u"'''
|-
| '''"u"''' || '''"A"'''
|-
| '''"u"''' || '''"u"'''
|-
| '''"B"''' || '''"B"'''
|-
| '''"B"''' || '''"i"'''
|-
| '''"i"''' || '''"B"'''
|-
| '''"i"''' || '''"i"'''
|}
|}
<br>
{| align="center" cellpadding="4" style="text-align:center; width:90%"
| proj<sub>''XY''</sub>('''L'''<sub>A</sub>) ≠ proj<sub>''XY''</sub>('''L'''<sub>B</sub>)
| proj<sub>''XZ''</sub>('''L'''<sub>A</sub>) ≠ proj<sub>''XZ''</sub>('''L'''<sub>B</sub>)
| proj<sub>''YZ''</sub>('''L'''<sub>A</sub>) ≠ proj<sub>''YZ''</sub>('''L'''<sub>B</sub>)
|}
<br>
==='"`UNIQ--h-46--QINU`"'Formatted Text Display===
: So in a triadic fact, say, the example <br>
{| align="center" cellspacing="8" style="width:72%"
| align="center" | ''A'' gives ''B'' to ''C''
|}
: we make no distinction in the ordinary logic of relations between the ''[[subject (grammar)|subject]] [[nominative]]'', the ''[[direct object]]'', and the ''[[indirect object]]''. We say that the proposition has three ''logical subjects''. We regard it as a mere affair of English grammar that there are six ways of expressing this: <br>
{| align="center" cellspacing="8" style="width:72%"
| style="width:36%" | ''A'' gives ''B'' to ''C''
| style="width:36%" | ''A'' benefits ''C'' with ''B''
|-
| ''B'' enriches ''C'' at expense of ''A''
| ''C'' receives ''B'' from ''A''
|-
| ''C'' thanks ''A'' for ''B''
| ''B'' leaves ''A'' for ''C''
|}
: These six sentences express one and the same indivisible phenomenon. (C.S. Peirce, "The Categories Defended", MS 308 (1903), EP 2, 170-171).
=='"`UNIQ--h-47--QINU`"'Work Area==
{| border="1" cellspacing="0" cellpadding="0" style="text-align:center"
|+ Binary Operations
|-
! style="width:2em" | x<sub>0</sub>
! style="width:2em" | x<sub>1</sub>
| style="width:2em" | <sup>2</sup>f<sub>0</sub>
| style="width:2em" | <sup>2</sup>f<sub>1</sub>
| style="width:2em" | <sup>2</sup>f<sub>2</sub>
| style="width:2em" | <sup>2</sup>f<sub>3</sub>
| style="width:2em" | <sup>2</sup>f<sub>4</sub>
| style="width:2em" | <sup>2</sup>f<sub>5</sub>
| style="width:2em" | <sup>2</sup>f<sub>6</sub>
| style="width:2em" | <sup>2</sup>f<sub>7</sub>
| style="width:2em" | <sup>2</sup>f<sub>8</sub>
| style="width:2em" | <sup>2</sup>f<sub>9</sub>
| style="width:2em" | <sup>2</sup>f<sub>10</sub>
| style="width:2em" | <sup>2</sup>f<sub>11</sub>
| style="width:2em" | <sup>2</sup>f<sub>12</sub>
| style="width:2em" | <sup>2</sup>f<sub>13</sub>
| style="width:2em" | <sup>2</sup>f<sub>14</sub>
| style="width:2em" | <sup>2</sup>f<sub>15</sub>
|-
| 0 || 0 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1
|-
| 1 || 0 || 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1
|-
| 0 || 1 || 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1
|-
| 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
|}
<br>
==='"`UNIQ--h-48--QINU`"'Draft 1===
<center><table>
<caption>TRUTH TABLES FOR THE BOOLEAN OPERATIONS OF ARITY UP TO 2</caption>
<tr valign="top">
<td><table border="5" cellspacing="0">
<caption>Constants</caption>
<tr><td></td>
<td><sup>0</sup>f<sub>0</sub></td> <td><sup>0</sup>f<sub>1</sub></td>
</tr> <tr><td></td>
<td align="center">0</td> <td align="center">1</td>
</tr></table></td><td> </td>
<td><table border="5" cellspacing="0"><caption>Unary Operations</caption><tr>
<td>x<sub>0</sub></td> <td></td>
<td><sup>1</sup>f<sub>0 </sub></td> <td><sup>1</sup>f<sub>1 </sub></td>
<td><sup>1</sup>f<sub>2 </sub></td> <td><sup>1</sup>f<sub>3 </sub></td>
</tr><tr> <td align="center">0</td> <td></td>
<td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td>
</tr> <tr> <td align="center">1</td> <td></td>
<td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td>
</tr></table></td><td> </td>
<td><table border="5" cellspacing="0"><caption>Binary Operations</caption><tr>
<td>x<sub>0</sub></td> <td>x<sub>1</sub></td>
<td></td>
<td><sup>2</sup>f<sub>0</sub></td> <td><sup>2</sup>f<sub>1 </sub></td>
<td><sup> 2</sup>f<sub>2 </sub></td> <td><sup>2</sup>f<sub>3 </sub></td>
<td><sup>2</sup>f<sub>4 </sub></td> <td><sup>2</sup>f<sub>5 </sub></td>
<td><sup>2</sup>f<sub>6 </sub></td> <td><sup>2</sup>f<sub>7 </sub></td>
<td><sup>2</sup>f<sub>8 </sub></td> <td><sup>2</sup>f<sub>9 </sub></td>
<td><sup>2</sup>f<sub>10</sub></td> <td><sup>2</sup>f<sub>11</sub></td>
<td><sup>2</sup>f<sub>12</sub></td> <td><sup>2</sup>f<sub>13</sub></td>
<td><sup>2</sup>f<sub>14</sub></td> <td><sup>2</sup>f<sub>15</sub></td>
</tr><tr> <td align="center">0</td> <td align="center">0</td> <td></td>
<td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td>
<td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td>
<td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td>
<td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td>
</tr> <tr> <td align="center">1</td> <td align="center">0</td> <td></td>
<td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td>
<td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td>
<td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td>
<td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td>
</tr> <tr> <td align="center">0</td> <td align="center">1</td> <td></td>
<td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td>
<td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td>
<td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td>
<td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td>
</tr> <tr> <td align="center">1</td> <td align="center">1</td> <td></td>
<td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td>
<td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td>
<td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td>
<td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td>
</tr> </table></td>
</table></center>
==='"`UNIQ--h-49--QINU`"'Draft 2===
<center><table>
<caption>TRUTH TABLES FOR THE BOOLEAN OPERATIONS OF ARITY UP TO 2</caption>
<tr valign="top">
<td><table border="5" cellspacing="0">
<caption>Constants</caption>
<tr><td></td>
<td><sup>0</sup>f<sub>0</sub></td> <td><sup>0</sup>f<sub>1</sub></td>
</tr> <tr><td></td>
<td align="center">0</td> <td align="center">1</td>
</tr></table></td><td> </td>
<td><table border="5" cellspacing="0"><caption>Unary Operations</caption><tr>
<td>x<sub>0</sub></td> <td></td>
<td><sup>1</sup>f<sub>0 </sub></td> <td><sup>1</sup>f<sub>1 </sub></td>
<td><sup>1</sup>f<sub>2 </sub></td> <td><sup>1</sup>f<sub>3 </sub></td>
</tr><tr> <td align="center">0</td> <td></td>
<td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td>
</tr> <tr> <td align="center">1</td> <td></td>
<td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td>
</tr></table></td><td> </td>
<td><table border="5" cellspacing="0"><caption>Binary Operations</caption><tr>
<td>x<sub>0</sub></td> <td>x<sub>1</sub></td>
<td></td>
<td><sup>2</sup>f<sub>0</sub></td> <td><sup>2</sup>f<sub>1 </sub></td>
<td><sup> 2</sup>f<sub>2 </sub></td> <td><sup>2</sup>f<sub>3 </sub></td>
<td><sup>2</sup>f<sub>4 </sub></td> <td><sup>2</sup>f<sub>5 </sub></td>
<td><sup>2</sup>f<sub>6 </sub></td> <td><sup>2</sup>f<sub>7 </sub></td>
<td><sup>2</sup>f<sub>8 </sub></td> <td><sup>2</sup>f<sub>9 </sub></td>
<td><sup>2</sup>f<sub>10</sub></td> <td><sup>2</sup>f<sub>11</sub></td>
<td><sup>2</sup>f<sub>12</sub></td> <td><sup>2</sup>f<sub>13</sub></td>
<td><sup>2</sup>f<sub>14</sub></td> <td><sup>2</sup>f<sub>15</sub></td>
</tr><tr> <td align="center">0</td> <td align="center">0</td> <td></td>
<td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td>
<td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td>
<td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td>
<td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td>
</tr> <tr> <td align="center">1</td> <td align="center">0</td> <td></td>
<td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td>
<td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td>
<td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td>
<td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td>
</tr> <tr> <td align="center">0</td> <td align="center">1</td> <td></td>
<td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td>
<td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td>
<td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td>
<td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td>
</tr> <tr> <td align="center">1</td> <td align="center">1</td> <td></td>
<td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td>
<td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td>
<td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td>
<td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td>
</tr> </table></td>
</table></center>
=='"`UNIQ--h-50--QINU`"'Inquiry and Analogy==
==='"`UNIQ--h-51--QINU`"'Test Patterns===
{| align="center"
| style="background:black; color:white" | 1
| style="background:white; color:black" | 0
| style="background:black; color:white" | 1
| style="background:white; color:black" | 0
| style="background:black; color:white" | 1
| style="background:white; color:black" | 0
| style="background:black; color:white" | 1
| style="background:white; color:black" | 0
|-
| style="background:white; color:black" | 0
| style="background:black; color:white" | 1
| style="background:white; color:black" | 0
| style="background:black; color:white" | 1
| style="background:white; color:black" | 0
| style="background:black; color:white" | 1
| style="background:white; color:black" | 0
| style="background:black; color:white" | 1
|}<br>
{| align="center"
| style="background:white; color:black" | 1
| style="background:black; color:white" | 0
| style="background:white; color:black" | 1
| style="background:black; color:white" | 0
| style="background:white; color:black" | 1
| style="background:black; color:white" | 0
| style="background:white; color:black" | 1
| style="background:black; color:white" | 0
|-
| style="background:black; color:white" | 0
| style="background:white; color:black" | 1
| style="background:black; color:white" | 0
| style="background:white; color:black" | 1
| style="background:black; color:white" | 0
| style="background:white; color:black" | 1
| style="background:black; color:white" | 0
| style="background:white; color:black" | 1
|}<br>
{| align="center"
| style="background:black; color:white" | 1
| style="background:white; color:black" | 0
| style="background:black; color:white" | 1
| style="background:white; color:black" | 0
| style="background:black; color:white" | 1
| style="background:white; color:black" | 0
| style="background:black; color:white" | 1
| style="background:white; color:black" | 0
|-
| style="background:white; color:black" | 0
| style="background:black; color:white" | 1
| style="background:white; color:black" | 0
| style="background:black; color:white" | 1
| style="background:white; color:black" | 0
| style="background:black; color:white" | 1
| style="background:white; color:black" | 0
| style="background:black; color:white" | 1
|}<br>
==='"`UNIQ--h-52--QINU`"'Table 10===
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|+ '''Table 10. Higher Order Propositions (''n'' = 1)'''
|- style="background:ghostwhite"
| align="right" | \(x\):
|
1 0
|
\(f\)
|
\(m_0\)
|
\(m_1\)
|
\(m_2\)
|
\(m_3\)
|
\(m_4\)
|
\(m_5\)
|
\(m_6\)
|
\(m_7\)
|
\(m_8\)
|
\(m_9\)
|
\(m_{10}\)
|
\(m_{11}\)
|
\(m_{12}\)
|
\(m_{13}\)
|
\(m_{14}\)
|
\(m_{15}\)
|
| \(f_0\)
|
0 0
|
\(0\!\)
|
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1
|
| \(f_1\)
|
0 1
|
\((x)\!\)
|
0 |
0 |
1 |
1 |
0 |
0 |
1 |
1 |
0 |
0 |
1 |
1 |
0 |
0 |
1 |
1
|
| \(f_2\)
|
1 0
|
\(x\!\)
|
0 |
0 |
0 |
0 |
1 |
1 |
1 |
1 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
1
|
| \(f_3\)
|
1 1
|
\(1\!\)
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1
|
Table 10. Higher Order Propositions (n = 1)
| \(x:\)
|
1 0
|
\(f\!\)
|
\(m_0\)
|
\(m_1\)
|
\(m_2\)
|
\(m_3\)
|
\(m_4\)
|
\(m_5\)
|
\(m_6\)
|
\(m_7\)
|
\(m_8\)
|
\(m_9\)
|
\(m_{10}\)
|
\(m_{11}\)
|
\(m_{12}\)
|
\(m_{13}\)
|
\(m_{14}\)
|
\(m_{15}\)
|
| \(f_0\)
|
0 0
|
\(0\!\)
|
0 |
1
|
0 |
1
|
0 |
1
|
0 |
1
|
0 |
1
|
0 |
1
|
0 |
1
|
0 |
1
|
| \(f_1\)
|
0 1
|
\((x)\!\)
|
0 |
0
|
1
|
1
|
0 |
0
|
1
|
1
|
0 |
0
|
1
|
1
|
0 |
0
|
1
|
1
|
| \(f_2\)
|
1 0
|
\(x\!\)
|
0 |
0 |
0 |
0
|
1
|
1
|
1
|
1
|
0 |
0 |
0 |
0
|
1
|
1
|
1
|
1
|
| \(f_3\)
|
1 1
|
\(1\!\)
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
Table 11
Table 11. Interpretive Categories for Higher Order Propositions (n = 1)
| Measure
|
Happening
|
Exactness
|
Existence
|
Linearity
|
Uniformity
|
Information
|
| \(m_0\!\)
|
Nothing happens
|
|
|
|
|
|
| \(m_1\!\)
|
|
Just false
|
Nothing exists
|
|
|
|
| \(m_2\!\)
|
|
Just not \(x\!\)
|
|
|
|
|
| \(m_3\!\)
|
|
|
Nothing is \(x\!\)
|
|
|
|
| \(m_4\!\)
|
|
Just \(x\!\)
|
|
|
|
|
| \(m_5\!\)
|
|
|
Everything is \(x\!\)
|
\(f\!\) is linear
|
|
|
| \(m_6\!\)
|
|
|
|
|
\(f\!\) is not uniform
|
\(f\!\) is informed
|
| \(m_7\!\)
|
|
Not just true
|
|
|
|
|
| \(m_8\!\)
|
|
Just true
|
|
|
|
|
| \(m_9\!\)
|
|
|
|
|
\(f\!\) is uniform
|
\(f\!\) is not informed
|
| \(m_{10}\!\)
|
|
|
Something is not \(x\!\)
|
\(f\!\) is not linear
|
|
|
| \(m_{11}\!\)
|
|
Not just \(x\!\)
|
|
|
|
|
| \(m_{12}\!\)
|
|
|
Something is \(x\!\)
|
|
|
|
| \(m_{13}\!\)
|
|
Not just not \(x\!\)
|
|
|
|
|
| \(m_{14}\!\)
|
|
Not just false
|
Something exists
|
|
|
|
| \(m_{15}\!\)
|
Anything happens
|
|
|
|
|
|
Table 12
Table 12. Higher Order Propositions (n = 2)
\(x:\)
\(y:\)
|
1100
1010
|
\(f\!\)
|
\(m_0\)
|
\(m_1\)
|
\(m_2\)
|
\(m_3\)
|
\(m_4\)
|
\(m_5\)
|
\(m_6\)
|
\(m_7\)
|
\(m_8\)
|
\(m_9\)
|
\(m_{10}\)
|
\(m_{11}\)
|
\(m_{12}\)
|
\(m_{13}\)
|
\(m_{14}\)
|
\(m_{15}\)
|
\(m_{16}\)
|
\(m_{17}\)
|
\(m_{18}\)
|
\(m_{19}\)
|
\(m_{20}\)
|
\(m_{21}\)
|
\(m_{22}\)
|
\(m_{23}\)
|
| \(f_0\) |
0000 |
\((~)\)
|
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1
|
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1
|
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1
|
| \(f_1\) |
0001 |
\((x)(y)\!\)
|
|
|
1 |
1 |
0 |
0 |
1 |
1
|
0 |
0 |
1 |
1 |
0 |
0 |
1 |
1
|
0 |
0 |
1 |
1 |
0 |
0 |
1 |
1
|
| \(f_2\) |
0010 |
\((x) y\!\)
|
|
|
|
|
1 |
1 |
1 |
1
|
0 |
0 |
0 |
0 |
1 |
1 |
1 |
1
|
0 |
0 |
0 |
0 |
1 |
1 |
1 |
1
|
| \(f_3\) |
0011 |
\((x)\!\)
|
|
|
|
|
|
|
|
|
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
| \(f_4\) |
0100 |
\(x (y)\!\)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1
|
| \(f_5\) |
0101 |
\((y)\!\)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| \(f_6\) |
0110 |
\((x, y)\!\)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| \(f_7\) |
0111 |
\((x y)\!\)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| \(f_8\) |
1000 |
\(x y\!\)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| \(f_9\) |
1001 |
\(((x, y))\!\)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| \(f_{10}\) |
1010 |
\(y\!\)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| \(f_{11}\) |
1011 |
\((x (y))\!\)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| \(f_{12}\) |
1100 |
\(x\!\)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| \(f_{13}\) |
1101 |
\(((x) y)\!\)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| \(f_{14}\) |
1110 |
\(((x)(y))\!\)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| \(f_{15}\) |
1111 |
\(((~))\!\)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Table 12. Higher Order Propositions (n = 2)
\(u:\)
\(v:\)
|
1100
1010
|
\(f\!\)
|
\(m_0\)
|
\(m_1\)
|
\(m_2\)
|
\(m_3\)
|
\(m_4\)
|
\(m_5\)
|
\(m_6\)
|
\(m_7\)
|
\(m_8\)
|
\(m_9\)
|
\(m_{10}\)
|
\(m_{11}\)
|
\(m_{12}\)
|
\(m_{13}\)
|
\(m_{14}\)
|
\(m_{15}\)
|
\(m_{16}\)
|
\(m_{17}\)
|
\(m_{18}\)
|
\(m_{19}\)
|
\(m_{20}\)
|
\(m_{21}\)
|
\(m_{22}\)
|
\(m_{23}\)
|
| \(f_0\)
|
0000
|
\((~)\)
|
0 |
1
|
0 |
1
|
0 |
1
|
0 |
1
|
0 |
1
|
0 |
1
|
0 |
1
|
0 |
1
|
0 |
1
|
0 |
1
|
0 |
1
|
0 |
1
|
| \(f_1\)
|
0001
|
\((u)(v)\!\)
|
0 |
0
|
1
|
1
|
0 |
0
|
1
|
1
|
0 |
0
|
1
|
1
|
0 |
0
|
1
|
1
|
0 |
0
|
1
|
1
|
0 |
0
|
1
|
1
|
| \(f_2\)
|
0010
|
\((u) v\!\)
|
0 |
0 |
0 |
0
|
1
|
1
|
1
|
1
|
0 |
0 |
0 |
0
|
1
|
1
|
1
|
1
|
0 |
0 |
0 |
0
|
1
|
1
|
1
|
1
|
| \(f_3\)
|
0011
|
\((u)\!\)
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
| \(f_4\)
|
0100
|
\(u (v)\!\)
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
| \(f_5\)
|
0101
|
\((v)\!\)
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
| \(f_6\)
|
0110
|
\((u, v)\!\)
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
| \(f_7\)
|
0111
|
\((u v)\!\)
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
| \(f_8\)
|
1000
|
\(u v\!\)
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
| \(f_9\)
|
1001
|
\(((u, v))\!\)
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
| \(f_{10}\)
|
1010
|
\(v\!\)
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
| \(f_{11}\)
|
1011
|
\((u (v))\!\)
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
| \(f_{12}\)
|
1100
|
\(u\!\)
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
| \(f_{13}\)
|
1101
|
\(((u) v)\!\)
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
| \(f_{14}\)
|
1110
|
\(((u)(v))\!\)
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
| \(f_{15}\)
|
1111
|
\(((~))\!\)
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
Table 13
Table 13. Qualifiers of Implication Ordering: \(\alpha_i f = \Upsilon (f_i, f) = \Upsilon (f_i \Rightarrow f)\)
\(u:\)
\(v:\)
|
1100
1010
|
\(f\!\)
|
\(\alpha_0\)
|
\(\alpha_1\)
|
\(\alpha_2\)
|
\(\alpha_3\)
|
\(\alpha_4\)
|
\(\alpha_5\)
|
\(\alpha_6\)
|
\(\alpha_7\)
|
\(\alpha_8\)
|
\(\alpha_9\)
|
\(\alpha_{10}\)
|
\(\alpha_{11}\)
|
\(\alpha_{12}\)
|
\(\alpha_{13}\)
|
\(\alpha_{14}\)
|
\(\alpha_{15}\)
|
| \(f_0\)
|
0000
|
\((~)\)
|
1
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
| \(f_1\)
|
0001
|
\((u)(v)\!\)
|
1
|
1
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
| \(f_2\)
|
0010
|
\((u) v\!\)
|
1
|
0
|
1
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
| \(f_3\)
|
0011
|
\((u)\!\)
|
1
|
1
|
1
|
1
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
| \(f_4\)
|
0100
|
\(u (v)\!\)
|
1
|
0
|
0
|
0
|
1
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
| \(f_5\)
|
0101
|
\((v)\!\)
|
1
|
1
|
0
|
0
|
1
|
1
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
| \(f_6\)
|
0110
|
\((u, v)\!\)
|
1
|
0
|
1
|
0
|
1
|
0
|
1
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
| \(f_7\)
|
0111
|
\((u v)\!\)
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
| \(f_8\)
|
1000
|
\(u v\!\)
|
1
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
1
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
| \(f_9\)
|
1001
|
\(((u, v))\!\)
|
1
|
1
|
0
|
0
|
0
|
0
|
0
|
0
|
1
|
1
|
0
|
0
|
0
|
0
|
0
|
0
|
| \(f_{10}\)
|
1010
|
\(v\!\)
|
1
|
0
|
1
|
0
|
0
|
0
|
0
|
0
|
1
|
0
|
1
|
0
|
0
|
0
|
0
|
0
|
| \(f_{11}\)
|
1011
|
\((u (v))\!\)
|
1
|
1
|
1
|
1
|
0
|
0
|
0
|
0
|
1
|
1
|
1
|
1
|
0
|
0
|
0
|
0
|
| \(f_{12}\)
|
1100
|
\(u\!\)
|
1
|
0
|
0
|
0
|
1
|
0
|
0
|
0
|
1
|
0
|
0
|
0
|
1
|
0
|
0
|
0
|
| \(f_{13}\)
|
1101
|
\(((u) v)\!\)
|
1
|
1
|
0
|
0
|
1
|
1
|
0
|
0
|
1
|
1
|
0
|
0
|
1
|
1
|
0
|
0
|
| \(f_{14}\)
|
1110
|
\(((u)(v))\!\)
|
1
|
0
|
1
|
0
|
1
|
0
|
1
|
0
|
1
|
0
|
1
|
0
|
1
|
0
|
1
|
0
|
| \(f_{15}\)
|
1111
|
\(((~))\)
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
Table 14
Table 14. Qualifiers of Implication Ordering: \(\beta_i f = \Upsilon (f, f_i) = \Upsilon (f \Rightarrow f_i)\)
\(u:\)
\(v:\)
|
1100
1010
|
\(f\!\)
|
\(\beta_0\)
|
\(\beta_1\)
|
\(\beta_2\)
|
\(\beta_3\)
|
\(\beta_4\)
|
\(\beta_5\)
|
\(\beta_6\)
|
\(\beta_7\)
|
\(\beta_8\)
|
\(\beta_9\)
|
\(\beta_{10}\)
|
\(\beta_{11}\)
|
\(\beta_{12}\)
|
\(\beta_{13}\)
|
\(\beta_{14}\)
|
\(\beta_{15}\)
|
| \(f_0\)
|
0000
|
\((~)\)
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
| \(f_1\)
|
0001
|
\((u)(v)\!\)
|
0
|
1
|
0
|
1
|
0
|
1
|
0
|
1
|
0
|
1
|
0
|
1
|
0
|
1
|
0
|
1
|
| \(f_2\)
|
0010
|
\((u) v\!\)
|
0
|
0
|
1
|
1
|
0
|
0
|
1
|
1
|
0
|
0
|
1
|
1
|
0
|
0
|
1
|
1
|
| \(f_3\)
|
0011
|
\((u)\!\)
|
0
|
0
|
0
|
1
|
0
|
0
|
0
|
1
|
0
|
0
|
0
|
1
|
0
|
0
|
0
|
1
|
| \(f_4\)
|
0100
|
\(u (v)\!\)
|
0
|
0
|
0
|
0
|
1
|
1
|
1
|
1
|
0
|
0
|
0
|
0
|
1
|
1
|
1
|
1
|
| \(f_5\)
|
0101
|
\((v)\!\)
|
0
|
0
|
0
|
0
|
0
|
1
|
0
|
1
|
0
|
0
|
0
|
0
|
0
|
1
|
0
|
1
|
| \(f_6\)
|
0110
|
\((u, v)\!\)
|
0
|
0
|
0
|
0
|
0
|
0
|
1
|
1
|
0
|
0
|
0
|
0
|
0
|
0
|
1
|
1
|
| \(f_7\)
|
0111
|
\((u v)\!\)
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
1
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
1
|
| \(f_8\)
|
1000
|
\(u v\!\)
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
| \(f_9\)
|
1001
|
\(((u, v))\!\)
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
1
|
0
|
1
|
0
|
1
|
0
|
1
|
| \(f_{10}\)
|
1010
|
\(v\!\)
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
1
|
1
|
0
|
0
|
1
|
1
|
| \(f_{11}\)
|
1011
|
\((u (v))\!\)
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
1
|
0
|
0
|
0
|
1
|
| \(f_{12}\)
|
1100
|
\(u\!\)
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
1
|
1
|
1
|
1
|
| \(f_{13}\)
|
1101
|
\(((u) v)\!\)
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
1
|
0
|
1
|
| \(f_{14}\)
|
1110
|
\(((u)(v))\!\)
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
1
|
1
|
| \(f_{15}\)
|
1111
|
\(((~))\!\)
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
1
|
Figure 15
Table 16
Table 16. Syllogistic Premisses as Higher Order Indicator Functions
|
\(\begin{array}{clcl}
\mathrm{A} &
\mathrm{Universal~Affirmative} &
\mathrm{All}\ u\ \mathrm{is}\ v &
\mathrm{Indicator~of}\ u (v) = 0 \\
\mathrm{E} &
\mathrm{Universal~Negative} &
\mathrm{All}\ u\ \mathrm{is}\ (v) &
\mathrm{Indicator~of}\ u \cdot v = 0 \\
\mathrm{I} &
\mathrm{Particular~Affirmative} &
\mathrm{Some}\ u\ \mathrm{is}\ v &
\mathrm{Indicator~of}\ u \cdot v = 1 \\
\mathrm{O} &
\mathrm{Particular~Negative} &
\mathrm{Some}\ u\ \mathrm{is}\ (v) &
\mathrm{Indicator~of}\ u (v) = 1 \\
\end{array}\)
|
Table 17
Table 17. Simple Qualifiers of Propositions (Version 1)
\(u:\)
\(v:\)
|
1100
1010
|
\(f\!\)
|
\((\ell_{11})\)
\(\text{No } u \)
\(\text{is } v \)
|
\((\ell_{10})\)
\(\text{No } u \)
\(\text{is }(v)\)
|
\((\ell_{01})\)
\(\text{No }(u)\)
\(\text{is } v \)
|
\((\ell_{00})\)
\(\text{No }(u)\)
\(\text{is }(v)\)
|
\( \ell_{00} \)
\(\text{Some }(u)\)
\(\text{is }(v)\)
|
\( \ell_{01} \)
\(\text{Some }(u)\)
\(\text{is } v \)
|
\( \ell_{10} \)
\(\text{Some } u \)
\(\text{is }(v)\)
|
\( \ell_{11} \)
\(\text{Some } u \)
\(\text{is } v \)
|
| \(f_0\)
|
0000
|
\((~)\)
|
1
|
1
|
1
|
1
|
0
|
0
|
0
|
0
|
| \(f_1\)
|
0001
|
\((u)(v)\!\)
|
1
|
1
|
1
|
0
|
1
|
0
|
0
|
0
|
| \(f_2\)
|
0010
|
\((u) v\!\)
|
1
|
1
|
0
|
1
|
0
|
1
|
0
|
0
|
| \(f_3\)
|
0011
|
\((u)\!\)
|
1
|
1
|
0
|
0
|
1
|
1
|
0
|
0
|
| \(f_4\)
|
0100
|
\(u (v)\!\)
|
1
|
0
|
1
|
1
|
0
|
0
|
1
|
0
|
| \(f_5\)
|
0101
|
\((v)\!\)
|
1
|
0
|
1
|
0
|
1
|
0
|
1
|
0
|
| \(f_6\)
|
0110
|
\((u, v)\!\)
|
1
|
0
|
0
|
1
|
0
|
1
|
1
|
0
|
| \(f_7\)
|
0111
|
\((u v)\!\)
|
1
|
0
|
0
|
0
|
1
|
1
|
1
|
0
|
| \(f_8\)
|
1000
|
\(u v\!\)
|
0
|
1
|
1
|
1
|
0
|
0
|
0
|
1
|
| \(f_9\)
|
1001
|
\(((u, v))\!\)
|
0
|
1
|
1
|
0
|
1
|
0
|
0
|
1
|
| \(f_{10}\)
|
1010
|
\(v\!\)
|
0
|
1
|
0
|
1
|
0
|
1
|
0
|
1
|
| \(f_{11}\)
|
1011
|
\((u (v))\!\)
|
0
|
1
|
0
|
0
|
1
|
1
|
0
|
1
|
| \(f_{12}\)
|
1100
|
\(u\!\)
|
0
|
0
|
1
|
1
|
0
|
0
|
1
|
1
|
| \(f_{13}\)
|
1101
|
\(((u) v)\!\)
|
0
|
0
|
1
|
0
|
1
|
0
|
1
|
1
|
| \(f_{14}\)
|
1110
|
\(((u)(v))\!\)
|
0
|
0
|
0
|
1
|
0
|
1
|
1
|
1
|
| \(f_{15}\)
|
1111
|
\(((~))\)
|
0
|
0
|
0
|
0
|
1
|
1
|
1
|
1
|
Table 18
Table 18. Simple Qualifiers of Propositions (Version 2)
\(u:\)
\(v:\)
|
1100
1010
|
\(f\!\)
|
\((\ell_{11})\)
\(\text{No } u \)
\(\text{is } v \)
|
\((\ell_{10})\)
\(\text{No } u \)
\(\text{is }(v)\)
|
\((\ell_{01})\)
\(\text{No }(u)\)
\(\text{is } v \)
|
\((\ell_{00})\)
\(\text{No }(u)\)
\(\text{is }(v)\)
|
\( \ell_{00} \)
\(\text{Some }(u)\)
\(\text{is }(v)\)
|
\( \ell_{01} \)
\(\text{Some }(u)\)
\(\text{is } v \)
|
\( \ell_{10} \)
\(\text{Some } u \)
\(\text{is }(v)\)
|
\( \ell_{11} \)
\(\text{Some } u \)
\(\text{is } v \)
|
| \(f_0\)
|
0000
|
\((~)\)
|
1
|
1
|
1
|
1
|
0
|
0
|
0
|
0
|
| \(f_1\)
|
0001
|
\((u)(v)\!\)
|
1
|
1
|
1
|
0
|
1
|
0
|
0
|
0
|
| \(f_2\)
|
0010
|
\((u) v\!\)
|
1
|
1
|
0
|
1
|
0
|
1
|
0
|
0
|
| \(f_4\)
|
0100
|
\(u (v)\!\)
|
1
|
0
|
1
|
1
|
0
|
0
|
1
|
0
|
| \(f_8\)
|
1000
|
\(u v\!\)
|
0
|
1
|
1
|
1
|
0
|
0
|
0
|
1
|
| \(f_3\)
|
0011
|
\((u)\!\)
|
1
|
1
|
0
|
0
|
1
|
1
|
0
|
0
|
| \(f_{12}\)
|
1100
|
\(u\!\)
|
0
|
0
|
1
|
1
|
0
|
0
|
1
|
1
|
| \(f_6\)
|
0110
|
\((u, v)\!\)
|
1
|
0
|
0
|
1
|
0
|
1
|
1
|
0
|
| \(f_9\)
|
1001
|
\(((u, v))\!\)
|
0
|
1
|
1
|
0
|
1
|
0
|
0
|
1
|
| \(f_5\)
|
0101
|
\((v)\!\)
|
1
|
0
|
1
|
0
|
1
|
0
|
1
|
0
|
| \(f_{10}\)
|
1010
|
\(v\!\)
|
0
|
1
|
0
|
1
|
0
|
1
|
0
|
1
|
| \(f_7\)
|
0111
|
\((u v)\!\)
|
1
|
0
|
0
|
0
|
1
|
1
|
1
|
0
|
| \(f_{11}\)
|
1011
|
\((u (v))\!\)
|
0
|
1
|
0
|
0
|
1
|
1
|
0
|
1
|
| \(f_{13}\)
|
1101
|
\(((u) v)\!\)
|
0
|
0
|
1
|
0
|
1
|
0
|
1
|
1
|
| \(f_{14}\)
|
1110
|
\(((u)(v))\!\)
|
0
|
0
|
0
|
1
|
0
|
1
|
1
|
1
|
| \(f_{15}\)
|
1111
|
\(((~))\)
|
0
|
0
|
0
|
0
|
1
|
1
|
1
|
1
|
Table 19
Table 19. Relation of Quantifiers to Higher Order Propositions
| \(\text{Mnemonic}\)
|
\(\text{Category}\)
|
\(\text{Classical Form}\)
|
\(\text{Alternate Form}\)
|
\(\text{Symmetric Form}\)
|
\(\text{Operator}\)
|
\(\text{E}\!\)
\(\text{Exclusive}\)
|
\(\text{Universal}\)
\(\text{Negative}\)
|
\(\text{All}\ u\ \text{is}\ (v)\)
|
|
\(\text{No}\ u\ \text{is}\ v \)
|
\((\ell_{11})\)
|
\(\text{A}\!\)
\(\text{Absolute}\)
|
\(\text{Universal}\)
\(\text{Affirmative}\)
|
\(\text{All}\ u\ \text{is}\ v \)
|
|
\(\text{No}\ u\ \text{is}\ (v)\)
|
\((\ell_{10})\)
|
|
|
|
\(\text{All}\ v\ \text{is}\ u \)
|
\(\text{No}\ v\ \text{is}\ (u)\)
|
\(\text{No}\ (u)\ \text{is}\ v \)
|
\((\ell_{01})\)
|
|
|
|
\(\text{All}\ (v)\ \text{is}\ u \)
|
\(\text{No}\ (v)\ \text{is}\ (u)\)
|
\(\text{No}\ (u)\ \text{is}\ (v)\)
|
\((\ell_{00})\)
|
|
|
|
\(\text{Some}\ (u)\ \text{is}\ (v)\)
|
|
\(\text{Some}\ (u)\ \text{is}\ (v)\)
|
\(\ell_{00}\!\)
|
|
|
|
\(\text{Some}\ (u)\ \text{is}\ v\)
|
|
\(\text{Some}\ (u)\ \text{is}\ v\)
|
\(\ell_{01}\!\)
|
\(\text{O}\!\)
\(\text{Obtrusive}\)
|
\(\text{Particular}\)
\(\text{Negative}\)
|
\(\text{Some}\ u\ \text{is}\ (v)\)
|
|
\(\text{Some}\ u\ \text{is}\ (v)\)
|
\(\ell_{10}\!\)
|
\(\text{I}\!\)
\(\text{Indefinite}\)
|
\(\text{Particular}\)
\(\text{Affirmative}\)
|
\(\text{Some}\ u\ \text{is}\ v\)
|
|
\(\text{Some}\ u\ \text{is}\ v\)
|
\(\ell_{11}\!\)
|
