Differential Logic
Ascii Tables
Table A1. Propositional Forms On Two Variables
o---------o---------o---------o----------o------------------o----------o
| L_1 | L_2 | L_3 | L_4 | L_5 | L_6 |
| | | | | | |
| Decimal | Binary | Vector | Cactus | English | Ordinary |
o---------o---------o---------o----------o------------------o----------o
| | x : 1 1 0 0 | | | |
| | y : 1 0 1 0 | | | |
o---------o---------o---------o----------o------------------o----------o
| | | | | | |
| f_0 | f_0000 | 0 0 0 0 | () | false | 0 |
| | | | | | |
| f_1 | f_0001 | 0 0 0 1 | (x)(y) | neither x nor y | ~x & ~y |
| | | | | | |
| f_2 | f_0010 | 0 0 1 0 | (x) y | y and not x | ~x & y |
| | | | | | |
| f_3 | f_0011 | 0 0 1 1 | (x) | not x | ~x |
| | | | | | |
| f_4 | f_0100 | 0 1 0 0 | x (y) | x and not y | x & ~y |
| | | | | | |
| f_5 | f_0101 | 0 1 0 1 | (y) | not y | ~y |
| | | | | | |
| f_6 | f_0110 | 0 1 1 0 | (x, y) | x not equal to y | x + y |
| | | | | | |
| f_7 | f_0111 | 0 1 1 1 | (x y) | not both x and y | ~x v ~y |
| | | | | | |
| f_8 | f_1000 | 1 0 0 0 | x y | x and y | x & y |
| | | | | | |
| f_9 | f_1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y |
| | | | | | |
| f_10 | f_1010 | 1 0 1 0 | y | y | y |
| | | | | | |
| f_11 | f_1011 | 1 0 1 1 | (x (y)) | not x without y | x => y |
| | | | | | |
| f_12 | f_1100 | 1 1 0 0 | x | x | x |
| | | | | | |
| f_13 | f_1101 | 1 1 0 1 | ((x) y) | not y without x | x <= y |
| | | | | | |
| f_14 | f_1110 | 1 1 1 0 | ((x)(y)) | x or y | x v y |
| | | | | | |
| f_15 | f_1111 | 1 1 1 1 | (()) | true | 1 |
| | | | | | |
o---------o---------o---------o----------o------------------o----------o
Table A2. Propositional Forms On Two Variables
o---------o---------o---------o----------o------------------o----------o
| L_1 | L_2 | L_3 | L_4 | L_5 | L_6 |
| | | | | | |
| Decimal | Binary | Vector | Cactus | English | Ordinary |
o---------o---------o---------o----------o------------------o----------o
| | x : 1 1 0 0 | | | |
| | y : 1 0 1 0 | | | |
o---------o---------o---------o----------o------------------o----------o
| | | | | | |
| f_0 | f_0000 | 0 0 0 0 | () | false | 0 |
| | | | | | |
o---------o---------o---------o----------o------------------o----------o
| | | | | | |
| f_1 | f_0001 | 0 0 0 1 | (x)(y) | neither x nor y | ~x & ~y |
| | | | | | |
| f_2 | f_0010 | 0 0 1 0 | (x) y | y and not x | ~x & y |
| | | | | | |
| f_4 | f_0100 | 0 1 0 0 | x (y) | x and not y | x & ~y |
| | | | | | |
| f_8 | f_1000 | 1 0 0 0 | x y | x and y | x & y |
| | | | | | |
o---------o---------o---------o----------o------------------o----------o
| | | | | | |
| f_3 | f_0011 | 0 0 1 1 | (x) | not x | ~x |
| | | | | | |
| f_12 | f_1100 | 1 1 0 0 | x | x | x |
| | | | | | |
o---------o---------o---------o----------o------------------o----------o
| | | | | | |
| f_6 | f_0110 | 0 1 1 0 | (x, y) | x not equal to y | x + y |
| | | | | | |
| f_9 | f_1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y |
| | | | | | |
o---------o---------o---------o----------o------------------o----------o
| | | | | | |
| f_5 | f_0101 | 0 1 0 1 | (y) | not y | ~y |
| | | | | | |
| f_10 | f_1010 | 1 0 1 0 | y | y | y |
| | | | | | |
o---------o---------o---------o----------o------------------o----------o
| | | | | | |
| f_7 | f_0111 | 0 1 1 1 | (x y) | not both x and y | ~x v ~y |
| | | | | | |
| f_11 | f_1011 | 1 0 1 1 | (x (y)) | not x without y | x => y |
| | | | | | |
| f_13 | f_1101 | 1 1 0 1 | ((x) y) | not y without x | x <= y |
| | | | | | |
| f_14 | f_1110 | 1 1 1 0 | ((x)(y)) | x or y | x v y |
| | | | | | |
o---------o---------o---------o----------o------------------o----------o
| | | | | | |
| f_15 | f_1111 | 1 1 1 1 | (()) | true | 1 |
| | | | | | |
o---------o---------o---------o----------o------------------o----------o
Table A3. Ef Expanded Over Differential Features {dx, dy}
o------o------------o------------o------------o------------o------------o
| | | | | | |
| | f | T_11 f | T_10 f | T_01 f | T_00 f |
| | | | | | |
| | | Ef| dx dy | Ef| dx(dy) | Ef| (dx)dy | Ef|(dx)(dy)|
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_0 | () | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_1 | (x)(y) | x y | x (y) | (x) y | (x)(y) |
| | | | | | |
| f_2 | (x) y | x (y) | x y | (x)(y) | (x) y |
| | | | | | |
| f_4 | x (y) | (x) y | (x)(y) | x y | x (y) |
| | | | | | |
| f_8 | x y | (x)(y) | (x) y | x (y) | x y |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_3 | (x) | x | x | (x) | (x) |
| | | | | | |
| f_12 | x | (x) | (x) | x | x |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_6 | (x, y) | (x, y) | ((x, y)) | ((x, y)) | (x, y) |
| | | | | | |
| f_9 | ((x, y)) | ((x, y)) | (x, y) | (x, y) | ((x, y)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_5 | (y) | y | (y) | y | (y) |
| | | | | | |
| f_10 | y | (y) | y | (y) | y |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_7 | (x y) | ((x)(y)) | ((x) y) | (x (y)) | (x y) |
| | | | | | |
| f_11 | (x (y)) | ((x) y) | ((x)(y)) | (x y) | (x (y)) |
| | | | | | |
| f_13 | ((x) y) | (x (y)) | (x y) | ((x)(y)) | ((x) y) |
| | | | | | |
| f_14 | ((x)(y)) | (x y) | (x (y)) | ((x) y) | ((x)(y)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_15 | (()) | (()) | (()) | (()) | (()) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | |
| Fixed Point Total | 4 | 4 | 4 | 16 |
| | | | | |
o-------------------o------------o------------o------------o------------o
Table A4. Df Expanded Over Differential Features {dx, dy}
o------o------------o------------o------------o------------o------------o
| | | | | | |
| | f | Df| dx dy | Df| dx(dy) | Df| (dx)dy | Df|(dx)(dy)|
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_0 | () | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_1 | (x)(y) | ((x, y)) | (y) | (x) | () |
| | | | | | |
| f_2 | (x) y | (x, y) | y | (x) | () |
| | | | | | |
| f_4 | x (y) | (x, y) | (y) | x | () |
| | | | | | |
| f_8 | x y | ((x, y)) | y | x | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_3 | (x) | (()) | (()) | () | () |
| | | | | | |
| f_12 | x | (()) | (()) | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_6 | (x, y) | () | (()) | (()) | () |
| | | | | | |
| f_9 | ((x, y)) | () | (()) | (()) | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_5 | (y) | (()) | () | (()) | () |
| | | | | | |
| f_10 | y | (()) | () | (()) | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_7 | (x y) | ((x, y)) | y | x | () |
| | | | | | |
| f_11 | (x (y)) | (x, y) | (y) | x | () |
| | | | | | |
| f_13 | ((x) y) | (x, y) | y | (x) | () |
| | | | | | |
| f_14 | ((x)(y)) | ((x, y)) | (y) | (x) | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_15 | (()) | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
Table A5. Ef Expanded Over Ordinary Features {x, y}
o------o------------o------------o------------o------------o------------o
| | | | | | |
| | f | Ef | xy | Ef | x(y) | Ef | (x)y | Ef | (x)(y)|
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_0 | () | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_1 | (x)(y) | dx dy | dx (dy) | (dx) dy | (dx)(dy) |
| | | | | | |
| f_2 | (x) y | dx (dy) | dx dy | (dx)(dy) | (dx) dy |
| | | | | | |
| f_4 | x (y) | (dx) dy | (dx)(dy) | dx dy | dx (dy) |
| | | | | | |
| f_8 | x y | (dx)(dy) | (dx) dy | dx (dy) | dx dy |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_3 | (x) | dx | dx | (dx) | (dx) |
| | | | | | |
| f_12 | x | (dx) | (dx) | dx | dx |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_6 | (x, y) | (dx, dy) | ((dx, dy)) | ((dx, dy)) | (dx, dy) |
| | | | | | |
| f_9 | ((x, y)) | ((dx, dy)) | (dx, dy) | (dx, dy) | ((dx, dy)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_5 | (y) | dy | (dy) | dy | (dy) |
| | | | | | |
| f_10 | y | (dy) | dy | (dy) | dy |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_7 | (x y) | ((dx)(dy)) | ((dx) dy) | (dx (dy)) | (dx dy) |
| | | | | | |
| f_11 | (x (y)) | ((dx) dy) | ((dx)(dy)) | (dx dy) | (dx (dy)) |
| | | | | | |
| f_13 | ((x) y) | (dx (dy)) | (dx dy) | ((dx)(dy)) | ((dx) dy) |
| | | | | | |
| f_14 | ((x)(y)) | (dx dy) | (dx (dy)) | ((dx) dy) | ((dx)(dy)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_15 | (()) | (()) | (()) | (()) | (()) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
Table A6. Df Expanded Over Ordinary Features {x, y}
o------o------------o------------o------------o------------o------------o
| | | | | | |
| | f | Df | xy | Df | x(y) | Df | (x)y | Df | (x)(y)|
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_0 | () | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_1 | (x)(y) | dx dy | dx (dy) | (dx) dy | ((dx)(dy)) |
| | | | | | |
| f_2 | (x) y | dx (dy) | dx dy | ((dx)(dy)) | (dx) dy |
| | | | | | |
| f_4 | x (y) | (dx) dy | ((dx)(dy)) | dx dy | dx (dy) |
| | | | | | |
| f_8 | x y | ((dx)(dy)) | (dx) dy | dx (dy) | dx dy |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_3 | (x) | dx | dx | dx | dx |
| | | | | | |
| f_12 | x | dx | dx | dx | dx |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_6 | (x, y) | (dx, dy) | (dx, dy) | (dx, dy) | (dx, dy) |
| | | | | | |
| f_9 | ((x, y)) | (dx, dy) | (dx, dy) | (dx, dy) | (dx, dy) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_5 | (y) | dy | dy | dy | dy |
| | | | | | |
| f_10 | y | dy | dy | dy | dy |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_7 | (x y) | ((dx)(dy)) | (dx) dy | dx (dy) | dx dy |
| | | | | | |
| f_11 | (x (y)) | (dx) dy | ((dx)(dy)) | dx dy | dx (dy) |
| | | | | | |
| f_13 | ((x) y) | dx (dy) | dx dy | ((dx)(dy)) | (dx) dy |
| | | | | | |
| f_14 | ((x)(y)) | dx dy | dx (dy) | (dx) dy | ((dx)(dy)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_15 | (()) | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
Wiki Tables : New Versions
Propositional Forms on Two Variables
Table A1. Propositional Forms on Two Variables
| L1
|
L2
|
L3
|
L4
|
L5
|
L6
|
|
|
x :
|
1 1 0 0
|
|
|
|
|
|
y :
|
1 0 1 0
|
|
|
|
| f0
|
f0000
|
0 0 0 0
|
( )
|
false
|
0
|
| f1
|
f0001
|
0 0 0 1
|
(x)(y)
|
neither x nor y
|
¬x ∧ ¬y
|
| f2
|
f0010
|
0 0 1 0
|
(x) y
|
y and not x
|
¬x ∧ y
|
| f3
|
f0011
|
0 0 1 1
|
(x)
|
not x
|
¬x
|
| f4
|
f0100
|
0 1 0 0
|
x (y)
|
x and not y
|
x ∧ ¬y
|
| f5
|
f0101
|
0 1 0 1
|
(y)
|
not y
|
¬y
|
| f6
|
f0110
|
0 1 1 0
|
(x, y)
|
x not equal to y
|
x ≠ y
|
| f7
|
f0111
|
0 1 1 1
|
(x y)
|
not both x and y
|
¬x ∨ ¬y
|
| f8
|
f1000
|
1 0 0 0
|
x y
|
x and y
|
x ∧ y
|
| f9
|
f1001
|
1 0 0 1
|
((x, y))
|
x equal to y
|
x = y
|
| f10
|
f1010
|
1 0 1 0
|
y
|
y
|
y
|
| f11
|
f1011
|
1 0 1 1
|
(x (y))
|
not x without y
|
x ⇒ y
|
| f12
|
f1100
|
1 1 0 0
|
x
|
x
|
x
|
| f13
|
f1101
|
1 1 0 1
|
((x) y)
|
not y without x
|
x ⇐ y
|
| f14
|
f1110
|
1 1 1 0
|
((x)(y))
|
x or y
|
x ∨ y
|
| f15
|
f1111
|
1 1 1 1
|
(( ))
|
true |
1
|
Table A2. Propositional Forms on Two Variables
| L1
|
L2
|
L3
|
L4
|
L5
|
L6
|
|
|
x :
|
1 1 0 0
|
|
|
|
|
|
y :
|
1 0 1 0
|
|
|
|
| f0
|
f0000
|
0 0 0 0
|
( )
|
false
|
0
|
|
|
|
|
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
|
|
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¬A ∧ ¬dA
¬A ∧ dA
A ∧ ¬dA
A ∧ dA
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A not equal to dA
A equal to dA
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0 1 1 1
1 0 1 1
1 1 0 1
1 1 1 0
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(A dA)
(A (dA))
((A) dA)
((A)(dA))
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Not both A and dA
Not A without dA
Not dA without A
A or dA
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¬A ∨ ¬dA
A → dA
A ← dA
A ∨ dA
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| f3
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g15
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1 1 1 1
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(( ))
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True
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1
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Wiki TeX Tables
\(\text{Table A1.}~~\text{Propositional Forms on Two Variables}\)
|
\(\mathcal{L}_1\)
\(\text{Decimal}\)
|
\(\mathcal{L}_2\)
\(\text{Binary}\)
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\(\mathcal{L}_3\)
\(\text{Vector}\)
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\(\mathcal{L}_4\)
\(\text{Cactus}\)
|
\(\mathcal{L}_5\)
\(\text{English}\)
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\(\mathcal{L}_6\)
\(\text{Ordinary}\)
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|
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\(x\colon\!\)
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\(1~1~0~0\!\)
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\(y\colon\!\)
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\(1~0~1~0\!\)
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| \(f_{0}\!\)
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\(f_{0000}\!\)
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\(0~0~0~0\!\)
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\((~)\!\)
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\(\text{false}\!\)
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\(0\!\)
|
| \(f_{1}\!\)
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\(f_{0001}\!\)
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\(0~0~0~1\!\)
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\((x)(y)\!\)
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\(\text{neither}~ x ~\text{nor}~ y\!\)
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\(\lnot x \land \lnot y\!\)
|
| \(f_{2}\!\)
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\(f_{0010}\!\)
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\(0~0~1~0\!\)
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\((x)~y\!\)
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\(y ~\text{without}~ x\!\)
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\(\lnot x \land y\!\)
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| \(f_{3}\!\)
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\(f_{0011}\!\)
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\(0~0~1~1\!\)
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\((x)\!\)
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\(\text{not}~ x\!\)
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\(\lnot x\!\)
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| \(f_{4}\!\)
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\(f_{0100}\!\)
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\(0~1~0~0\!\)
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\(x~(y)\!\)
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\(x ~\text{without}~ y\!\)
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\(x \land \lnot y\!\)
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| \(f_{5}\!\)
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\(f_{0101}\!\)
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\(0~1~0~1\!\)
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\((y)\!\)
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\(\text{not}~ y\!\)
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\(\lnot y\!\)
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| \(f_{6}\!\)
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\(f_{0110}\!\)
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\(0~1~1~0\!\)
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\((x,~y)\!\)
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\(x ~\text{not equal to}~ y\!\)
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\(x \ne y\!\)
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| \(f_{7}\!\)
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\(f_{0111}\!\)
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\(0~1~1~1\!\)
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\((x~y)\!\)
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\(\text{not both}~ x ~\text{and}~ y\!\)
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\(\lnot x \lor \lnot y\!\)
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| \(f_{8}\!\)
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\(f_{1000}\!\)
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\(1~0~0~0\!\)
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\(x~y\!\)
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\(x ~\text{and}~ y\!\)
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\(x \land y\!\)
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| \(f_{9}\!\)
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\(f_{1001}\!\)
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\(1~0~0~1\!\)
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\(((x,~y))\!\)
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\(x ~\text{equal to}~ y\!\)
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\(x = y\!\)
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| \(f_{10}\!\)
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\(f_{1010}\!\)
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\(1~0~1~0\!\)
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\(y\!\)
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\(y\!\)
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\(y\!\)
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| \(f_{11}\!\)
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\(f_{1011}\!\)
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\(1~0~1~1\!\)
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\((x~(y))\!\)
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\(\text{not}~ x ~\text{without}~ y\!\)
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\(x \Rightarrow y\!\)
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| \(f_{12}\!\)
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\(f_{1100}\!\)
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\(1~1~0~0\!\)
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\(x\!\)
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\(x\!\)
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\(x\!\)
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| \(f_{13}\!\)
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\(f_{1101}\!\)
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\(1~1~0~1\!\)
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\(((x)~y)\!\)
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\(\text{not}~ y ~\text{without}~ x\!\)
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\(x \Leftarrow y\!\)
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| \(f_{14}\!\)
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\(f_{1110}\!\)
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\(1~1~1~0\!\)
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\(((x)(y))\!\)
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\(x ~\text{or}~ y\!\)
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\(x \lor y\!\)
|
| \(f_{15}\!\)
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\(f_{1111}\!\)
|
\(1~1~1~1\!\)
|
\(((~))\!\)
|
\(\text{true}\!\)
|
\(1\!\)
|
\(\text{Table A2.}~~\text{Propositional Forms on Two Variables}\)
|
\(\mathcal{L}_1\)
\(\text{Decimal}\)
|
\(\mathcal{L}_2\)
\(\text{Binary}\)
|
\(\mathcal{L}_3\)
\(\text{Vector}\)
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\(\mathcal{L}_4\)
\(\text{Cactus}\)
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\(\mathcal{L}_5\)
\(\text{English}\)
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\(\mathcal{L}_6\)
\(\text{Ordinary}\)
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|
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\(x\colon\!\)
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\(1~1~0~0\!\)
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\(y\colon\!\)
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\(1~0~1~0\!\)
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| f0
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f0000
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0 0 0 0
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( )
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false
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0
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0 0 0 1
0 0 1 0
0 1 0 0
1 0 0 0
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neither x nor y
not x but y
x but not y
x and y
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¬x ∧ ¬y
¬x ∧ y
x ∧ ¬y
x ∧ y
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x not equal to y
x equal to y
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0 1 1 1
1 0 1 1
1 1 0 1
1 1 1 0
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(x y)
(x (y))
((x) y)
((x)(y))
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not both x and y
not x without y
not y without x
x or y
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¬x ∨ ¬y
x ⇒ y
x ⇐ y
x ∨ y
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| f15
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f1111
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1 1 1 1
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(( ))
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true
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1
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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
|
|
|
|
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
|
|
|
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
|
|
|
|
|
|
|
|
| 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
|
|
|
|
|
| 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
|
|
|
|
|
|
|
|
| 0000
|
| 0001
|
| 0010
|
| 0011
|
| 0100
|
| 0101
|
| 0110
|
| 0111
|
|
| ()
|
| (u)(v)
|
| (u) v
|
| (u)
|
| u (v)
|
| (v)
|
| (u, v)
|
| (u v)
|
|
|
|
| f8
|
| f9
|
| f10
|
| f11
|
| f12
|
| f13
|
| f14
|
| f15
|
|
| 1000
|
| 1001
|
| 1010
|
| 1011
|
| 1100
|
| 1101
|
| 1110
|
| 1111
|
|
| u v
|
| ((u, v))
|
| v
|
| (u (v))
|
| u
|
| ((u) v)
|
| ((u)(v))
|
| (())
|
|
| f8
|
| f9
|
| f10
|
| f11
|
| f12
|
| f13
|
| f14
|
| f15
|
|
| f8
|
| f9
|
| f10
|
| f11
|
| f12
|
| f13
|
| f14
|
| f15
|
|
Higher Order Propositions
Table 7. Higher Order Propositions (n = 1)
| \ x |
1 0 |
F
|
m |
m |
m |
m |
m |
m |
m |
m
|
m |
m |
m |
m |
m |
m |
m |
m
|
| F \ |
|
|
00 |
01 |
02 |
03 |
04 |
05 |
06 |
07 |
08 |
09 |
10 |
11 |
12 |
13 |
14 |
15
|
| F0 |
0 0 |
0 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1
|
| F1 |
0 1 |
(x) |
0 |
0 |
1 |
1 |
0 |
0 |
1 |
1 |
0 |
0 |
1 |
1 |
0 |
0 |
1 |
1
|
| F2 |
1 0 |
x |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
1 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
1
|
| F3 |
1 1 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1
|
Table 8. Interpretive Categories for Higher Order Propositions (n = 1)
| Measure |
Happening |
Exactness |
Existence |
Linearity |
Uniformity |
Information
|
| m0 |
nothing happens |
|
|
|
|
|
| m1 |
|
just false |
nothing exists |
|
|
|
| m2 |
|
just not x |
|
|
|
|
| m3 |
|
|
nothing is x |
|
|
|
| m4 |
|
just x |
|
|
|
|
| m5 |
|
|
everything is x |
F is linear |
|
|
| m6 |
|
|
|
|
F is not uniform |
F is informed
|
| m7 |
|
not just true |
|
|
|
|
| m8 |
|
just true |
|
|
|
|
| m9 |
|
|
|
|
F is uniform |
F is not informed
|
| m10 |
|
|
something is not x |
F is not linear |
|
|
| m11 |
|
not just x |
|
|
|
|
| m12 |
|
|
something is x |
|
|
|
| m13 |
|
not just not x |
|
|
|
|
| m14 |
|
not just false |
something exists |
|
|
|
| m15 |
anything happens |
|
|
|
|
|
Table 9. Higher Order Propositions (n = 2)
| x : |
1100 |
f
|
m |
m |
m |
m |
m |
m |
m |
m
|
m |
m |
m |
m |
m |
m |
m |
m
|
m |
m |
m |
m |
m |
m |
m |
m
|
| y : |
1010 |
|
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12
|
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
21 |
22 |
23
|
| f0 |
0000 |
( )
|
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1
|
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1
|
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1
|
| f1 |
0001 |
(x)(y)
|
|
|
1 |
1 |
0 |
0 |
1 |
1
|
0 |
0 |
1 |
1 |
0 |
0 |
1 |
1
|
0 |
0 |
1 |
1 |
0 |
0 |
1 |
1
|
| f2 |
0010 |
(x) y
|
|
|
|
|
1 |
1 |
1 |
1
|
0 |
0 |
0 |
0 |
1 |
1 |
1 |
1
|
0 |
0 |
0 |
0 |
1 |
1 |
1 |
1
|
| f3 |
0011 |
(x)
|
|
|
|
|
|
|
|
|
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0
|
| f4 |
0100 |
x (y)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1
|
| f5 |
0101 |
(y)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| f6 |
0110 |
(x, y)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| f7 |
0111 |
(x y)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| f8 |
1000 |
x y
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| f9 |
1001 |
((x, y))
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| f10 |
1010 |
y
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| f11 |
1011 |
(x (y))
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| f12 |
1100 |
x
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| f13 |
1101 |
((x) y)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| f14 |
1110 |
((x)(y))
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| f15 |
1111 |
(( ))
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Table 10. Qualifiers of Implication Ordering: αi f = Υ(fi ⇒ f)
| x : |
1100 |
f
|
α |
α |
α |
α |
α |
α |
α |
α
|
α |
α |
α |
α |
α |
α |
α |
α
|
| y : |
1010 |
|
15 |
14 |
13 |
12 |
11 |
10 |
9 |
8 |
7 |
6 |
5 |
4 |
3 |
2 |
1 |
0
|
| f0 |
0000 |
( )
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1
|
| f1 |
0001 |
(x)(y)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
1
|
| f2 |
0010 |
(x) y
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
|
1
|
| f3 |
0011 |
(x)
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
1 |
1 |
1
|
| f4 |
0100 |
x (y)
|
|
|
|
|
|
|
|
|
|
|
|
1 |
|
|
|
1
|
| f5 |
0101 |
(y)
|
|
|
|
|
|
|
|
|
|
|
1 |
1 |
|
|
1 |
1
|
| f6 |
0110 |
(x, y)
|
|
|
|
|
|
|
|
|
|
1 |
|
1 |
|
1 |
|
1
|
| f7 |
0111 |
(x y)
|
|
|
|
|
|
|
|
|
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1
|
| f8 |
1000 |
x y
|
|
|
|
|
|
|
|
1
|
|
|
|
|
|
|
|
1
|
| f9 |
1001 |
((x, y))
|
|
|
|
|
|
|
1 |
1
|
|
|
|
|
|
|
1 |
1
|
| f10 |
1010 |
y
|
|
|
|
|
|
1 |
|
1
|
|
|
|
|
|
1 |
|
1
|
| f11 |
1011 |
(x (y))
|
|
|
|
|
1 |
1 |
1 |
1
|
|
|
|
|
1 |
1 |
1 |
1
|
| f12 |
1100 |
x
|
|
|
|
1 |
|
|
|
1
|
|
|
|
1 |
|
|
|
1
|
| f13 |
1101 |
((x) y)
|
|
|
1 |
1 |
|
|
1 |
1
|
|
|
1 |
1 |
|
|
1 |
1
|
| f14 |
1110 |
((x)(y))
|
|
1 |
|
1 |
|
1 |
|
1
|
|
1 |
|
1 |
|
1 |
|
1
|
| f15 |
1111 |
(( ))
|
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1
|
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1
|
Table 11. Qualifiers of Implication Ordering: βi f = Υ(f ⇒ fi)
| x : |
1100 |
f
|
β |
β |
β |
β |
β |
β |
β |
β
|
β |
β |
β |
β |
β |
β |
β |
β
|
| y : |
1010 |
|
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15
|
| f0 |
0000 |
( )
|
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1
|
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1
|
| f1 |
0001 |
(x)(y)
|
|
1 |
|
1 |
|
1 |
|
1
|
|
1 |
|
1 |
|
1 |
|
1
|
| f2 |
0010 |
(x) y
|
|
|
1 |
1 |
|
|
1 |
1
|
|
|
1 |
1 |
|
|
1 |
1
|
| f3 |
0011 |
(x)
|
|
|
|
1 |
|
|
|
1
|
|
|
|
1 |
|
|
|
1
|
| f4 |
0100 |
x (y)
|
|
|
|
|
1 |
1 |
1 |
1
|
|
|
|
|
1 |
1 |
1 |
1
|
| f5 |
0101 |
(y)
|
|
|
|
|
|
1 |
|
1
|
|
|
|
|
|
1 |
|
1
|
| f6 |
0110 |
(x, y)
|
|
|
|
|
|
|
1 |
1
|
|
|
|
|
|
|
1 |
1
|
| f7 |
0111 |
(x y)
|
|
|
|
|
|
|
|
1
|
|
|
|
|
|
|
|
1
|
| f8 |
1000 |
x y
|
|
|
|
|
|
|
|
|
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1
|
| f9 |
1001 |
((x, y))
|
|
|
|
|
|
|
|
|
|
1 |
|
1 |
|
1 |
|
1
|
| f10 |
1010 |
y
|
|
|
|
|
|
|
|
|
|
|
1 |
1 |
|
|
1 |
1
|
| f11 |
1011 |
(x (y))
|
|
|
|
|
|
|
|
|
|
|
|
1 |
|
|
|
1
|
| f12 |
1100 |
x
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
1 |
1 |
1
|
| f13 |
1101 |
((x) y)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
|
1
|
| f14 |
1110 |
((x)(y))
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
1
|
| f15 |
1111 |
(( ))
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1
|
Table 13. Syllogistic Premisses as Higher Order Indicator Functions
| A
|
Universal Affirmative
|
All
|
x |
is |
y
|
Indicator of " x (y)" = 0
|
| E
|
Universal Negative
|
All
|
x |
is |
(y)
|
Indicator of " x y " = 0
|
| I
|
Particular Affirmative
|
Some
|
x |
is |
y
|
Indicator of " x y " = 1
|
| O
|
Particular Negative
|
Some
|
x |
is |
(y)
|
Indicator of " x (y)" = 1
|
Table 14. Relation of Quantifiers to Higher Order Propositions
| Mnemonic |
Category |
Classical Form |
Alternate Form |
Symmetric Form |
Operator
|
E
Exclusive
|
Universal
Negative
|
All x is (y)
|
|
No x is y
|
(L11)
|
A
Absolute
|
Universal
Affirmative
|
All x is y
|
|
No x is (y)
|
(L10)
|
|
|
|
All y is x
|
No y is (x)
|
No (x) is y
|
(L01)
|
|
|
|
All (y) is x
|
No (y) is (x)
|
No (x) is (y)
|
(L00)
|
|
|
|
Some (x) is (y)
|
|
Some (x) is (y)
|
L00
|
|
|
|
Some (x) is y
|
|
Some (x) is y
|
L01
|
O
Obtrusive
|
Particular
Negative
|
Some x is (y)
|
|
Some x is (y)
|
L10
|
I
Indefinite
|
Particular
Affirmative
|
Some x is y
|
|
Some x is y
|
L11
|
Table 15. Simple Qualifiers of Propositions (n = 2)
| x : |
1100 |
f
|
(L11)
|
(L10)
|
(L01)
|
(L00)
|
L00
|
L01
|
L10
|
L11
|
| y : |
1010 |
|
no x
is y
|
no x
is (y)
|
no (x)
is y
|
no (x)
is (y)
|
some (x)
is (y)
|
some (x)
is y
|
some x
is (y)
|
some x
is y
|
| f0 |
0000 |
( )
|
1 |
1 |
1 |
1 |
0 |
0 |
0 |
0
|
| f1 |
0001 |
(x)(y)
|
1 |
1 |
1 |
0 |
1 |
0 |
0 |
0
|
| f2 |
0010 |
(x) y
|
1 |
1 |
0 |
1 |
0 |
1 |
0 |
0
|
| f3 |
0011 |
(x)
|
1 |
1 |
0 |
0 |
1 |
1 |
0 |
0
|
| f4 |
0100 |
x (y)
|
1 |
0 |
1 |
1 |
0 |
0 |
1 |
0
|
| f5 |
0101 |
(y)
|
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0
|
| f6 |
0110 |
(x, y)
|
1 |
0 |
0 |
1 |
0 |
1 |
1 |
0
|
| f7 |
0111 |
(x y)
|
1 |
0 |
0 |
0 |
1 |
1 |
1 |
0
|
| f8 |
1000 |
x y
|
0 |
1 |
1 |
1 |
0 |
0 |
0 |
1
|
| f9 |
1001 |
((x, y))
|
0 |
1 |
1 |
0 |
1 |
0 |
0 |
1
|
| f10 |
1010 |
y
|
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1
|
| f11 |
1011 |
(x (y))
|
0 |
1 |
0 |
0 |
1 |
1 |
0 |
1
|
| f12 |
1100 |
x
|
0 |
0 |
1 |
1 |
0 |
0 |
1 |
1
|
| f13 |
1101 |
((x) y)
|
0 |
0 |
1 |
0 |
1 |
0 |
1 |
1
|
| f14 |
1110 |
((x)(y))
|
0 |
0 |
0 |
1 |
0 |
1 |
1 |
1
|
| f15 |
1111 |
(( ))
|
0 |
0 |
0 |
0 |
1 |
1 |
1 |
1
|
Table 7. Higher Order Propositions (n = 1)
o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o
| \ x | 1 0 | F |m |m |m |m |m |m |m |m |m |m |m |m |m |m |m |m |
| F \ | | |00|01|02|03|04|05|06|07|08|09|10|11|12|13|14|15 |
o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o
| | | | |
| F_0 | 0 0 | 0 | 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 |
| | | | |
| F_1 | 0 1 | (x) | 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 |
| | | | |
| F_2 | 1 0 | x | 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 |
| | | | |
| F_3 | 1 1 | 1 | 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 |
| | | | |
o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o
Table 8. Interpretive Categories for Higher Order Propositions (n = 1)
o-------o----------o------------o------------o----------o----------o-----------o
|Measure| Happening| Exactness | Existence | Linearity|Uniformity|Information|
o-------o----------o------------o------------o----------o----------o-----------o
| m_0 | nothing | | | | | |
| | happens | | | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_1 | | | nothing | | | |
| | | just false | exists | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_2 | | | | | | |
| | | just not x | | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_3 | | | nothing | | | |
| | | | is x | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_4 | | | | | | |
| | | just x | | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_5 | | | everything | F is | | |
| | | | is x | linear | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_6 | | | | | F is not | F is |
| | | | | | uniform | informed |
o-------o----------o------------o------------o----------o----------o-----------o
| m_7 | | not | | | | |
| | | just true | | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_8 | | | | | | |
| | | just true | | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_9 | | | | | F is | F is not |
| | | | | | uniform | informed |
o-------o----------o------------o------------o----------o----------o-----------o
| m_10 | | | something | F is not | | |
| | | | is not x | linear | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_11 | | not | | | | |
| | | just x | | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_12 | | | something | | | |
| | | | is x | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_13 | | not | | | | |
| | | just not x | | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_14 | | not | something | | | |
| | | just false | exists | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_15 | anything | | | | | |
| | happens | | | | | |
o-------o----------o------------o------------o----------o----------o-----------o
Table 9. Higher Order Propositions (n = 2)
o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
| | x | 1100 | f |m|m|m|m|m|m|m|m|m|m|m|m|m|m|m|m|.|
| | y | 1010 | |0|0|0|0|0|0|0|0|0|0|1|1|1|1|1|1|.|
| f \ | | |0|1|2|3|4|5|6|7|8|9|0|1|2|3|4|5|.|
o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
| | | | |
| f_0 | 0000 | () |0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 |
| | | | |
| f_1 | 0001 | (x)(y) | 1 1 0 0 1 1 0 0 1 1 0 0 1 1 |
| | | | |
| f_2 | 0010 | (x) y | 1 1 1 1 0 0 0 0 1 1 1 1 |
| | | | |
| f_3 | 0011 | (x) | 1 1 1 1 1 1 1 1 |
| | | | |
| f_4 | 0100 | x (y) | |
| | | | |
| f_5 | 0101 | (y) | |
| | | | |
| f_6 | 0110 | (x, y) | |
| | | | |
| f_7 | 0111 | (x y) | |
| | | | |
o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
| | | | |
| f_8 | 1000 | x y | |
| | | | |
| f_9 | 1001 | ((x, y)) | |
| | | | |
| f_10 | 1010 | y | |
| | | | |
| f_11 | 1011 | (x (y)) | |
| | | | |
| f_12 | 1100 | x | |
| | | | |
| f_13 | 1101 | ((x) y) | |
| | | | |
| f_14 | 1110 | ((x)(y)) | |
| | | | |
| f_15 | 1111 | (()) | |
| | | | |
o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
Table 10. Qualifiers of Implication Ordering: !a!_i f = !Y!(f_i => f)
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
| | x | 1100 | f |a |a |a |a |a |a |a |a |a |a |a |a |a |a |a |a |
| | y | 1010 | |1 |1 |1 |1 |1 |1 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |
| f \ | | |5 |4 |3 |2 |1 |0 |9 |8 |7 |6 |5 |4 |3 |2 |1 |0 |
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
| | | | |
| f_0 | 0000 | () | 1 |
| | | | |
| f_1 | 0001 | (x)(y) | 1 1 |
| | | | |
| f_2 | 0010 | (x) y | 1 1 |
| | | | |
| f_3 | 0011 | (x) | 1 1 1 1 |
| | | | |
| f_4 | 0100 | x (y) | 1 1 |
| | | | |
| f_5 | 0101 | (y) | 1 1 1 1 |
| | | | |
| f_6 | 0110 | (x, y) | 1 1 1 1 |
| | | | |
| f_7 | 0111 | (x y) | 1 1 1 1 1 1 1 1 |
| | | | |
| f_8 | 1000 | x y | 1 1 |
| | | | |
| f_9 | 1001 | ((x, y)) | 1 1 1 1 |
| | | | |
| f_10 | 1010 | y | 1 1 1 1 |
| | | | |
| f_11 | 1011 | (x (y)) | 1 1 1 1 1 1 1 1 |
| | | | |
| f_12 | 1100 | x | 1 1 1 1 |
| | | | |
| f_13 | 1101 | ((x) y) | 1 1 1 1 1 1 1 1 |
| | | | |
| f_14 | 1110 | ((x)(y)) | 1 1 1 1 1 1 1 1 |
| | | | |
| f_15 | 1111 | (()) |1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
| | | | |
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
Table 11. Qualifiers of Implication Ordering: !b!_i f = !Y!(f => f_i)
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
| | x | 1100 | f |b |b |b |b |b |b |b |b |b |b |b |b |b |b |b |b |
| | y | 1010 | |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |1 |1 |1 |1 |1 |1 |
| f \ | | |0 |1 |2 |3 |4 |5 |6 |7 |8 |9 |0 |1 |2 |3 |4 |5 |
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
| | | | |
| f_0 | 0000 | () |1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
| | | | |
| f_1 | 0001 | (x)(y) | 1 1 1 1 1 1 1 1 |
| | | | |
| f_2 | 0010 | (x) y | 1 1 1 1 1 1 1 1 |
| | | | |
| f_3 | 0011 | (x) | 1 1 1 1 |
| | | | |
| f_4 | 0100 | x (y) | 1 1 1 1 1 1 1 1 |
| | | | |
| f_5 | 0101 | (y) | 1 1 1 1 |
| | | | |
| f_6 | 0110 | (x, y) | 1 1 1 1 |
| | | | |
| f_7 | 0111 | (x y) | 1 1 |
| | | | |
| f_8 | 1000 | x y | 1 1 1 1 1 1 1 1 |
| | | | |
| f_9 | 1001 | ((x, y)) | 1 1 1 1 |
| | | | |
| f_10 | 1010 | y | 1 1 1 1 |
| | | | |
| f_11 | 1011 | (x (y)) | 1 1 |
| | | | |
| f_12 | 1100 | x | 1 1 1 1 |
| | | | |
| f_13 | 1101 | ((x) y) | 1 1 |
| | | | |
| f_14 | 1110 | ((x)(y)) | 1 1 |
| | | | |
| f_15 | 1111 | (()) | 1 |
| | | | |
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
Table 13. Syllogistic Premisses as Higher Order Indicator Functions
o---o------------------------o-----------------o---------------------------o
| | | | |
| A | Universal Affirmative | All x is y | Indicator of " x (y)" = 0 |
| | | | |
| E | Universal Negative | All x is (y) | Indicator of " x y " = 0 |
| | | | |
| I | Particular Affirmative | Some x is y | Indicator of " x y " = 1 |
| | | | |
| O | Particular Negative | Some x is (y) | Indicator of " x (y)" = 1 |
| | | | |
o---o------------------------o-----------------o---------------------------o
Table 14. Relation of Quantifiers to Higher Order Propositions
o------------o------------o-----------o-----------o-----------o-----------o
| Mnemonic | Category | Classical | Alternate | Symmetric | Operator |
| | | Form | Form | Form | |
o============o============o===========o===========o===========o===========o
| E | Universal | All x | | No x | (L_11) |
| Exclusive | Negative | is (y) | | is y | |
o------------o------------o-----------o-----------o-----------o-----------o
| A | Universal | All x | | No x | (L_10) |
| Absolute | Affrmtve | is y | | is (y) | |
o------------o------------o-----------o-----------o-----------o-----------o
| | | All y | No y | No (x) | (L_01) |
| | | is x | is (x) | is y | |
o------------o------------o-----------o-----------o-----------o-----------o
| | | All (y) | No (y) | No (x) | (L_00) |
| | | is x | is (x) | is (y) | |
o------------o------------o-----------o-----------o-----------o-----------o
| | | Some (x) | | Some (x) | L_00 |
| | | is (y) | | is (y) | |
o------------o------------o-----------o-----------o-----------o-----------o
| | | Some (x) | | Some (x) | L_01 |
| | | is y | | is y | |
o------------o------------o-----------o-----------o-----------o-----------o
| O | Particular | Some x | | Some x | L_10 |
| Obtrusive | Negative | is (y) | | is (y) | |
o------------o------------o-----------o-----------o-----------o-----------o
| I | Particular | Some x | | Some x | L_11 |
| Indefinite | Affrmtve | is y | | is y | |
o------------o------------o-----------o-----------o-----------o-----------o
Table 15. Simple Qualifiers of Propositions (n = 2)
o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o
| | x | 1100 | f |(L11)|(L10)|(L01)|(L00)| L00 | L01 | L10 | L11 |
| | y | 1010 | |no x|no x|no ~x|no ~x|sm ~x|sm ~x|sm x|sm x|
| f \ | | |is y|is ~y|is y|is ~y|is ~y|is y|is ~y|is y|
o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o
| | | | |
| f_0 | 0000 | () | 1 1 1 1 0 0 0 0 |
| | | | |
| f_1 | 0001 | (x)(y) | 1 1 1 0 1 0 0 0 |
| | | | |
| f_2 | 0010 | (x) y | 1 1 0 1 0 1 0 0 |
| | | | |
| f_3 | 0011 | (x) | 1 1 0 0 1 1 0 0 |
| | | | |
| f_4 | 0100 | x (y) | 1 0 1 1 0 0 1 0 |
| | | | |
| f_5 | 0101 | (y) | 1 0 1 0 1 0 1 0 |
| | | | |
| f_6 | 0110 | (x, y) | 1 0 0 1 0 1 1 0 |
| | | | |
| f_7 | 0111 | (x y) | 1 0 0 0 1 1 1 0 |
| | | | |
| f_8 | 1000 | x y | 0 1 1 1 0 0 0 1 |
| | | | |
| f_9 | 1001 | ((x, y)) | 0 1 1 0 1 0 0 1 |
| | | | |
| f_10 | 1010 | y | 0 1 0 1 0 1 0 1 |
| | | | |
| f_11 | 1011 | (x (y)) | 0 1 0 0 1 1 0 1 |
| | | | |
| f_12 | 1100 | x | 0 0 1 1 0 0 1 1 |
| | | | |
| f_13 | 1101 | ((x) y) | 0 0 1 0 1 0 1 1 |
| | | | |
| f_14 | 1110 | ((x)(y)) | 0 0 0 1 0 1 1 1 |
| | | | |
| f_15 | 1111 | (()) | 0 0 0 0 1 1 1 1 |
| | | | |
o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o
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 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 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 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 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)
\[\begin{matrix}
p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\
& = & ((p \land \lnot q) \lor \lnot p) & \and & ((p \land \lnot q) \lor q) \\
& = & ((p \lor \lnot q) \land (\lnot q \lor \lnot p)) & \land & ((p \lor q) \land (\lnot q \lor q)) \\
& = & (\lnot p \lor \lnot q) & \land & (p \lor q) \\
& = & \lnot (p \land q) & \land & (p \lor q)
\end{matrix}\]
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
|
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
|
The NNOR operation is a logical operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both of its operands are false. In other words, it produces a value of false if and only if at least one of its operands is true.
The truth table of p NNOR q (also written as p ⊥ q or p ↓ q) is as follows:
Logical NOR
| p
|
q
|
p ↓ q
|
| F |
F |
T
|
| F |
T |
F
|
| T |
F |
F
|
| T |
T |
F
|
Relational Tables
Sign Relations
|
|
O |
= |
Object Domain
|
|
|
S |
= |
Sign Domain
|
|
|
I |
= |
Interpretant Domain
|
|
|
O
|
=
|
{Ann, Bob}
|
=
|
{A, B}
|
|
|
S
|
=
|
{"Ann", "Bob", "I", "You"}
|
=
|
{"A", "B", "i", "u"}
|
|
|
I
|
=
|
{"Ann", "Bob", "I", "You"}
|
=
|
{"A", "B", "i", "u"}
|
LA = Sign Relation of Interpreter A
| Object
|
Sign
|
Interpretant
|
| A |
"A" |
"A"
|
| A |
"A" |
"i"
|
| A |
"i" |
"A"
|
| A |
"i" |
"i"
|
| B |
"B" |
"B"
|
| B |
"B" |
"u"
|
| B |
"u" |
"B"
|
| B |
"u" |
"u"
|
LB = Sign Relation of Interpreter B
| Object
|
Sign
|
Interpretant
|
| A |
"A" |
"A"
|
| A |
"A" |
"u"
|
| A |
"u" |
"A"
|
| A |
"u" |
"u"
|
| B |
"B" |
"B"
|
| B |
"B" |
"i"
|
| B |
"i" |
"B"
|
| B |
"i" |
"i"
|
Triadic Relations
Algebraic Examples
L0 = {(x, y, z) ∈ B3 : x + y + z = 0}
| X |
Y |
Z
|
| 0 |
0 |
0
|
| 0 |
1 |
1
|
| 1 |
0 |
1
|
| 1 |
1 |
0
|
L1 = {(x, y, z) ∈ B3 : x + y + z = 1}
| X |
Y |
Z
|
| 0 |
0 |
1
|
| 0 |
1 |
0
|
| 1 |
0 |
0
|
| 1 |
1 |
1
|
Semiotic Examples
LA = Sign Relation of Interpreter A
| Object
|
Sign
|
Interpretant
|
| A |
"A" |
"A"
|
| A |
"A" |
"i"
|
| A |
"i" |
"A"
|
| A |
"i" |
"i"
|
| B |
"B" |
"B"
|
| B |
"B" |
"u"
|
| B |
"u" |
"B"
|
| B |
"u" |
"u"
|
LB = Sign Relation of Interpreter B
| Object
|
Sign
|
Interpretant
|
| A |
"A" |
"A"
|
| A |
"A" |
"u"
|
| A |
"u" |
"A"
|
| A |
"u" |
"u"
|
| B |
"B" |
"B"
|
| B |
"B" |
"i"
|
| B |
"i" |
"B"
|
| B |
"i" |
"i"
|
Dyadic Projections
|
|
LOS
|
=
|
projOS(L)
|
=
|
{ (o, s) ∈ O × S : (o, s, i) ∈ L for some i ∈ I }
|
|
|
LSO
|
=
|
projSO(L)
|
=
|
{ (s, o) ∈ S × O : (o, s, i) ∈ L for some i ∈ I }
|
|
|
LIS
|
=
|
projIS(L)
|
=
|
{ (i, s) ∈ I × S : (o, s, i) ∈ L for some o ∈ O }
|
|
|
LSI
|
=
|
projSI(L)
|
=
|
{ (s, i) ∈ S × I : (o, s, i) ∈ L for some o ∈ O }
|
|
|
LOI
|
=
|
projOI(L)
|
=
|
{ (o, i) ∈ O × I : (o, s, i) ∈ L for some s ∈ S }
|
|
|
LIO
|
=
|
projIO(L)
|
=
|
{ (i, o) ∈ I × O : (o, s, i) ∈ L for some s ∈ S }
|
Method 1 : Subtitles as Captions
projOS(LA)
| Object
|
Sign
|
| A |
"A"
|
| A |
"i"
|
| B |
"B"
|
| B |
"u"
|
|
projOS(LB)
| Object
|
Sign
|
| A |
"A"
|
| A |
"u"
|
| B |
"B"
|
| B |
"i"
|
|
projSI(LA)
| Sign
|
Interpretant
|
| "A" |
"A"
|
| "A" |
"i"
|
| "i" |
"A"
|
| "i" |
"i"
|
| "B" |
"B"
|
| "B" |
"u"
|
| "u" |
"B"
|
| "u" |
"u"
|
|
projSI(LB)
| Sign
|
Interpretant
|
| "A" |
"A"
|
| "A" |
"u"
|
| "u" |
"A"
|
| "u" |
"u"
|
| "B" |
"B"
|
| "B" |
"i"
|
| "i" |
"B"
|
| "i" |
"i"
|
|
projOI(LA)
| Object
|
Interpretant
|
| A |
"A"
|
| A |
"i"
|
| B |
"B"
|
| B |
"u"
|
|
projOI(LB)
| Object
|
Interpretant
|
| A |
"A"
|
| A |
"u"
|
| B |
"B"
|
| B |
"i"
|
|
Method 2 : Subtitles as Top Rows
projOS(LA)
| Object
|
Sign
|
| A |
"A"
|
| A |
"i"
|
| B |
"B"
|
| B |
"u"
|
|
projOS(LB)
| Object
|
Sign
|
| A |
"A"
|
| A |
"u"
|
| B |
"B"
|
| B |
"i"
|
|
projSI(LA)
| Sign
|
Interpretant
|
| "A" |
"A"
|
| "A" |
"i"
|
| "i" |
"A"
|
| "i" |
"i"
|
| "B" |
"B"
|
| "B" |
"u"
|
| "u" |
"B"
|
| "u" |
"u"
|
|
projSI(LB)
| Sign
|
Interpretant
|
| "A" |
"A"
|
| "A" |
"u"
|
| "u" |
"A"
|
| "u" |
"u"
|
| "B" |
"B"
|
| "B" |
"i"
|
| "i" |
"B"
|
| "i" |
"i"
|
|
projOI(LA)
| Object
|
Interpretant
|
| A |
"A"
|
| A |
"i"
|
| B |
"B"
|
| B |
"u"
|
|
projOI(LB)
| Object
|
Interpretant
|
| A |
"A"
|
| A |
"u"
|
| B |
"B"
|
| B |
"i"
|
|
Relation Reduction
Method 1 : Subtitles as Captions
L0 = {(x, y, z) ∈ B3 : x + y + z = 0}
| X |
Y |
Z
|
| 0 |
0 |
0
|
| 0 |
1 |
1
|
| 1 |
0 |
1
|
| 1 |
1 |
0
|
L1 = {(x, y, z) ∈ B3 : x + y + z = 1}
| X |
Y |
Z
|
| 0 |
0 |
1
|
| 0 |
1 |
0
|
| 1 |
0 |
0
|
| 1 |
1 |
1
|
projXY(L0)
| X |
Y
|
| 0 |
0
|
| 0 |
1
|
| 1 |
0
|
| 1 |
1
|
|
projXZ(L0)
| X |
Z
|
| 0 |
0
|
| 0 |
1
|
| 1 |
1
|
| 1 |
0
|
|
projYZ(L0)
| Y |
Z
|
| 0 |
0
|
| 1 |
1
|
| 0 |
1
|
| 1 |
0
|
|
projXY(L1)
| X |
Y
|
| 0 |
0
|
| 0 |
1
|
| 1 |
0
|
| 1 |
1
|
|
projXZ(L1)
| X |
Z
|
| 0 |
1
|
| 0 |
0
|
| 1 |
0
|
| 1 |
1
|
|
projYZ(L1)
| Y |
Z
|
| 0 |
1
|
| 1 |
0
|
| 0 |
0
|
| 1 |
1
|
|
| projXY(L0) = projXY(L1)
|
projXZ(L0) = projXZ(L1)
|
projYZ(L0) = projYZ(L1)
|
LA = Sign Relation of Interpreter A
| Object
|
Sign
|
Interpretant
|
| A |
"A" |
"A"
|
| A |
"A" |
"i"
|
| A |
"i" |
"A"
|
| A |
"i" |
"i"
|
| B |
"B" |
"B"
|
| B |
"B" |
"u"
|
| B |
"u" |
"B"
|
| B |
"u" |
"u"
|
LB = Sign Relation of Interpreter B
| Object
|
Sign
|
Interpretant
|
| A |
"A" |
"A"
|
| A |
"A" |
"u"
|
| A |
"u" |
"A"
|
| A |
"u" |
"u"
|
| B |
"B" |
"B"
|
| B |
"B" |
"i"
|
| B |
"i" |
"B"
|
| B |
"i" |
"i"
|
projXY(LA)
| Object
|
Sign
|
| A |
"A"
|
| A |
"i"
|
| B |
"B"
|
| B |
"u"
|
|
projXZ(LA)
| Object
|
Interpretant
|
| A |
"A"
|
| A |
"i"
|
| B |
"B"
|
| B |
"u"
|
|
projYZ(LA)
| Sign
|
Interpretant
|
| "A" |
"A"
|
| "A" |
"i"
|
| "i" |
"A"
|
| "i" |
"i"
|
| "B" |
"B"
|
| "B" |
"u"
|
| "u" |
"B"
|
| "u" |
"u"
|
|
projXY(LB)
| Object
|
Sign
|
| A |
"A"
|
| A |
"u"
|
| B |
"B"
|
| B |
"i"
|
|
projXZ(LB)
| Object
|
Interpretant
|
| A |
"A"
|
| A |
"u"
|
| B |
"B"
|
| B |
"i"
|
|
projYZ(LB)
| Sign
|
Interpretant
|
| "A" |
"A"
|
| "A" |
"u"
|
| "u" |
"A"
|
| "u" |
"u"
|
| "B" |
"B"
|
| "B" |
"i"
|
| "i" |
"B"
|
| "i" |
"i"
|
|
| projXY(LA) ≠ projXY(LB)
|
projXZ(LA) ≠ projXZ(LB)
|
projYZ(LA) ≠ projYZ(LB)
|
Method 2 : Subtitles as Top Rows
L0 = {(x, y, z) ∈ B3 : x + y + z = 0}
| X |
Y |
Z
|
| 0 |
0 |
0
|
| 0 |
1 |
1
|
| 1 |
0 |
1
|
| 1 |
1 |
0
|
L1 = {(x, y, z) ∈ B3 : x + y + z = 1}
| X |
Y |
Z
|
| 0 |
0 |
1
|
| 0 |
1 |
0
|
| 1 |
0 |
0
|
| 1 |
1 |
1
|
| projXY(L0)
|
projXZ(L0)
|
projYZ(L0)
|
| projXY(L1)
|
projXZ(L1)
|
projYZ(L1)
|
| projXY(L0) = projXY(L1)
|
projXZ(L0) = projXZ(L1)
|
projYZ(L0) = projYZ(L1)
|
LA = Sign Relation of Interpreter A
| Object
|
Sign
|
Interpretant
|
| A |
"A" |
"A"
|
| A |
"A" |
"i"
|
| A |
"i" |
"A"
|
| A |
"i" |
"i"
|
| B |
"B" |
"B"
|
| B |
"B" |
"u"
|
| B |
"u" |
"B"
|
| B |
"u" |
"u"
|
LB = Sign Relation of Interpreter B
| Object
|
Sign
|
Interpretant
|
| A |
"A" |
"A"
|
| A |
"A" |
"u"
|
| A |
"u" |
"A"
|
| A |
"u" |
"u"
|
| B |
"B" |
"B"
|
| B |
"B" |
"i"
|
| B |
"i" |
"B"
|
| B |
"i" |
"i"
|
projXY(LA)
| Object
|
Sign
|
| A |
"A"
|
| A |
"i"
|
| B |
"B"
|
| B |
"u"
|
|
projXZ(LA)
| Object
|
Interpretant
|
| A |
"A"
|
| A |
"i"
|
| B |
"B"
|
| B |
"u"
|
|
projYZ(LA)
| Sign
|
Interpretant
|
| "A" |
"A"
|
| "A" |
"i"
|
| "i" |
"A"
|
| "i" |
"i"
|
| "B" |
"B"
|
| "B" |
"u"
|
| "u" |
"B"
|
| "u" |
"u"
|
|
projXY(LB)
| Object
|
Sign
|
| A |
"A"
|
| A |
"u"
|
| B |
"B"
|
| B |
"i"
|
|
projXZ(LB)
| Object
|
Interpretant
|
| A |
"A"
|
| A |
"u"
|
| B |
"B"
|
| B |
"i"
|
|
projYZ(LB)
| Sign
|
Interpretant
|
| "A" |
"A"
|
| "A" |
"u"
|
| "u" |
"A"
|
| "u" |
"u"
|
| "B" |
"B"
|
| "B" |
"i"
|
| "i" |
"B"
|
| "i" |
"i"
|
|
| projXY(LA) ≠ projXY(LB)
|
projXZ(LA) ≠ projXZ(LB)
|
projYZ(LA) ≠ projYZ(LB)
|
Formatted Text Display
- So in a triadic fact, say, the example
- 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
| |
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
| |
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
| \(\text{Mnemonic}\)
|
\(\text{Category}\)
|
\(\text{Classical Form}\)
|
\(\text{Alternate Form}\)
|
\(\text{Symmetric Form}\)
|
\(\text{Operator}\)
|
\(\text{E}\!\)
\(\text{Exclusive}\)
|
\(\text{Universal}\)
\(\text{Negative}\)
|
\(\text{All}\ u\ \text{is}\ (v)\)
|
|
\(\text{No}\ u\ \text{is}\ v \)
|
\((\ell_{11})\)
|
\(\text{A}\!\)
\(\text{Absolute}\)
|
\(\text{Universal}\)
\(\text{Affirmative}\)
|
\(\text{All}\ u\ \text{is}\ v \)
|
|
\(\text{No}\ u\ \text{is}\ (v)\)
|
\((\ell_{10})\)
|
|
|
|
\(\text{All}\ v\ \text{is}\ u \)
|
\(\text{No}\ v\ \text{is}\ (u)\)
|
\(\text{No}\ (u)\ \text{is}\ v \)
|
\((\ell_{01})\)
|
|
|
|
\(\text{All}\ (v)\ \text{is}\ u \)
|
\(\text{No}\ (v)\ \text{is}\ (u)\)
|
\(\text{No}\ (u)\ \text{is}\ (v)\)
|
\((\ell_{00})\)
|
|
|
|
\(\text{Some}\ (u)\ \text{is}\ (v)\)
|
|
\(\text{Some}\ (u)\ \text{is}\ (v)\)
|
\(\ell_{00}\!\)
|
|
|
|
\(\text{Some}\ (u)\ \text{is}\ v\)
|
|
\(\text{Some}\ (u)\ \text{is}\ v\)
|
\(\ell_{01}\!\)
|
\(\text{O}\!\)
\(\text{Obtrusive}\)
|
\(\text{Particular}\)
\(\text{Negative}\)
|
\(\text{Some}\ u\ \text{is}\ (v)\)
|
|
\(\text{Some}\ u\ \text{is}\ (v)\)
|
\(\ell_{10}\!\)
|
\(\text{I}\!\)
\(\text{Indefinite}\)
|
\(\text{Particular}\)
\(\text{Affirmative}\)
|
\(\text{Some}\ u\ \text{is}\ v\)
|
|
\(\text{Some}\ u\ \text{is}\ v\)
|
\(\ell_{11}\!\)
|
