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\end{matrix}</math> | \end{matrix}</math> | ||
|} | |} | ||
| + | |||
| + | <br> | ||
| + | |||
| + | <pre> | ||
| + | Matrix Representations of Permutations in Sym(3) | ||
| + | o---------o---------o---------o---------o---------o---------o | ||
| + | | | | | | | | | ||
| + | | e | f | g | h | i | j | | ||
| + | | | | | | | | | ||
| + | o=========o=========o=========o=========o=========o=========o | ||
| + | | | | | | | | | ||
| + | | 1 0 0 | 0 0 1 | 0 1 0 | 1 0 0 | 0 0 1 | 0 1 0 | | ||
| + | | 0 1 0 | 1 0 0 | 0 0 1 | 0 0 1 | 0 1 0 | 1 0 0 | | ||
| + | | 0 0 1 | 0 1 0 | 1 0 0 | 0 1 0 | 1 0 0 | 0 0 1 | | ||
| + | | | | | | | | | ||
| + | o---------o---------o---------o---------o---------o---------o | ||
| + | </pre> | ||
<br> | <br> | ||
Revision as of 18:14, 11 June 2009
Differential Logic
Ascii Tables
Table A1. Propositional Forms On Two Variables o---------o---------o---------o----------o------------------o----------o | L_1 | L_2 | L_3 | L_4 | L_5 | L_6 | | | | | | | | | Decimal | Binary | Vector | Cactus | English | Ordinary | o---------o---------o---------o----------o------------------o----------o | | x : 1 1 0 0 | | | | | | y : 1 0 1 0 | | | | o---------o---------o---------o----------o------------------o----------o | | | | | | | | f_0 | f_0000 | 0 0 0 0 | () | false | 0 | | | | | | | | | f_1 | f_0001 | 0 0 0 1 | (x)(y) | neither x nor y | ~x & ~y | | | | | | | | | f_2 | f_0010 | 0 0 1 0 | (x) y | y and not x | ~x & y | | | | | | | | | f_3 | f_0011 | 0 0 1 1 | (x) | not x | ~x | | | | | | | | | f_4 | f_0100 | 0 1 0 0 | x (y) | x and not y | x & ~y | | | | | | | | | f_5 | f_0101 | 0 1 0 1 | (y) | not y | ~y | | | | | | | | | f_6 | f_0110 | 0 1 1 0 | (x, y) | x not equal to y | x + y | | | | | | | | | f_7 | f_0111 | 0 1 1 1 | (x y) | not both x and y | ~x v ~y | | | | | | | | | f_8 | f_1000 | 1 0 0 0 | x y | x and y | x & y | | | | | | | | | f_9 | f_1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y | | | | | | | | | f_10 | f_1010 | 1 0 1 0 | y | y | y | | | | | | | | | f_11 | f_1011 | 1 0 1 1 | (x (y)) | not x without y | x => y | | | | | | | | | f_12 | f_1100 | 1 1 0 0 | x | x | x | | | | | | | | | f_13 | f_1101 | 1 1 0 1 | ((x) y) | not y without x | x <= y | | | | | | | | | f_14 | f_1110 | 1 1 1 0 | ((x)(y)) | x or y | x v y | | | | | | | | | f_15 | f_1111 | 1 1 1 1 | (()) | true | 1 | | | | | | | | o---------o---------o---------o----------o------------------o----------o
Table A2. Propositional Forms On Two Variables o---------o---------o---------o----------o------------------o----------o | L_1 | L_2 | L_3 | L_4 | L_5 | L_6 | | | | | | | | | Decimal | Binary | Vector | Cactus | English | Ordinary | o---------o---------o---------o----------o------------------o----------o | | x : 1 1 0 0 | | | | | | y : 1 0 1 0 | | | | o---------o---------o---------o----------o------------------o----------o | | | | | | | | f_0 | f_0000 | 0 0 0 0 | () | false | 0 | | | | | | | | o---------o---------o---------o----------o------------------o----------o | | | | | | | | f_1 | f_0001 | 0 0 0 1 | (x)(y) | neither x nor y | ~x & ~y | | | | | | | | | f_2 | f_0010 | 0 0 1 0 | (x) y | y and not x | ~x & y | | | | | | | | | f_4 | f_0100 | 0 1 0 0 | x (y) | x and not y | x & ~y | | | | | | | | | f_8 | f_1000 | 1 0 0 0 | x y | x and y | x & y | | | | | | | | o---------o---------o---------o----------o------------------o----------o | | | | | | | | f_3 | f_0011 | 0 0 1 1 | (x) | not x | ~x | | | | | | | | | f_12 | f_1100 | 1 1 0 0 | x | x | x | | | | | | | | o---------o---------o---------o----------o------------------o----------o | | | | | | | | f_6 | f_0110 | 0 1 1 0 | (x, y) | x not equal to y | x + y | | | | | | | | | f_9 | f_1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y | | | | | | | | o---------o---------o---------o----------o------------------o----------o | | | | | | | | f_5 | f_0101 | 0 1 0 1 | (y) | not y | ~y | | | | | | | | | f_10 | f_1010 | 1 0 1 0 | y | y | y | | | | | | | | o---------o---------o---------o----------o------------------o----------o | | | | | | | | f_7 | f_0111 | 0 1 1 1 | (x y) | not both x and y | ~x v ~y | | | | | | | | | f_11 | f_1011 | 1 0 1 1 | (x (y)) | not x without y | x => y | | | | | | | | | f_13 | f_1101 | 1 1 0 1 | ((x) y) | not y without x | x <= y | | | | | | | | | f_14 | f_1110 | 1 1 1 0 | ((x)(y)) | x or y | x v y | | | | | | | | o---------o---------o---------o----------o------------------o----------o | | | | | | | | f_15 | f_1111 | 1 1 1 1 | (()) | true | 1 | | | | | | | | o---------o---------o---------o----------o------------------o----------o
Table A3. Ef Expanded Over Differential Features {dx, dy}
o------o------------o------------o------------o------------o------------o
| | | | | | |
| | f | T_11 f | T_10 f | T_01 f | T_00 f |
| | | | | | |
| | | Ef| dx dy | Ef| dx(dy) | Ef| (dx)dy | Ef|(dx)(dy)|
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_0 | () | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_1 | (x)(y) | x y | x (y) | (x) y | (x)(y) |
| | | | | | |
| f_2 | (x) y | x (y) | x y | (x)(y) | (x) y |
| | | | | | |
| f_4 | x (y) | (x) y | (x)(y) | x y | x (y) |
| | | | | | |
| f_8 | x y | (x)(y) | (x) y | x (y) | x y |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_3 | (x) | x | x | (x) | (x) |
| | | | | | |
| f_12 | x | (x) | (x) | x | x |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_6 | (x, y) | (x, y) | ((x, y)) | ((x, y)) | (x, y) |
| | | | | | |
| f_9 | ((x, y)) | ((x, y)) | (x, y) | (x, y) | ((x, y)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_5 | (y) | y | (y) | y | (y) |
| | | | | | |
| f_10 | y | (y) | y | (y) | y |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_7 | (x y) | ((x)(y)) | ((x) y) | (x (y)) | (x y) |
| | | | | | |
| f_11 | (x (y)) | ((x) y) | ((x)(y)) | (x y) | (x (y)) |
| | | | | | |
| f_13 | ((x) y) | (x (y)) | (x y) | ((x)(y)) | ((x) y) |
| | | | | | |
| f_14 | ((x)(y)) | (x y) | (x (y)) | ((x) y) | ((x)(y)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_15 | (()) | (()) | (()) | (()) | (()) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | |
| Fixed Point Total | 4 | 4 | 4 | 16 |
| | | | | |
o-------------------o------------o------------o------------o------------o
Table A4. Df Expanded Over Differential Features {dx, dy}
o------o------------o------------o------------o------------o------------o
| | | | | | |
| | f | Df| dx dy | Df| dx(dy) | Df| (dx)dy | Df|(dx)(dy)|
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_0 | () | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_1 | (x)(y) | ((x, y)) | (y) | (x) | () |
| | | | | | |
| f_2 | (x) y | (x, y) | y | (x) | () |
| | | | | | |
| f_4 | x (y) | (x, y) | (y) | x | () |
| | | | | | |
| f_8 | x y | ((x, y)) | y | x | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_3 | (x) | (()) | (()) | () | () |
| | | | | | |
| f_12 | x | (()) | (()) | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_6 | (x, y) | () | (()) | (()) | () |
| | | | | | |
| f_9 | ((x, y)) | () | (()) | (()) | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_5 | (y) | (()) | () | (()) | () |
| | | | | | |
| f_10 | y | (()) | () | (()) | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_7 | (x y) | ((x, y)) | y | x | () |
| | | | | | |
| f_11 | (x (y)) | (x, y) | (y) | x | () |
| | | | | | |
| f_13 | ((x) y) | (x, y) | y | (x) | () |
| | | | | | |
| f_14 | ((x)(y)) | ((x, y)) | (y) | (x) | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_15 | (()) | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
Table A5. Ef Expanded Over Ordinary Features {x, y}
o------o------------o------------o------------o------------o------------o
| | | | | | |
| | f | Ef | xy | Ef | x(y) | Ef | (x)y | Ef | (x)(y)|
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_0 | () | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_1 | (x)(y) | dx dy | dx (dy) | (dx) dy | (dx)(dy) |
| | | | | | |
| f_2 | (x) y | dx (dy) | dx dy | (dx)(dy) | (dx) dy |
| | | | | | |
| f_4 | x (y) | (dx) dy | (dx)(dy) | dx dy | dx (dy) |
| | | | | | |
| f_8 | x y | (dx)(dy) | (dx) dy | dx (dy) | dx dy |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_3 | (x) | dx | dx | (dx) | (dx) |
| | | | | | |
| f_12 | x | (dx) | (dx) | dx | dx |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_6 | (x, y) | (dx, dy) | ((dx, dy)) | ((dx, dy)) | (dx, dy) |
| | | | | | |
| f_9 | ((x, y)) | ((dx, dy)) | (dx, dy) | (dx, dy) | ((dx, dy)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_5 | (y) | dy | (dy) | dy | (dy) |
| | | | | | |
| f_10 | y | (dy) | dy | (dy) | dy |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_7 | (x y) | ((dx)(dy)) | ((dx) dy) | (dx (dy)) | (dx dy) |
| | | | | | |
| f_11 | (x (y)) | ((dx) dy) | ((dx)(dy)) | (dx dy) | (dx (dy)) |
| | | | | | |
| f_13 | ((x) y) | (dx (dy)) | (dx dy) | ((dx)(dy)) | ((dx) dy) |
| | | | | | |
| f_14 | ((x)(y)) | (dx dy) | (dx (dy)) | ((dx) dy) | ((dx)(dy)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_15 | (()) | (()) | (()) | (()) | (()) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
Table A6. Df Expanded Over Ordinary Features {x, y}
o------o------------o------------o------------o------------o------------o
| | | | | | |
| | f | Df | xy | Df | x(y) | Df | (x)y | Df | (x)(y)|
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_0 | () | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_1 | (x)(y) | dx dy | dx (dy) | (dx) dy | ((dx)(dy)) |
| | | | | | |
| f_2 | (x) y | dx (dy) | dx dy | ((dx)(dy)) | (dx) dy |
| | | | | | |
| f_4 | x (y) | (dx) dy | ((dx)(dy)) | dx dy | dx (dy) |
| | | | | | |
| f_8 | x y | ((dx)(dy)) | (dx) dy | dx (dy) | dx dy |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_3 | (x) | dx | dx | dx | dx |
| | | | | | |
| f_12 | x | dx | dx | dx | dx |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_6 | (x, y) | (dx, dy) | (dx, dy) | (dx, dy) | (dx, dy) |
| | | | | | |
| f_9 | ((x, y)) | (dx, dy) | (dx, dy) | (dx, dy) | (dx, dy) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_5 | (y) | dy | dy | dy | dy |
| | | | | | |
| f_10 | y | dy | dy | dy | dy |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_7 | (x y) | ((dx)(dy)) | (dx) dy | dx (dy) | dx dy |
| | | | | | |
| f_11 | (x (y)) | (dx) dy | ((dx)(dy)) | dx dy | dx (dy) |
| | | | | | |
| f_13 | ((x) y) | dx (dy) | dx dy | ((dx)(dy)) | (dx) dy |
| | | | | | |
| f_14 | ((x)(y)) | dx dy | dx (dy) | (dx) dy | ((dx)(dy)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_15 | (()) | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
o----------o----------o----------o----------o----------o | % | | | | | · % T_00 | T_01 | T_10 | T_11 | | % | | | | o==========o==========o==========o==========o==========o | % | | | | | T_00 % T_00 | T_01 | T_10 | T_11 | | % | | | | o----------o----------o----------o----------o----------o | % | | | | | T_01 % T_01 | T_00 | T_11 | T_10 | | % | | | | o----------o----------o----------o----------o----------o | % | | | | | T_10 % T_10 | T_11 | T_00 | T_01 | | % | | | | o----------o----------o----------o----------o----------o | % | | | | | T_11 % T_11 | T_10 | T_01 | T_00 | | % | | | | o----------o----------o----------o----------o----------o
o---------o---------o---------o---------o---------o | % | | | | | · % e | f | g | h | | % | | | | o=========o=========o=========o=========o=========o | % | | | | | e % e | f | g | h | | % | | | | o---------o---------o---------o---------o---------o | % | | | | | f % f | e | h | g | | % | | | | o---------o---------o---------o---------o---------o | % | | | | | g % g | h | e | f | | % | | | | o---------o---------o---------o---------o---------o | % | | | | | h % h | g | f | e | | % | | | | o---------o---------o---------o---------o---------o
Permutation Substitutions in Sym {A, B, C}
o---------o---------o---------o---------o---------o---------o
| | | | | | |
| e | f | g | h | i | j |
| | | | | | |
o=========o=========o=========o=========o=========o=========o
| | | | | | |
| A B C | A B C | A B C | A B C | A B C | A B C |
| | | | | | |
| | | | | | | | | | | | | | | | | | | | | | | | |
| v v v | v v v | v v v | v v v | v v v | v v v |
| | | | | | |
| A B C | C A B | B C A | A C B | C B A | B A C |
| | | | | | |
o---------o---------o---------o---------o---------o---------o
Matrix Representations of Permutations in Sym(3) o---------o---------o---------o---------o---------o---------o | | | | | | | | e | f | g | h | i | j | | | | | | | | o=========o=========o=========o=========o=========o=========o | | | | | | | | 1 0 0 | 0 0 1 | 0 1 0 | 1 0 0 | 0 0 1 | 0 1 0 | | 0 1 0 | 1 0 0 | 0 0 1 | 0 0 1 | 0 1 0 | 1 0 0 | | 0 0 1 | 0 1 0 | 1 0 0 | 0 1 0 | 1 0 0 | 0 0 1 | | | | | | | | o---------o---------o---------o---------o---------o---------o
Symmetric Group S_3 o-------------------------------------------------o | | | ^ | | e / \ e | | / \ | | / e \ | | f / \ / \ f | | / \ / \ | | / f \ f \ | | g / \ / \ / \ g | | / \ / \ / \ | | / g \ g \ g \ | | h / \ / \ / \ / \ h | | / \ / \ / \ / \ | | / h \ e \ e \ h \ | | i / \ / \ / \ / \ / \ i | | / \ / \ / \ / \ / \ | | / i \ i \ f \ j \ i \ | | j / \ / \ / \ / \ / \ / \ j | | / \ / \ / \ / \ / \ / \ | | ( j \ j \ j \ i \ h \ j ) | | \ / \ / \ / \ / \ / \ / | | \ / \ / \ / \ / \ / \ / | | \ h \ h \ e \ j \ i / | | \ / \ / \ / \ / \ / | | \ / \ / \ / \ / \ / | | \ i \ g \ f \ h / | | \ / \ / \ / \ / | | \ / \ / \ / \ / | | \ f \ e \ g / | | \ / \ / \ / | | \ / \ / \ / | | \ g \ f / | | \ / \ / | | \ / \ / | | \ e / | | \ / | | \ / | | v | | | o-------------------------------------------------o
Wiki Tables : New Versions
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 |
| L1 | L2 | L3 | L4 | L5 | L6 | ||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| x : | 1 1 0 0 | ||||||||||
| y : | 1 0 1 0 | ||||||||||
| f0 | f0000 | 0 0 0 0 | ( ) | false | 0 | ||||||
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| f15 | f1111 | 1 1 1 1 | (( )) | true | 1 |
Differential Propositions
| A : | 1 1 0 0 | ||||||||||
| dA : | 1 0 1 0 | ||||||||||
| f0 | g0 | 0 0 0 0 | ( ) | False | 0 | ||||||
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| f3 | g15 | 1 1 1 1 | (( )) | True | 1 |
Wiki Tables : Old Versions
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
| A : | 1 1 0 0 | ||||||||||
| dA : | 1 0 1 0 | ||||||||||
| f0 | g0 | 0 0 0 0 | ( ) | False | 0 | ||||||
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Wiki TeX Tables
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\(\mathcal{L}_1\) \(\text{Decimal}\) |
\(\mathcal{L}_2\) \(\text{Binary}\) |
\(\mathcal{L}_3\) \(\text{Vector}\) |
\(\mathcal{L}_4\) \(\text{Cactus}\) |
\(\mathcal{L}_5\) \(\text{English}\) |
\(\mathcal{L}_6\) \(\text{Ordinary}\) |
| \(x\colon\!\) | \(1~1~0~0\!\) | ||||
| \(y\colon\!\) | \(1~0~1~0\!\) | ||||
| \(f_{0}\!\) | \(f_{0000}\!\) | \(0~0~0~0\!\) | \((~)\!\) | \(\text{false}\!\) | \(0\!\) |
| \(f_{1}\!\) | \(f_{0001}\!\) | \(0~0~0~1\!\) | \((x)(y)\!\) | \(\text{neither}~ x ~\text{nor}~ y\!\) | \(\lnot x \land \lnot y\!\) |
| \(f_{2}\!\) | \(f_{0010}\!\) | \(0~0~1~0\!\) | \((x)~y\!\) | \(y ~\text{without}~ x\!\) | \(\lnot x \land y\!\) |
| \(f_{3}\!\) | \(f_{0011}\!\) | \(0~0~1~1\!\) | \((x)\!\) | \(\text{not}~ x\!\) | \(\lnot x\!\) |
| \(f_{4}\!\) | \(f_{0100}\!\) | \(0~1~0~0\!\) | \(x~(y)\!\) | \(x ~\text{without}~ y\!\) | \(x \land \lnot y\!\) |
| \(f_{5}\!\) | \(f_{0101}\!\) | \(0~1~0~1\!\) | \((y)\!\) | \(\text{not}~ y\!\) | \(\lnot y\!\) |
| \(f_{6}\!\) | \(f_{0110}\!\) | \(0~1~1~0\!\) | \((x,~y)\!\) | \(x ~\text{not equal to}~ y\!\) | \(x \ne y\!\) |
| \(f_{7}\!\) | \(f_{0111}\!\) | \(0~1~1~1\!\) | \((x~y)\!\) | \(\text{not both}~ x ~\text{and}~ y\!\) | \(\lnot x \lor \lnot y\!\) |
| \(f_{8}\!\) | \(f_{1000}\!\) | \(1~0~0~0\!\) | \(x~y\!\) | \(x ~\text{and}~ y\!\) | \(x \land y\!\) |
| \(f_{9}\!\) | \(f_{1001}\!\) | \(1~0~0~1\!\) | \(((x,~y))\!\) | \(x ~\text{equal to}~ y\!\) | \(x = y\!\) |
| \(f_{10}\!\) | \(f_{1010}\!\) | \(1~0~1~0\!\) | \(y\!\) | \(y\!\) | \(y\!\) |
| \(f_{11}\!\) | \(f_{1011}\!\) | \(1~0~1~1\!\) | \((x~(y))\!\) | \(\text{not}~ x ~\text{without}~ y\!\) | \(x \Rightarrow y\!\) |
| \(f_{12}\!\) | \(f_{1100}\!\) | \(1~1~0~0\!\) | \(x\!\) | \(x\!\) | \(x\!\) |
| \(f_{13}\!\) | \(f_{1101}\!\) | \(1~1~0~1\!\) | \(((x)~y)\!\) | \(\text{not}~ y ~\text{without}~ x\!\) | \(x \Leftarrow y\!\) |
| \(f_{14}\!\) | \(f_{1110}\!\) | \(1~1~1~0\!\) | \(((x)(y))\!\) | \(x ~\text{or}~ y\!\) | \(x \lor y\!\) |
| \(f_{15}\!\) | \(f_{1111}\!\) | \(1~1~1~1\!\) | \(((~))\!\) | \(\text{true}\!\) | \(1\!\) |
|
\(\mathcal{L}_1\) \(\text{Decimal}\) |
\(\mathcal{L}_2\) \(\text{Binary}\) |
\(\mathcal{L}_3\) \(\text{Vector}\) |
\(\mathcal{L}_4\) \(\text{Cactus}\) |
\(\mathcal{L}_5\) \(\text{English}\) |
\(\mathcal{L}_6\) \(\text{Ordinary}\) | |||||||||||||
| \(x\colon\!\) | \(1~1~0~0\!\) | |||||||||||||||||
| \(y\colon\!\) | \(1~0~1~0\!\) | |||||||||||||||||
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\(\begin{matrix} f_0 \'"`UNIQ-MathJax1-QINU`"' '''Generalized''' or '''n-ary''' XOR is true when the number of 1-bits is odd. '"`UNIQ--pre-00000017-QINU`"' '"`UNIQ--pre-00000018-QINU`"' '"`UNIQ--pre-00000019-QINU`"' '"`UNIQ-MathJax2-QINU`"' ===='"`UNIQ--h-33--QINU`"'[[Logical implication]]==== The '''material conditional''' and '''logical implication''' are both associated with an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if the first operand is true and the second operand is false. The [[truth table]] associated with the material conditional '''if p then q''' (symbolized as '''p → q''') and the logical implication '''p implies q''' (symbolized as '''p ⇒ q''') is as follows: {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%" |+ '''Logical Implication''' |- style="background:aliceblue" ! style="width:15%" | p ! style="width:15%" | q ! style="width:15%" | p ⇒ q |- | F || F || T |- | F || T || T |- | T || F || F |- | T || T || T |} <br> ===='"`UNIQ--h-34--QINU`"'[[Logical NAND]]==== The '''NAND operation''' is a [[logical operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if both of its operands are true. In other words, it produces a value of ''true'' if and only if at least one of its operands is false. The [[truth table]] of '''p NAND q''' (also written as '''p | q''' or '''p ↑ q''') is as follows: {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%" |+ '''Logical NAND''' |- style="background:aliceblue" ! style="width:15%" | p ! style="width:15%" | q ! style="width:15%" | p ↑ q |- | F || F || T |- | F || T || T |- | T || F || T |- | T || T || F |} <br> ===='"`UNIQ--h-35--QINU`"'[[Logical NNOR]]==== The '''NNOR operation''' is a [[logical operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' if and only if both of its operands are false. In other words, it produces a value of ''false'' if and only if at least one of its operands is true. The [[truth table]] of '''p NNOR q''' (also written as '''p ⊥ q''' or '''p ↓ q''') is as follows: {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%" |+ '''Logical NOR''' |- style="background:aliceblue" ! style="width:15%" | p ! style="width:15%" | q ! style="width:15%" | p ↓ q |- | F || F || T |- | F || T || F |- | T || F || F |- | T || T || F |} <br> =='"`UNIQ--h-36--QINU`"'Relational Tables== ==='"`UNIQ--h-37--QINU`"'Sign Relations=== {| cellpadding="4" | width="20px" | | align="center" | '''O''' || = || Object Domain |- | width="20px" | | align="center" | '''S''' || = || Sign Domain |- | width="20px" | | align="center" | '''I''' || = || Interpretant Domain |} <br> {| cellpadding="4" | width="20px" | | align="center" | '''O''' | = | {Ann, Bob} | = | {A, B} |- | width="20px" | | align="center" | '''S''' | = | {"Ann", "Bob", "I", "You"} | = | {"A", "B", "i", "u"} |- | width="20px" | | align="center" | '''I''' | = | {"Ann", "Bob", "I", "You"} | = | {"A", "B", "i", "u"} |} <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A |- style="background:paleturquoise" ! style="width:20%" | Object ! style="width:20%" | Sign ! style="width:20%" | Interpretant |- | '''A''' || '''"A"''' || '''"A"''' |- | '''A''' || '''"A"''' || '''"i"''' |- | '''A''' || '''"i"''' || '''"A"''' |- | '''A''' || '''"i"''' || '''"i"''' |- | '''B''' || '''"B"''' || '''"B"''' |- | '''B''' || '''"B"''' || '''"u"''' |- | '''B''' || '''"u"''' || '''"B"''' |- | '''B''' || '''"u"''' || '''"u"''' |} <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B |- style="background:paleturquoise" ! style="width:20%" | Object ! style="width:20%" | Sign ! style="width:20%" | Interpretant |- | '''A''' || '''"A"''' || '''"A"''' |- | '''A''' || '''"A"''' || '''"u"''' |- | '''A''' || '''"u"''' || '''"A"''' |- | '''A''' || '''"u"''' || '''"u"''' |- | '''B''' || '''"B"''' || '''"B"''' |- | '''B''' || '''"B"''' || '''"i"''' |- | '''B''' || '''"i"''' || '''"B"''' |- | '''B''' || '''"i"''' || '''"i"''' |} <br> ==='"`UNIQ--h-38--QINU`"'Triadic Relations=== ===='"`UNIQ--h-39--QINU`"'Algebraic Examples==== {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |+ '''L'''<sub>0</sub> = {(''x'', ''y'', ''z'') ∈ '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 0} |- style="background:paleturquoise" ! X !! Y !! Z |- | '''0''' || '''0''' || '''0''' |- | '''0''' || '''1''' || '''1''' |- | '''1''' || '''0''' || '''1''' |- | '''1''' || '''1''' || '''0''' |} <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |+ '''L'''<sub>1</sub> = {(''x'', ''y'', ''z'') ∈ '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 1} |- style="background:paleturquoise" ! X !! Y !! Z |- | '''0''' || '''0''' || '''1''' |- | '''0''' || '''1''' || '''0''' |- | '''1''' || '''0''' || '''0''' |- | '''1''' || '''1''' || '''1''' |} <br> ===='"`UNIQ--h-40--QINU`"'Semiotic Examples==== {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A |- style="background:paleturquoise" ! style="width:20%" | Object ! style="width:20%" | Sign ! style="width:20%" | Interpretant |- | '''A''' || '''"A"''' || '''"A"''' |- | '''A''' || '''"A"''' || '''"i"''' |- | '''A''' || '''"i"''' || '''"A"''' |- | '''A''' || '''"i"''' || '''"i"''' |- | '''B''' || '''"B"''' || '''"B"''' |- | '''B''' || '''"B"''' || '''"u"''' |- | '''B''' || '''"u"''' || '''"B"''' |- | '''B''' || '''"u"''' || '''"u"''' |} <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B |- style="background:paleturquoise" ! style="width:20%" | Object ! style="width:20%" | Sign ! style="width:20%" | Interpretant |- | '''A''' || '''"A"''' || '''"A"''' |- | '''A''' || '''"A"''' || '''"u"''' |- | '''A''' || '''"u"''' || '''"A"''' |- | '''A''' || '''"u"''' || '''"u"''' |- | '''B''' || '''"B"''' || '''"B"''' |- | '''B''' || '''"B"''' || '''"i"''' |- | '''B''' || '''"i"''' || '''"B"''' |- | '''B''' || '''"i"''' || '''"i"''' |} <br> ==='"`UNIQ--h-41--QINU`"'Dyadic Projections=== {| cellpadding="4" | width="20px" | | '''L'''<sub>OS</sub> | = | ''proj''<sub>OS</sub>('''L''') | = | { (''o'', ''s'') ∈ '''O''' × '''S''' : (''o'', ''s'', ''i'') ∈ '''L''' for some ''i'' ∈ '''I''' } |- | width="20px" | | '''L'''<sub>SO</sub> | = | ''proj''<sub>SO</sub>('''L''') | = | { (''s'', ''o'') ∈ '''S''' × '''O''' : (''o'', ''s'', ''i'') ∈ '''L''' for some ''i'' ∈ '''I''' } |- | width="20px" | | '''L'''<sub>IS</sub> | = | ''proj''<sub>IS</sub>('''L''') | = | { (''i'', ''s'') ∈ '''I''' × '''S''' : (''o'', ''s'', ''i'') ∈ '''L''' for some ''o'' ∈ '''O''' } |- | width="20px" | | '''L'''<sub>SI</sub> | = | ''proj''<sub>SI</sub>('''L''') | = | { (''s'', ''i'') ∈ '''S''' × '''I''' : (''o'', ''s'', ''i'') ∈ '''L''' for some ''o'' ∈ '''O''' } |- | width="20px" | | '''L'''<sub>OI</sub> | = | ''proj''<sub>OI</sub>('''L''') | = | { (''o'', ''i'') ∈ '''O''' × '''I''' : (''o'', ''s'', ''i'') ∈ '''L''' for some ''s'' ∈ '''S''' } |- | width="20px" | | '''L'''<sub>IO</sub> | = | ''proj''<sub>IO</sub>('''L''') | = | { (''i'', ''o'') ∈ '''I''' × '''O''' : (''o'', ''s'', ''i'') ∈ '''L''' for some ''s'' ∈ '''S''' } |} <br> ===='"`UNIQ--h-42--QINU`"'Method 1 : Subtitles as Captions==== {| align="center" style="width:90%" | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ ''proj''<sub>OS</sub>('''L'''<sub>A</sub>) |- style="background:paleturquoise" ! style="width:50%" | Object ! style="width:50%" | Sign |- | '''A''' || '''"A"''' |- | '''A''' || '''"i"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"u"''' |} | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ ''proj''<sub>OS</sub>('''L'''<sub>B</sub>) |- style="background:paleturquoise" ! style="width:50%" | Object ! style="width:50%" | Sign |- | '''A''' || '''"A"''' |- | '''A''' || '''"u"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"i"''' |} |} <br> {| align="center" style="width:90%" | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ ''proj''<sub>SI</sub>('''L'''<sub>A</sub>) |- style="background:paleturquoise" ! style="width:50%" | Sign ! style="width:50%" | Interpretant |- | '''"A"''' || '''"A"''' |- | '''"A"''' || '''"i"''' |- | '''"i"''' || '''"A"''' |- | '''"i"''' || '''"i"''' |- | '''"B"''' || '''"B"''' |- | '''"B"''' || '''"u"''' |- | '''"u"''' || '''"B"''' |- | '''"u"''' || '''"u"''' |} | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ ''proj''<sub>SI</sub>('''L'''<sub>B</sub>) |- style="background:paleturquoise" ! style="width:50%" | Sign ! style="width:50%" | Interpretant |- | '''"A"''' || '''"A"''' |- | '''"A"''' || '''"u"''' |- | '''"u"''' || '''"A"''' |- | '''"u"''' || '''"u"''' |- | '''"B"''' || '''"B"''' |- | '''"B"''' || '''"i"''' |- | '''"i"''' || '''"B"''' |- | '''"i"''' || '''"i"''' |} |} <br> {| align="center" style="width:90%" | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ ''proj''<sub>OI</sub>('''L'''<sub>A</sub>) |- style="background:paleturquoise" ! style="width:50%" | Object ! style="width:50%" | Interpretant |- | '''A''' || '''"A"''' |- | '''A''' || '''"i"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"u"''' |} | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ ''proj''<sub>OI</sub>('''L'''<sub>B</sub>) |- style="background:paleturquoise" ! style="width:50%" | Object ! style="width:50%" | Interpretant |- | '''A''' || '''"A"''' |- | '''A''' || '''"u"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"i"''' |} |} <br> ===='"`UNIQ--h-43--QINU`"'Method 2 : Subtitles as Top Rows==== {| align="center" style="width:90%" | align="center" style="width:45%" | ''proj''<sub>OS</sub>('''L'''<sub>A</sub>) {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! style="width:50%" | Object ! style="width:50%" | Sign |- | '''A''' || '''"A"''' |- | '''A''' || '''"i"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"u"''' |} | align="center" style="width:45%" | ''proj''<sub>OS</sub>('''L'''<sub>B</sub>) {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! style="width:50%" | Object ! style="width:50%" | Sign |- | '''A''' || '''"A"''' |- | '''A''' || '''"u"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"i"''' |} |} <br> {| align="center" style="width:90%" | align="center" style="width:45%" | ''proj''<sub>SI</sub>('''L'''<sub>A</sub>) {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! style="width:50%" | Sign ! style="width:50%" | Interpretant |- | '''"A"''' || '''"A"''' |- | '''"A"''' || '''"i"''' |- | '''"i"''' || '''"A"''' |- | '''"i"''' || '''"i"''' |- | '''"B"''' || '''"B"''' |- | '''"B"''' || '''"u"''' |- | '''"u"''' || '''"B"''' |- | '''"u"''' || '''"u"''' |} | align="center" style="width:45%" | ''proj''<sub>SI</sub>('''L'''<sub>B</sub>) {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! style="width:50%" | Sign ! style="width:50%" | Interpretant |- | '''"A"''' || '''"A"''' |- | '''"A"''' || '''"u"''' |- | '''"u"''' || '''"A"''' |- | '''"u"''' || '''"u"''' |- | '''"B"''' || '''"B"''' |- | '''"B"''' || '''"i"''' |- | '''"i"''' || '''"B"''' |- | '''"i"''' || '''"i"''' |} |} <br> {| align="center" style="width:90%" | align="center" style="width:45%" | ''proj''<sub>OI</sub>('''L'''<sub>A</sub>) {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! style="width:50%" | Object ! style="width:50%" | Interpretant |- | '''A''' || '''"A"''' |- | '''A''' || '''"i"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"u"''' |} | align="center" style="width:45%" | ''proj''<sub>OI</sub>('''L'''<sub>B</sub>) {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! style="width:50%" | Object ! style="width:50%" | Interpretant |- | '''A''' || '''"A"''' |- | '''A''' || '''"u"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"i"''' |} |} <br> ==='"`UNIQ--h-44--QINU`"'Relation Reduction=== ===='"`UNIQ--h-45--QINU`"'Method 1 : Subtitles as Captions==== {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |+ '''L'''<sub>0</sub> = {(''x'', ''y'', ''z'') ∈ '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 0} |- style="background:paleturquoise" ! X !! Y !! Z |- | '''0''' || '''0''' || '''0''' |- | '''0''' || '''1''' || '''1''' |- | '''1''' || '''0''' || '''1''' |- | '''1''' || '''1''' || '''0''' |} <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |+ '''L'''<sub>1</sub> = {(''x'', ''y'', ''z'') ∈ '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 1} |- style="background:paleturquoise" ! X !! Y !! Z |- | '''0''' || '''0''' || '''1''' |- | '''0''' || '''1''' || '''0''' |- | '''1''' || '''0''' || '''0''' |- | '''1''' || '''1''' || '''1''' |} <br> {| align="center" style="width:90%" | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ proj<sub>''XY''</sub>('''L'''<sub>0</sub>) |- style="background:paleturquoise" ! X !! Y |- | '''0''' || '''0''' |- | '''0''' || '''1''' |- | '''1''' || '''0''' |- | '''1''' || '''1''' |} | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ proj<sub>''XZ''</sub>('''L'''<sub>0</sub>) |- style="background:paleturquoise" ! X !! Z |- | '''0''' || '''0''' |- | '''0''' || '''1''' |- | '''1''' || '''1''' |- | '''1''' || '''0''' |} | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ proj<sub>''YZ''</sub>('''L'''<sub>0</sub>) |- style="background:paleturquoise" ! Y !! Z |- | '''0''' || '''0''' |- | '''1''' || '''1''' |- | '''0''' || '''1''' |- | '''1''' || '''0''' |} |} <br> {| align="center" style="width:90%" | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ proj<sub>''XY''</sub>('''L'''<sub>1</sub>) |- style="background:paleturquoise" ! X !! Y |- | '''0''' || '''0''' |- | '''0''' || '''1''' |- | '''1''' || '''0''' |- | '''1''' || '''1''' |} | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ proj<sub>''XZ''</sub>('''L'''<sub>1</sub>) |- style="background:paleturquoise" ! X !! Z |- | '''0''' || '''1''' |- | '''0''' || '''0''' |- | '''1''' || '''0''' |- | '''1''' || '''1''' |} | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ proj<sub>''YZ''</sub>('''L'''<sub>1</sub>) |- style="background:paleturquoise" ! Y !! Z |- | '''0''' || '''1''' |- | '''1''' || '''0''' |- | '''0''' || '''0''' |- | '''1''' || '''1''' |} |} <br> {| align="center" cellpadding="4" style="text-align:center; width:90%" | proj<sub>''XY''</sub>('''L'''<sub>0</sub>) = proj<sub>''XY''</sub>('''L'''<sub>1</sub>) | proj<sub>''XZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''XZ''</sub>('''L'''<sub>1</sub>) | proj<sub>''YZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''YZ''</sub>('''L'''<sub>1</sub>) |} <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A |- style="background:paleturquoise" ! style="width:20%" | Object ! style="width:20%" | Sign ! style="width:20%" | Interpretant |- | '''A''' || '''"A"''' || '''"A"''' |- | '''A''' || '''"A"''' || '''"i"''' |- | '''A''' || '''"i"''' || '''"A"''' |- | '''A''' || '''"i"''' || '''"i"''' |- | '''B''' || '''"B"''' || '''"B"''' |- | '''B''' || '''"B"''' || '''"u"''' |- | '''B''' || '''"u"''' || '''"B"''' |- | '''B''' || '''"u"''' || '''"u"''' |} <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B |- style="background:paleturquoise" ! style="width:20%" | Object ! style="width:20%" | Sign ! style="width:20%" | Interpretant |- | '''A''' || '''"A"''' || '''"A"''' |- | '''A''' || '''"A"''' || '''"u"''' |- | '''A''' || '''"u"''' || '''"A"''' |- | '''A''' || '''"u"''' || '''"u"''' |- | '''B''' || '''"B"''' || '''"B"''' |- | '''B''' || '''"B"''' || '''"i"''' |- | '''B''' || '''"i"''' || '''"B"''' |- | '''B''' || '''"i"''' || '''"i"''' |} <br> {| align="center" style="width:90%" | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ proj<sub>''XY''</sub>('''L'''<sub>A</sub>) |- style="background:paleturquoise" ! style="width:50%" | Object ! style="width:50%" | Sign |- | '''A''' || '''"A"''' |- | '''A''' || '''"i"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"u"''' |} | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ proj<sub>''XZ''</sub>('''L'''<sub>A</sub>) |- style="background:paleturquoise" ! style="width:50%" | Object ! style="width:50%" | Interpretant |- | '''A''' || '''"A"''' |- | '''A''' || '''"i"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"u"''' |} | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ proj<sub>''YZ''</sub>('''L'''<sub>A</sub>) |- style="background:paleturquoise" ! style="width:50%" | Sign ! style="width:50%" | Interpretant |- | '''"A"''' || '''"A"''' |- | '''"A"''' || '''"i"''' |- | '''"i"''' || '''"A"''' |- | '''"i"''' || '''"i"''' |- | '''"B"''' || '''"B"''' |- | '''"B"''' || '''"u"''' |- | '''"u"''' || '''"B"''' |- | '''"u"''' || '''"u"''' |} |} <br> {| align="center" style="width:90%" | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ proj<sub>''XY''</sub>('''L'''<sub>B</sub>) |- style="background:paleturquoise" ! style="width:50%" | Object ! style="width:50%" | Sign |- | '''A''' || '''"A"''' |- | '''A''' || '''"u"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"i"''' |} | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ proj<sub>''XZ''</sub>('''L'''<sub>B</sub>) |- style="background:paleturquoise" ! style="width:50%" | Object ! style="width:50%" | Interpretant |- | '''A''' || '''"A"''' |- | '''A''' || '''"u"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"i"''' |} | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ proj<sub>''YZ''</sub>('''L'''<sub>B</sub>) |- style="background:paleturquoise" ! style="width:50%" | Sign ! style="width:50%" | Interpretant |- | '''"A"''' || '''"A"''' |- | '''"A"''' || '''"u"''' |- | '''"u"''' || '''"A"''' |- | '''"u"''' || '''"u"''' |- | '''"B"''' || '''"B"''' |- | '''"B"''' || '''"i"''' |- | '''"i"''' || '''"B"''' |- | '''"i"''' || '''"i"''' |} |} <br> {| align="center" cellpadding="4" style="text-align:center; width:90%" | proj<sub>''XY''</sub>('''L'''<sub>A</sub>) ≠ proj<sub>''XY''</sub>('''L'''<sub>B</sub>) | proj<sub>''XZ''</sub>('''L'''<sub>A</sub>) ≠ proj<sub>''XZ''</sub>('''L'''<sub>B</sub>) | proj<sub>''YZ''</sub>('''L'''<sub>A</sub>) ≠ proj<sub>''YZ''</sub>('''L'''<sub>B</sub>) |} <br> ===='"`UNIQ--h-46--QINU`"'Method 2 : Subtitles as Top Rows==== {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |+ '''L'''<sub>0</sub> = {(''x'', ''y'', ''z'') ∈ '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 0} |- style="background:paleturquoise" ! X !! Y !! Z |- | '''0''' || '''0''' || '''0''' |- | '''0''' || '''1''' || '''1''' |- | '''1''' || '''0''' || '''1''' |- | '''1''' || '''1''' || '''0''' |} <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |+ '''L'''<sub>1</sub> = {(''x'', ''y'', ''z'') ∈ '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 1} |- style="background:paleturquoise" ! X !! Y !! Z |- | '''0''' || '''0''' || '''1''' |- | '''0''' || '''1''' || '''0''' |- | '''1''' || '''0''' || '''0''' |- | '''1''' || '''1''' || '''1''' |} <br> {| align="center" style="width:90%" | align="center" | proj<sub>''XY''</sub>('''L'''<sub>0</sub>) {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! X !! Y |- | '''0''' || '''0''' |- | '''0''' || '''1''' |- | '''1''' || '''0''' |- | '''1''' || '''1''' |} | align="center" | proj<sub>''XZ''</sub>('''L'''<sub>0</sub>) {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! X !! Z |- | '''0''' || '''0''' |- | '''0''' || '''1''' |- | '''1''' || '''1''' |- | '''1''' || '''0''' |} | align="center" | proj<sub>''YZ''</sub>('''L'''<sub>0</sub>) {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! Y !! Z |- | '''0''' || '''0''' |- | '''1''' || '''1''' |- | '''0''' || '''1''' |- | '''1''' || '''0''' |} |} <br> {| align="center" style="width:90%" | align="center" | proj<sub>''XY''</sub>('''L'''<sub>1</sub>) {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! X !! Y |- | '''0''' || '''0''' |- | '''0''' || '''1''' |- | '''1''' || '''0''' |- | '''1''' || '''1''' |} | align="center" | proj<sub>''XZ''</sub>('''L'''<sub>1</sub>) {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! X !! Z |- | '''0''' || '''1''' |- | '''0''' || '''0''' |- | '''1''' || '''0''' |- | '''1''' || '''1''' |} | align="center" | proj<sub>''YZ''</sub>('''L'''<sub>1</sub>) {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! Y !! Z |- | '''0''' || '''1''' |- | '''1''' || '''0''' |- | '''0''' || '''0''' |- | '''1''' || '''1''' |} |} <br> {| align="center" cellpadding="4" style="text-align:center; width:90%" | proj<sub>''XY''</sub>('''L'''<sub>0</sub>) = proj<sub>''XY''</sub>('''L'''<sub>1</sub>) | proj<sub>''XZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''XZ''</sub>('''L'''<sub>1</sub>) | proj<sub>''YZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''YZ''</sub>('''L'''<sub>1</sub>) |} <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A |- style="background:paleturquoise" ! style="width:20%" | Object ! style="width:20%" | Sign ! style="width:20%" | Interpretant |- | '''A''' || '''"A"''' || '''"A"''' |- | '''A''' || '''"A"''' || '''"i"''' |- | '''A''' || '''"i"''' || '''"A"''' |- | '''A''' || '''"i"''' || '''"i"''' |- | '''B''' || '''"B"''' || '''"B"''' |- | '''B''' || '''"B"''' || '''"u"''' |- | '''B''' || '''"u"''' || '''"B"''' |- | '''B''' || '''"u"''' || '''"u"''' |} <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B |- style="background:paleturquoise" ! style="width:20%" | Object ! style="width:20%" | Sign ! style="width:20%" | Interpretant |- | '''A''' || '''"A"''' || '''"A"''' |- | '''A''' || '''"A"''' || '''"u"''' |- | '''A''' || '''"u"''' || '''"A"''' |- | '''A''' || '''"u"''' || '''"u"''' |- | '''B''' || '''"B"''' || '''"B"''' |- | '''B''' || '''"B"''' || '''"i"''' |- | '''B''' || '''"i"''' || '''"B"''' |- | '''B''' || '''"i"''' || '''"i"''' |} <br> {| align="center" style="width:90%" | align="center" style="width:30%" | proj<sub>''XY''</sub>('''L'''<sub>A</sub>) {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! style="width:50%" | Object ! style="width:50%" | Sign |- | '''A''' || '''"A"''' |- | '''A''' || '''"i"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"u"''' |} | align="center" style="width:30%" | proj<sub>''XZ''</sub>('''L'''<sub>A</sub>) {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! style="width:50%" | Object ! style="width:50%" | Interpretant |- | '''A''' || '''"A"''' |- | '''A''' || '''"i"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"u"''' |} | align="center" style="width:30%" | proj<sub>''YZ''</sub>('''L'''<sub>A</sub>) {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! style="width:50%" | Sign ! style="width:50%" | Interpretant |- | '''"A"''' || '''"A"''' |- | '''"A"''' || '''"i"''' |- | '''"i"''' || '''"A"''' |- | '''"i"''' || '''"i"''' |- | '''"B"''' || '''"B"''' |- | '''"B"''' || '''"u"''' |- | '''"u"''' || '''"B"''' |- | '''"u"''' || '''"u"''' |} |} <br> {| align="center" style="width:90%" | align="center" style="width:30%" | proj<sub>''XY''</sub>('''L'''<sub>B</sub>) {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! style="width:50%" | Object ! style="width:50%" | Sign |- | '''A''' || '''"A"''' |- | '''A''' || '''"u"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"i"''' |} | align="center" style="width:30%" | proj<sub>''XZ''</sub>('''L'''<sub>B</sub>) {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! style="width:50%" | Object ! style="width:50%" | Interpretant |- | '''A''' || '''"A"''' |- | '''A''' || '''"u"''' |- | '''B''' || '''"B"''' |- | '''B''' || '''"i"''' |} | align="center" style="width:30%" | proj<sub>''YZ''</sub>('''L'''<sub>B</sub>) {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |- style="background:paleturquoise" ! style="width:50%" | Sign ! style="width:50%" | Interpretant |- | '''"A"''' || '''"A"''' |- | '''"A"''' || '''"u"''' |- | '''"u"''' || '''"A"''' |- | '''"u"''' || '''"u"''' |- | '''"B"''' || '''"B"''' |- | '''"B"''' || '''"i"''' |- | '''"i"''' || '''"B"''' |- | '''"i"''' || '''"i"''' |} |} <br> {| align="center" cellpadding="4" style="text-align:center; width:90%" | proj<sub>''XY''</sub>('''L'''<sub>A</sub>) ≠ proj<sub>''XY''</sub>('''L'''<sub>B</sub>) | proj<sub>''XZ''</sub>('''L'''<sub>A</sub>) ≠ proj<sub>''XZ''</sub>('''L'''<sub>B</sub>) | proj<sub>''YZ''</sub>('''L'''<sub>A</sub>) ≠ proj<sub>''YZ''</sub>('''L'''<sub>B</sub>) |} <br> ==='"`UNIQ--h-47--QINU`"'Formatted Text Display=== : So in a triadic fact, say, the example <br> {| align="center" cellspacing="8" style="width:72%" | align="center" | ''A'' gives ''B'' to ''C'' |} : we make no distinction in the ordinary logic of relations between the ''[[subject (grammar)|subject]] [[nominative]]'', the ''[[direct object]]'', and the ''[[indirect object]]''. We say that the proposition has three ''logical subjects''. We regard it as a mere affair of English grammar that there are six ways of expressing this: <br> {| align="center" cellspacing="8" style="width:72%" | style="width:36%" | ''A'' gives ''B'' to ''C'' | style="width:36%" | ''A'' benefits ''C'' with ''B'' |- | ''B'' enriches ''C'' at expense of ''A'' | ''C'' receives ''B'' from ''A'' |- | ''C'' thanks ''A'' for ''B'' | ''B'' leaves ''A'' for ''C'' |} : These six sentences express one and the same indivisible phenomenon. (C.S. Peirce, "The Categories Defended", MS 308 (1903), EP 2, 170-171). =='"`UNIQ--h-48--QINU`"'Work Area== {| border="1" cellspacing="0" cellpadding="0" style="text-align:center" |+ Binary Operations |- ! style="width:2em" | x<sub>0</sub> ! style="width:2em" | x<sub>1</sub> | style="width:2em" | <sup>2</sup>f<sub>0</sub> | style="width:2em" | <sup>2</sup>f<sub>1</sub> | style="width:2em" | <sup>2</sup>f<sub>2</sub> | style="width:2em" | <sup>2</sup>f<sub>3</sub> | style="width:2em" | <sup>2</sup>f<sub>4</sub> | style="width:2em" | <sup>2</sup>f<sub>5</sub> | style="width:2em" | <sup>2</sup>f<sub>6</sub> | style="width:2em" | <sup>2</sup>f<sub>7</sub> | style="width:2em" | <sup>2</sup>f<sub>8</sub> | style="width:2em" | <sup>2</sup>f<sub>9</sub> | style="width:2em" | <sup>2</sup>f<sub>10</sub> | style="width:2em" | <sup>2</sup>f<sub>11</sub> | style="width:2em" | <sup>2</sup>f<sub>12</sub> | style="width:2em" | <sup>2</sup>f<sub>13</sub> | style="width:2em" | <sup>2</sup>f<sub>14</sub> | style="width:2em" | <sup>2</sup>f<sub>15</sub> |- | 0 || 0 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 |- | 1 || 0 || 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 |- | 0 || 1 || 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 |- | 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 |} <br> ==='"`UNIQ--h-49--QINU`"'Draft 1=== <center><table> <caption>TRUTH TABLES FOR THE BOOLEAN OPERATIONS OF ARITY UP TO 2</caption> <tr valign="top"> <td><table border="5" cellspacing="0"> <caption>Constants</caption> <tr><td></td> <td><sup>0</sup>f<sub>0</sub></td> <td><sup>0</sup>f<sub>1</sub></td> </tr> <tr><td></td> <td align="center">0</td> <td align="center">1</td> </tr></table></td><td> </td> <td><table border="5" cellspacing="0"><caption>Unary Operations</caption><tr> <td>x<sub>0</sub></td> <td></td> <td><sup>1</sup>f<sub>0 </sub></td> <td><sup>1</sup>f<sub>1 </sub></td> <td><sup>1</sup>f<sub>2 </sub></td> <td><sup>1</sup>f<sub>3 </sub></td> </tr><tr> <td align="center">0</td> <td></td> <td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td> </tr> <tr> <td align="center">1</td> <td></td> <td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td> </tr></table></td><td> </td> <td><table border="5" cellspacing="0"><caption>Binary Operations</caption><tr> <td>x<sub>0</sub></td> <td>x<sub>1</sub></td> <td></td> <td><sup>2</sup>f<sub>0</sub></td> <td><sup>2</sup>f<sub>1 </sub></td> <td><sup> 2</sup>f<sub>2 </sub></td> <td><sup>2</sup>f<sub>3 </sub></td> <td><sup>2</sup>f<sub>4 </sub></td> <td><sup>2</sup>f<sub>5 </sub></td> <td><sup>2</sup>f<sub>6 </sub></td> <td><sup>2</sup>f<sub>7 </sub></td> <td><sup>2</sup>f<sub>8 </sub></td> <td><sup>2</sup>f<sub>9 </sub></td> <td><sup>2</sup>f<sub>10</sub></td> <td><sup>2</sup>f<sub>11</sub></td> <td><sup>2</sup>f<sub>12</sub></td> <td><sup>2</sup>f<sub>13</sub></td> <td><sup>2</sup>f<sub>14</sub></td> <td><sup>2</sup>f<sub>15</sub></td> </tr><tr> <td align="center">0</td> <td align="center">0</td> <td></td> <td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td> </tr> <tr> <td align="center">1</td> <td align="center">0</td> <td></td> <td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td> <td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td> <td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td> <td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td> </tr> <tr> <td align="center">0</td> <td align="center">1</td> <td></td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> </tr> <tr> <td align="center">1</td> <td align="center">1</td> <td></td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> </tr> </table></td> </table></center> ==='"`UNIQ--h-50--QINU`"'Draft 2=== <center><table> <caption>TRUTH TABLES FOR THE BOOLEAN OPERATIONS OF ARITY UP TO 2</caption> <tr valign="top"> <td><table border="5" cellspacing="0"> <caption>Constants</caption> <tr><td></td> <td><sup>0</sup>f<sub>0</sub></td> <td><sup>0</sup>f<sub>1</sub></td> </tr> <tr><td></td> <td align="center">0</td> <td align="center">1</td> </tr></table></td><td> </td> <td><table border="5" cellspacing="0"><caption>Unary Operations</caption><tr> <td>x<sub>0</sub></td> <td></td> <td><sup>1</sup>f<sub>0 </sub></td> <td><sup>1</sup>f<sub>1 </sub></td> <td><sup>1</sup>f<sub>2 </sub></td> <td><sup>1</sup>f<sub>3 </sub></td> </tr><tr> <td align="center">0</td> <td></td> <td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td> </tr> <tr> <td align="center">1</td> <td></td> <td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td> </tr></table></td><td> </td> <td><table border="5" cellspacing="0"><caption>Binary Operations</caption><tr> <td>x<sub>0</sub></td> <td>x<sub>1</sub></td> <td></td> <td><sup>2</sup>f<sub>0</sub></td> <td><sup>2</sup>f<sub>1 </sub></td> <td><sup> 2</sup>f<sub>2 </sub></td> <td><sup>2</sup>f<sub>3 </sub></td> <td><sup>2</sup>f<sub>4 </sub></td> <td><sup>2</sup>f<sub>5 </sub></td> <td><sup>2</sup>f<sub>6 </sub></td> <td><sup>2</sup>f<sub>7 </sub></td> <td><sup>2</sup>f<sub>8 </sub></td> <td><sup>2</sup>f<sub>9 </sub></td> <td><sup>2</sup>f<sub>10</sub></td> <td><sup>2</sup>f<sub>11</sub></td> <td><sup>2</sup>f<sub>12</sub></td> <td><sup>2</sup>f<sub>13</sub></td> <td><sup>2</sup>f<sub>14</sub></td> <td><sup>2</sup>f<sub>15</sub></td> </tr><tr> <td align="center">0</td> <td align="center">0</td> <td></td> <td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td> <td align="center">0</td> <td align="center">1</td> </tr> <tr> <td align="center">1</td> <td align="center">0</td> <td></td> <td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td> <td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td> <td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td> <td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td> </tr> <tr> <td align="center">0</td> <td align="center">1</td> <td></td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> </tr> <tr> <td align="center">1</td> <td align="center">1</td> <td></td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">0</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> <td align="center">1</td> </tr> </table></td> </table></center> =='"`UNIQ--h-51--QINU`"'Inquiry and Analogy== ==='"`UNIQ--h-52--QINU`"'Test Patterns=== {| align="center" | style="background:black; color:white" | 1 | style="background:white; color:black" | 0 | style="background:black; color:white" | 1 | style="background:white; color:black" | 0 | style="background:black; color:white" | 1 | style="background:white; color:black" | 0 | style="background:black; color:white" | 1 | style="background:white; color:black" | 0 |- | style="background:white; color:black" | 0 | style="background:black; color:white" | 1 | style="background:white; color:black" | 0 | style="background:black; color:white" | 1 | style="background:white; color:black" | 0 | style="background:black; color:white" | 1 | style="background:white; color:black" | 0 | style="background:black; color:white" | 1 |}<br> {| align="center" | style="background:white; color:black" | 1 | style="background:black; color:white" | 0 | style="background:white; color:black" | 1 | style="background:black; color:white" | 0 | style="background:white; color:black" | 1 | style="background:black; color:white" | 0 | style="background:white; color:black" | 1 | style="background:black; color:white" | 0 |- | style="background:black; color:white" | 0 | style="background:white; color:black" | 1 | style="background:black; color:white" | 0 | style="background:white; color:black" | 1 | style="background:black; color:white" | 0 | style="background:white; color:black" | 1 | style="background:black; color:white" | 0 | style="background:white; color:black" | 1 |}<br> {| align="center" | style="background:black; color:white" | 1 | style="background:white; color:black" | 0 | style="background:black; color:white" | 1 | style="background:white; color:black" | 0 | style="background:black; color:white" | 1 | style="background:white; color:black" | 0 | style="background:black; color:white" | 1 | style="background:white; color:black" | 0 |- | style="background:white; color:black" | 0 | style="background:black; color:white" | 1 | style="background:white; color:black" | 0 | style="background:black; color:white" | 1 | style="background:white; color:black" | 0 | style="background:black; color:white" | 1 | style="background:white; color:black" | 0 | style="background:black; color:white" | 1 |}<br> ==='"`UNIQ--h-53--QINU`"'Table 10=== {| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" |+ '''Table 10. Higher Order Propositions (''n'' = 1)''' |- style="background:ghostwhite" | align="right" | \(x\): |
1 0 | \(f\) | \(m_0\) | \(m_1\) | \(m_2\) | \(m_3\) | \(m_4\) | \(m_5\) | \(m_6\) | \(m_7\) | \(m_8\) | \(m_9\) | \(m_{10}\) | \(m_{11}\) | \(m_{12}\) | \(m_{13}\) | \(m_{14}\) | \(m_{15}\) |
| \(f_0\) | 0 0 | \(0\!\) | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| \(f_1\) | 0 1 | \((x)\!\) | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| \(f_2\) | 1 0 | \(x\!\) | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| \(f_3\) | 1 1 | \(1\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| \(x:\) | 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
| 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
| \(x:\) \(y:\) |
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 | \(((~))\!\) |
| \(u:\) \(v:\) |
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
| \(u:\) \(v:\) |
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
| \(u:\) \(v:\) |
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
|
\(\begin{array}{clcl} \mathrm{A} & \mathrm{Universal~Affirmative} & \mathrm{All}\ u\ \mathrm{is}\ v & \mathrm{Indicator~of}\ u (v) = 0 \\ \mathrm{E} & \mathrm{Universal~Negative} & \mathrm{All}\ u\ \mathrm{is}\ (v) & \mathrm{Indicator~of}\ u \cdot v = 0 \\ \mathrm{I} & \mathrm{Particular~Affirmative} & \mathrm{Some}\ u\ \mathrm{is}\ v & \mathrm{Indicator~of}\ u \cdot v = 1 \\ \mathrm{O} & \mathrm{Particular~Negative} & \mathrm{Some}\ u\ \mathrm{is}\ (v) & \mathrm{Indicator~of}\ u (v) = 1 \\ \end{array}\) |
Table 17
| \(u:\) \(v:\) |
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
| \(u:\) \(v:\) |
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
| \(\text{Mnemonic}\) | \(\text{Category}\) | \(\text{Classical Form}\) | \(\text{Alternate Form}\) | \(\text{Symmetric Form}\) | \(\text{Operator}\) |
| \(\text{E}\!\) \(\text{Exclusive}\) |
\(\text{Universal}\) \(\text{Negative}\) |
\(\text{All}\ u\ \text{is}\ (v)\) | \(\text{No}\ u\ \text{is}\ v \) | \((\ell_{11})\) | |
| \(\text{A}\!\) \(\text{Absolute}\) |
\(\text{Universal}\) \(\text{Affirmative}\) |
\(\text{All}\ u\ \text{is}\ v \) | \(\text{No}\ u\ \text{is}\ (v)\) | \((\ell_{10})\) | |
| \(\text{All}\ v\ \text{is}\ u \) | \(\text{No}\ v\ \text{is}\ (u)\) | \(\text{No}\ (u)\ \text{is}\ v \) | \((\ell_{01})\) | ||
| \(\text{All}\ (v)\ \text{is}\ u \) | \(\text{No}\ (v)\ \text{is}\ (u)\) | \(\text{No}\ (u)\ \text{is}\ (v)\) | \((\ell_{00})\) | ||
| \(\text{Some}\ (u)\ \text{is}\ (v)\) | \(\text{Some}\ (u)\ \text{is}\ (v)\) | \(\ell_{00}\!\) | |||
| \(\text{Some}\ (u)\ \text{is}\ v\) | \(\text{Some}\ (u)\ \text{is}\ v\) | \(\ell_{01}\!\) | |||
| \(\text{O}\!\) \(\text{Obtrusive}\) |
\(\text{Particular}\) \(\text{Negative}\) |
\(\text{Some}\ u\ \text{is}\ (v)\) | \(\text{Some}\ u\ \text{is}\ (v)\) | \(\ell_{10}\!\) | |
| \(\text{I}\!\) \(\text{Indefinite}\) |
\(\text{Particular}\) \(\text{Affirmative}\) |
\(\text{Some}\ u\ \text{is}\ v\) | \(\text{Some}\ u\ \text{is}\ v\) | \(\ell_{11}\!\) |
