Cactus Language
Ascii Tables
o-------------------o
| |
| @ |
| |
o-------------------o
| |
| o |
| | |
| @ |
| |
o-------------------o
| |
| a |
| @ |
| |
o-------------------o
| |
| a |
| o |
| | |
| @ |
| |
o-------------------o
| |
| a b c |
| @ |
| |
o-------------------o
| |
| a b c |
| o o o |
| \|/ |
| o |
| | |
| @ |
| |
o-------------------o
| |
| |
| a b |
| o---o |
| | |
| @ |
| |
o-------------------o
| |
| a b |
| o---o |
| \ / |
| @ |
| |
o-------------------o
| |
| a b |
| o---o |
| \ / |
| o |
| | |
| @ |
| |
o-------------------o
| |
| a b c |
| o--o--o |
| \ / |
| \ / |
| @ |
| |
o-------------------o
| |
| a b c |
| o o o |
| | | | |
| o--o--o |
| \ / |
| \ / |
| @ |
| |
o-------------------o
| |
| |
| b c |
| o o |
| a | | |
| o--o--o |
| \ / |
| \ / |
| @ |
| |
o-------------------o
Table 13. The Existential Interpretation
o----o-------------------o-------------------o-------------------o
| Ex | Cactus Graph | Cactus Expression | Existential |
| | | | Interpretation |
o----o-------------------o-------------------o-------------------o
| | | | |
| 1 | @ | " " | true. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | o | | |
| | | | | |
| 2 | @ | ( ) | untrue. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a | | |
| 3 | @ | a | a. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a | | |
| | o | | |
| | | | | |
| 4 | @ | (a) | not a. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a b c | | |
| 5 | @ | a b c | a and b and c. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a b c | | |
| | o o o | | |
| | \|/ | | |
| | o | | |
| | | | | |
| 6 | @ | ((a)(b)(c)) | a or b or c. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | | | a implies b. |
| | a b | | |
| | o---o | | if a then b. |
| | | | | |
| 7 | @ | ( a (b)) | no a sans b. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a b | | |
| | o---o | | a exclusive-or b. |
| | \ / | | |
| 8 | @ | ( a , b ) | a not equal to b. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a b | | |
| | o---o | | |
| | \ / | | |
| | o | | a if & only if b. |
| | | | | |
| 9 | @ | (( a , b )) | a equates with b. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a b c | | |
| | o--o--o | | |
| | \ / | | |
| | \ / | | just one false |
| 10 | @ | ( a , b , c ) | out of a, b, c. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a b c | | |
| | o o o | | |
| | | | | | | |
| | o--o--o | | |
| | \ / | | |
| | \ / | | just one true |
| 11 | @ | ((a),(b),(c)) | among a, b, c. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | | | genus a over |
| | b c | | species b, c. |
| | o o | | |
| | a | | | | partition a |
| | o--o--o | | among b & c. |
| | \ / | | |
| | \ / | | whole pie a: |
| 12 | @ | ( a ,(b),(c)) | slices b, c. |
| | | | |
o----o-------------------o-------------------o-------------------o
Table 14. The Entitative Interpretation
o----o-------------------o-------------------o-------------------o
| En | Cactus Graph | Cactus Expression | Entitative |
| | | | Interpretation |
o----o-------------------o-------------------o-------------------o
| | | | |
| 1 | @ | " " | untrue. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | o | | |
| | | | | |
| 2 | @ | ( ) | true. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a | | |
| 3 | @ | a | a. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a | | |
| | o | | |
| | | | | |
| 4 | @ | (a) | not a. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a b c | | |
| 5 | @ | a b c | a or b or c. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a b c | | |
| | o o o | | |
| | \|/ | | |
| | o | | |
| | | | | |
| 6 | @ | ((a)(b)(c)) | a and b and c. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | | | a implies b. |
| | | | |
| | o a | | if a then b. |
| | | | | |
| 7 | @ b | (a) b | not a, or b. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a b | | |
| | o---o | | a if & only if b. |
| | \ / | | |
| 8 | @ | ( a , b ) | a equates with b. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a b | | |
| | o---o | | |
| | \ / | | |
| | o | | a exclusive-or b. |
| | | | | |
| 9 | @ | (( a , b )) | a not equal to b. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a b c | | |
| | o--o--o | | |
| | \ / | | |
| | \ / | | not just one true |
| 10 | @ | ( a , b , c ) | out of a, b, c. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a b c | | |
| | o--o--o | | |
| | \ / | | |
| | \ / | | |
| | o | | |
| | | | | just one true |
| 11 | @ | (( a , b , c )) | among a, b, c. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a | | |
| | o | | genus a over |
| | | b c | | species b, c. |
| | o--o--o | | |
| | \ / | | partition a |
| | \ / | | among b & c. |
| | o | | |
| | | | | whole pie a: |
| 12 | @ | (((a), b , c )) | slices b, c. |
| | | | |
o----o-------------------o-------------------o-------------------o
Table 15. Existential & Entitative Interpretations of Cactus Structures
o-----------------o-----------------o-----------------o-----------------o
| Cactus Graph | Cactus String | Existential | Entitative |
| | | Interpretation | Interpretation |
o-----------------o-----------------o-----------------o-----------------o
| | | | |
| @ | " " | true | false |
| | | | |
o-----------------o-----------------o-----------------o-----------------o
| | | | |
| o | | | |
| | | | | |
| @ | ( ) | false | true |
| | | | |
o-----------------o-----------------o-----------------o-----------------o
| | | | |
| C_1 ... C_k | | | |
| @ | C_1 ... C_k | C_1 & ... & C_k | C_1 v ... v C_k |
| | | | |
o-----------------o-----------------o-----------------o-----------------o
| | | | |
| C_1 C_2 C_k | | Just one | Not just one |
| o---o-...-o | | | |
| \ / | | of the C_j, | of the C_j, |
| \ / | | | |
| \ / | | j = 1 to k, | j = 1 to k, |
| \ / | | | |
| @ | (C_1, ..., C_k) | is not true. | is true. |
| | | | |
o-----------------o-----------------o-----------------o-----------------o
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
Table A1. Propositional Forms on Two Variables
| L1
|
L2
|
L3
|
L4
|
L5
|
L6
|
|
|
x :
|
1 1 0 0
|
|
|
|
|
|
y :
|
1 0 1 0
|
|
|
|
| f0
|
f0000
|
0 0 0 0
|
( )
|
false
|
0
|
| f1
|
f0001
|
0 0 0 1
|
(x)(y)
|
neither x nor y
|
¬x ∧ ¬y
|
| f2
|
f0010
|
0 0 1 0
|
(x) y
|
y and not x
|
¬x ∧ y
|
| f3
|
f0011
|
0 0 1 1
|
(x)
|
not x
|
¬x
|
| f4
|
f0100
|
0 1 0 0
|
x (y)
|
x and not y
|
x ∧ ¬y
|
| f5
|
f0101
|
0 1 0 1
|
(y)
|
not y
|
¬y
|
| f6
|
f0110
|
0 1 1 0
|
(x, y)
|
x not equal to y
|
x ≠ y
|
| f7
|
f0111
|
0 1 1 1
|
(x y)
|
not both x and y
|
¬x ∨ ¬y
|
| f8
|
f1000
|
1 0 0 0
|
x y
|
x and y
|
x ∧ y
|
| f9
|
f1001
|
1 0 0 1
|
((x, y))
|
x equal to y
|
x = y
|
| f10
|
f1010
|
1 0 1 0
|
y
|
y
|
y
|
| f11
|
f1011
|
1 0 1 1
|
(x (y))
|
not x without y
|
x ⇒ y
|
| f12
|
f1100
|
1 1 0 0
|
x
|
x
|
x
|
| f13
|
f1101
|
1 1 0 1
|
((x) y)
|
not y without x
|
x ⇐ y
|
| f14
|
f1110
|
1 1 1 0
|
((x)(y))
|
x or y
|
x ∨ y
|
| f15
|
f1111
|
1 1 1 1
|
(( ))
|
true |
1
|
Table A2. Propositional Forms on Two Variables
| L1
|
L2
|
L3
|
L4
|
L5
|
L6
|
|
|
x :
|
1 1 0 0
|
|
|
|
|
|
y :
|
1 0 1 0
|
|
|
|
| f0
|
f0000
|
0 0 0 0
|
( )
|
false
|
0
|
|
|
|
|
0 0 0 1
0 0 1 0
0 1 0 0
1 0 0 0
|
|
|
|
neither x nor y
not x but y
x but not y
x and y
|
|
|
¬x ∧ ¬y
¬x ∧ y
x ∧ ¬y
x ∧ y
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
x not equal to y
x equal to y
|
|
|
|
|
|
|
|
|
|
|
|
|
|
0 1 1 1
1 0 1 1
1 1 0 1
1 1 1 0
|
|
|
(x y)
(x (y))
((x) y)
((x)(y))
|
|
|
not both x and y
not x without y
not y without x
x or y
|
|
|
¬x ∨ ¬y
x ⇒ y
x ⇐ y
x ∨ y
|
|
| f15
|
f1111
|
1 1 1 1
|
(( ))
|
true
|
1
|
Differential Propositions
Table 14. Differential Propositions
|
|
A :
|
1 1 0 0
|
|
|
|
|
|
dA :
|
1 0 1 0
|
|
|
|
| f0
|
g0
|
0 0 0 0
|
( )
|
False
|
0
|
|
|
|
|
0 0 0 1
0 0 1 0
0 1 0 0
1 0 0 0
|
|
|
(A)(dA)
(A) dA
A (dA)
A dA
|
|
|
Neither A nor dA
Not A but dA
A but not dA
A and dA
|
|
|
¬A ∧ ¬dA
¬A ∧ dA
A ∧ ¬dA
A ∧ dA
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
A not equal to dA
A equal to dA
|
|
|
|
|
|
|
|
|
|
|
|
|
|
0 1 1 1
1 0 1 1
1 1 0 1
1 1 1 0
|
|
|
(A dA)
(A (dA))
((A) dA)
((A)(dA))
|
|
|
Not both A and dA
Not A without dA
Not dA without A
A or dA
|
|
|
¬A ∨ ¬dA
A ⇒ dA
A ⇐ dA
A ∨ dA
|
|
| f3
|
g15
|
1 1 1 1
|
(( ))
|
True
|
1
|
Wiki Tables : Old Versions
Propositional Forms on Two Variables
Table 1. Propositional Forms on Two Variables
| L1
|
L2
|
L3
|
L4
|
L5
|
L6
|
|
|
x :
|
1 1 0 0
|
|
|
|
|
|
y :
|
1 0 1 0
|
|
|
|
| f0 |
f0000 |
0 0 0 0 |
( ) |
false |
0
|
| f1 |
f0001 |
0 0 0 1 |
(x)(y) |
neither x nor y |
¬x ∧ ¬y
|
| f2 |
f0010 |
0 0 1 0 |
(x) y |
y and not x |
¬x ∧ y
|
| f3 |
f0011 |
0 0 1 1 |
(x) |
not x |
¬x
|
| f4 |
f0100 |
0 1 0 0 |
x (y) |
x and not y |
x ∧ ¬y
|
| f5 |
f0101 |
0 1 0 1 |
(y) |
not y |
¬y
|
| f6 |
f0110 |
0 1 1 0 |
(x, y) |
x not equal to y |
x ≠ y
|
| f7 |
f0111 |
0 1 1 1 |
(x y) |
not both x and y |
¬x ∨ ¬y
|
| f8 |
f1000 |
1 0 0 0 |
x y |
x and y |
x ∧ y
|
| f9 |
f1001 |
1 0 0 1 |
((x, y)) |
x equal to y |
x = y
|
| f10 |
f1010 |
1 0 1 0 |
y |
y |
y
|
| f11 |
f1011 |
1 0 1 1 |
(x (y)) |
not x without y |
x → y
|
| f12 |
f1100 |
1 1 0 0 |
x |
x |
x
|
| f13 |
f1101 |
1 1 0 1 |
((x) y) |
not y without x |
x ← y
|
| f14 |
f1110 |
1 1 1 0 |
((x)(y)) |
x or y |
x ∨ y
|
| f15 |
f1111 |
1 1 1 1 |
(( )) |
true |
1
|
Differential Propositions
Table 14. Differential Propositions
|
|
A :
|
1 1 0 0
|
|
|
|
|
|
dA :
|
1 0 1 0
|
|
|
|
| f0
|
g0
|
0 0 0 0
|
( )
|
False
|
0
|
|
|
|
|
0 0 0 1
0 0 1 0
0 1 0 0
1 0 0 0
|
|
|
(A)(dA)
(A) dA
A (dA)
A dA
|
|
|
Neither A nor dA
Not A but dA
A but not dA
A and dA
|
|
|
¬A ∧ ¬dA
¬A ∧ dA
A ∧ ¬dA
A ∧ dA
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
A not equal to dA
A equal to dA
|
|
|
|
|
|
|
|
|
|
|
|
|
|
0 1 1 1
1 0 1 1
1 1 0 1
1 1 1 0
|
|
|
(A dA)
(A (dA))
((A) dA)
((A)(dA))
|
|
|
Not both A and dA
Not A without dA
Not dA without A
A or dA
|
|
|
¬A ∨ ¬dA
A → dA
A ← dA
A ∨ dA
|
|
| f3
|
g15
|
1 1 1 1
|
(( ))
|
True
|
1
|
Wiki TeX Tables : PQ
\(\text{Table A1.}~~\text{Propositional Forms on Two Variables}\)
|
\(\mathcal{L}_1\)
\(\text{Decimal}\)
|
\(\mathcal{L}_2\)
\(\text{Binary}\)
|
\(\mathcal{L}_3\)
\(\text{Vector}\)
|
\(\mathcal{L}_4\)
\(\text{Cactus}\)
|
\(\mathcal{L}_5\)
\(\text{English}\)
|
\(\mathcal{L}_6\)
\(\text{Ordinary}\)
|
|
|
\(p\colon\!\)
|
\(1~1~0~0\!\)
|
|
|
|
|
|
\(q\colon\!\)
|
\(1~0~1~0\!\)
|
|
|
|
|
\(\begin{matrix}
f_0
\'"`UNIQ-MathJax1-QINU`"'
'''Generalized''' or '''n-ary''' XOR is true when the number of 1-bits is odd.
'"`UNIQ--pre-0000001A-QINU`"'
'"`UNIQ--pre-0000001B-QINU`"'
'"`UNIQ--pre-0000001C-QINU`"'
'"`UNIQ-MathJax2-QINU`"'
===='"`UNIQ--h-36--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-37--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-38--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-39--QINU`"'Relational Tables==
==='"`UNIQ--h-40--QINU`"'Factorization===
{| align="center" style="text-align:center; width:60%"
|
{| align="center" style="text-align:center; width:100%"
| \(\text{Table 7. Plural Denotation}\!\)
|
|-
|
| \(\text{Object}\!\)
|
\(\text{Sign}\!\)
|
\(\text{Interpretant}\!\)
|
|
\(\begin{matrix}
o_1 \\ o_2 \\ o_3 \\ \ldots \\ o_k \\ \ldots
\end{matrix}\)
|
\(\begin{matrix}
s \\ s \\ s \\ \ldots \\ s \\ \ldots
\end{matrix}\)
|
\(\begin{matrix}
\ldots \\ \ldots \\ \ldots \\ \ldots \\ \ldots \\ \ldots
\end{matrix}\)
|
|}
| \(\text{Table 8. Sign Relation}~ L\)
|
|
| \(\text{Object}\!\)
|
\(\text{Sign}\!\)
|
\(\text{Interpretant}\!\)
|
|
\(\begin{matrix}
o_1 \\ o_2 \\ o_3
\end{matrix}\)
|
\(\begin{matrix}
s \\ s \\ s
\end{matrix}\)
|
\(\begin{matrix}
\ldots \\ \ldots \\ \ldots
\end{matrix}\)
|
|
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
| \(\text{Mnemonic}\)
|
\(\text{Category}\)
|
\(\text{Classical Form}\)
|
\(\text{Alternate Form}\)
|
\(\text{Symmetric Form}\)
|
\(\text{Operator}\)
|
\(\text{E}\!\)
\(\text{Exclusive}\)
|
\(\text{Universal}\)
\(\text{Negative}\)
|
\(\text{All}\ u\ \text{is}\ (v)\)
|
|
\(\text{No}\ u\ \text{is}\ v \)
|
\((\ell_{11})\)
|
\(\text{A}\!\)
\(\text{Absolute}\)
|
\(\text{Universal}\)
\(\text{Affirmative}\)
|
\(\text{All}\ u\ \text{is}\ v \)
|
|
\(\text{No}\ u\ \text{is}\ (v)\)
|
\((\ell_{10})\)
|
|
|
|
\(\text{All}\ v\ \text{is}\ u \)
|
\(\text{No}\ v\ \text{is}\ (u)\)
|
\(\text{No}\ (u)\ \text{is}\ v \)
|
\((\ell_{01})\)
|
|
|
|
\(\text{All}\ (v)\ \text{is}\ u \)
|
\(\text{No}\ (v)\ \text{is}\ (u)\)
|
\(\text{No}\ (u)\ \text{is}\ (v)\)
|
\((\ell_{00})\)
|
|
|
|
\(\text{Some}\ (u)\ \text{is}\ (v)\)
|
|
\(\text{Some}\ (u)\ \text{is}\ (v)\)
|
\(\ell_{00}\!\)
|
|
|
|
\(\text{Some}\ (u)\ \text{is}\ v\)
|
|
\(\text{Some}\ (u)\ \text{is}\ v\)
|
\(\ell_{01}\!\)
|
\(\text{O}\!\)
\(\text{Obtrusive}\)
|
\(\text{Particular}\)
\(\text{Negative}\)
|
\(\text{Some}\ u\ \text{is}\ (v)\)
|
|
\(\text{Some}\ u\ \text{is}\ (v)\)
|
\(\ell_{10}\!\)
|
\(\text{I}\!\)
\(\text{Indefinite}\)
|
\(\text{Particular}\)
\(\text{Affirmative}\)
|
\(\text{Some}\ u\ \text{is}\ v\)
|
|
\(\text{Some}\ u\ \text{is}\ v\)
|
\(\ell_{11}\!\)
|
