User:Jon Awbrey/TABLE

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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

f1

f2

f4

f8

f0001

f0010

f0100

f1000

0 0 0 1

0 0 1 0

0 1 0 0

1 0 0 0

(x)(y)

(x) y

x (y)

x y

neither x nor y

not x but y

x but not y

x and y

¬x ∧ ¬y

¬x ∧ y

x ∧ ¬y

x ∧ y

f3

f12

f0011

f1100

0 0 1 1

1 1 0 0

(x)

x

not x

x

¬x

x

f6

f9

f0110

f1001

0 1 1 0

1 0 0 1

(x, y)

((x, y))

x not equal to y

x equal to y

x ≠ y

x = y

f5

f10

f0101

f1010

0 1 0 1

1 0 1 0

(y)

y

not y

y

¬y

y

f7

f11

f13

f14

f0111

f1011

f1101

f1110

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

 
 
 
 

g1
g2
g4
g8

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

(A)(dA)
(A) dA
A (dA)
A dA

Neither A nor dA
Not A but dA
A but not dA
A and dA

¬A ∧ ¬dA
¬A ∧ dA
A ∧ ¬dA
A ∧ dA

f1
f2

g3
g12

0 0 1 1
1 1 0 0

(A)
A

Not A
A

¬A
A

 
 

g6
g9

0 1 1 0
1 0 0 1

(A, dA)
((A, dA))

A not equal to dA
A equal to dA

A ≠ dA
A = dA

 
 

g5
g10

0 1 0 1
1 0 1 0

(dA)
dA

Not dA
dA

¬dA
dA

 
 
 
 

g7
g11
g13
g14

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

(A dA)
(A (dA))
((A) dA)
((A)(dA))

Not both A and dA
Not A without dA
Not dA without A
A or dA

¬A ∨ ¬dA
A ⇒ dA
A ⇐ dA
A ∨ dA

f3 g15 1 1 1 1 (( )) True 1


Wiki 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

 
 
 
 

g1
g2
g4
g8

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

(A)(dA)
(A) dA
A (dA)
A dA

Neither A nor dA
Not A but dA
A but not dA
A and dA

¬A ∧ ¬dA
¬A ∧ dA
A ∧ ¬dA
A ∧ dA

f1
f2

g3
g12

0 0 1 1
1 1 0 0

(A)
A

Not A
A

¬A
A

 
 

g6
g9

0 1 1 0
1 0 0 1

(A, dA)
((A, dA))

A not equal to dA
A equal to dA

A ≠ dA
A = dA

 
 

g5
g10

0 1 0 1
1 0 1 0

(dA)
dA

Not dA
dA

¬dA
dA

 
 
 
 

g7
g11
g13
g14

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

(A dA)
(A (dA))
((A) dA)
((A)(dA))

Not both A and dA
Not A without dA
Not dA without A
A or dA

¬A ∨ ¬dA
A → dA
A ← dA
A ∨ dA

f3 g15 1 1 1 1 (( )) True 1


Wiki TeX Tables : 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-00000016-QINU`"' '"`UNIQ--pre-00000017-QINU`"' '"`UNIQ--pre-00000018-QINU`"' '"`UNIQ-MathJax2-QINU`"' ===='"`UNIQ--h-34--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-35--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-36--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-37--QINU`"'Relational Tables== ==='"`UNIQ--h-38--QINU`"'Factorization=== {| align="center" style="text-align:center; width:60%" | {| align="center" style="text-align:center; width:100%" | \(\text{Table 7. Plural Denotation}\!\)

|- |

\(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\)

\(\begin{matrix} o_1 \\ o_2 \\ o_3 \\ \ldots \\ o_k \\ \ldots \end{matrix}\)

\(\begin{matrix} s \\ s \\ s \\ \ldots \\ s \\ \ldots \end{matrix}\)

\(\begin{matrix} \ldots \\ \ldots \\ \ldots \\ \ldots \\ \ldots \\ \ldots \end{matrix}\)

|}

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)
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)


Formatted Text Display

So in a triadic fact, say, the example
A gives B to C
we make no distinction in the ordinary logic of relations between the subject nominative, the direct object, and the indirect object. We say that the proposition has three logical subjects. We regard it as a mere affair of English grammar that there are six ways of expressing this:
A gives B to C A benefits C with B
B enriches C at expense of A C receives B from A
C thanks A for B B leaves A for C
These six sentences express one and the same indivisible phenomenon. (C.S. Peirce, "The Categories Defended", MS 308 (1903), EP 2, 170-171).

Work Area

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


Draft 1

TRUTH TABLES FOR THE BOOLEAN OPERATIONS OF ARITY UP TO 2
Constants
0f0 0f1
0 1
    
Unary Operations
x0 1f0 1f1 1f2 1f3
0 0 1 0 1
1 0 0 1 1
    
Binary Operations
x0 x1 2f0 2f1 2f2 2f3 2f4 2f5 2f6 2f7 2f8 2f9 2f10 2f11 2f12 2f13 2f14 2f15
0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
0 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1

Draft 2

TRUTH TABLES FOR THE BOOLEAN OPERATIONS OF ARITY UP TO 2
Constants
0f0 0f1
0 1
    
Unary Operations
x0 1f0 1f1 1f2 1f3
0 0 1 0 1
1 0 0 1 1
    
Binary Operations
x0 x1 2f0 2f1 2f2 2f3 2f4 2f5 2f6 2f7 2f8 2f9 2f10 2f11 2f12 2f13 2f14 2f15
0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
0 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1

Inquiry and Analogy

Test Patterns

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


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


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


Table 10

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


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


Table 11

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


Table 12

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


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


Table 13

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


Table 14

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


Figure 15

Table 16

Table 16. Syllogistic Premisses as Higher Order Indicator Functions

\(\begin{array}{clcl} \mathrm{A} & \mathrm{Universal~Affirmative} & \mathrm{All}\ u\ \mathrm{is}\ v & \mathrm{Indicator~of}\ u (v) = 0 \\ \mathrm{E} & \mathrm{Universal~Negative} & \mathrm{All}\ u\ \mathrm{is}\ (v) & \mathrm{Indicator~of}\ u \cdot v = 0 \\ \mathrm{I} & \mathrm{Particular~Affirmative} & \mathrm{Some}\ u\ \mathrm{is}\ v & \mathrm{Indicator~of}\ u \cdot v = 1 \\ \mathrm{O} & \mathrm{Particular~Negative} & \mathrm{Some}\ u\ \mathrm{is}\ (v) & \mathrm{Indicator~of}\ u (v) = 1 \\ \end{array}\)


Table 17

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


Table 18

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


Table 19

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


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