User:Jon Awbrey/TABLE

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%" \| \(\text{Table 7. Plural Denotation}\!\)

|- |

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} o_1 \\ o_2 \\ o_3 \\ \ldots \\ o_k \\ \ldots \end{matrix}\)	\(\begin{matrix} s \\ s \\ s \\ \ldots \\ s \\ \ldots \end{matrix}\)	\(\begin{matrix} \ldots \\ \ldots \\ \ldots \\ \ldots \\ \ldots \\ \ldots \end{matrix}\)

|}

\(\text{Table 8. Sign Relation}~ L\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} o_1 \\ o_2 \\ o_3 \end{matrix}\)	\(\begin{matrix} s \\ s \\ s \end{matrix}\)	\(\begin{matrix} \ldots \\ \ldots \\ \ldots \end{matrix}\)

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"}

L_A = 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"

L_B = 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

L_A = 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"

L_B = 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

L_OS	=	proj_OS(L)	=	{ (o, s) ∈ O × S : (o, s, i) ∈ L for some i ∈ I }
L_SO	=	proj_SO(L)	=	{ (s, o) ∈ S × O : (o, s, i) ∈ L for some i ∈ I }
L_IS	=	proj_IS(L)	=	{ (i, s) ∈ I × S : (o, s, i) ∈ L for some o ∈ O }
L_SI	=	proj_SI(L)	=	{ (s, i) ∈ S × I : (o, s, i) ∈ L for some o ∈ O }
L_OI	=	proj_OI(L)	=	{ (o, i) ∈ O × I : (o, s, i) ∈ L for some s ∈ S }
L_IO	=	proj_IO(L)	=	{ (i, o) ∈ I × O : (o, s, i) ∈ L for some s ∈ S }

Method 1 : Subtitles as Captions

*proj*_OS(L_A)
Object	Sign
A	"A"
A	"i"
B	"B"
B	"u"

*proj*_OS(L_B)
Object	Sign
A	"A"
A	"u"
B	"B"
B	"i"

*proj*_SI(L_A)
Sign	Interpretant
"A"	"A"
"A"	"i"
"i"	"A"
"i"	"i"
"B"	"B"
"B"	"u"
"u"	"B"
"u"	"u"

*proj*_SI(L_B)
Sign	Interpretant
"A"	"A"
"A"	"u"
"u"	"A"
"u"	"u"
"B"	"B"
"B"	"i"
"i"	"B"
"i"	"i"

*proj*_OI(L_A)
Object	Interpretant
A	"A"
A	"i"
B	"B"
B	"u"

*proj*_OI(L_B)
Object	Interpretant
A	"A"
A	"u"
B	"B"
B	"i"

Method 2 : Subtitles as Top Rows

proj_OS(L_A)

Object	Sign
A	"A"
A	"i"
B	"B"
B	"u"

proj_OS(L_B)

Object	Sign
A	"A"
A	"u"
B	"B"
B	"i"

proj_SI(L_A)

Sign	Interpretant
"A"	"A"
"A"	"i"
"i"	"A"
"i"	"i"
"B"	"B"
"B"	"u"
"u"	"B"
"u"	"u"

proj_SI(L_B)

Sign	Interpretant
"A"	"A"
"A"	"u"
"u"	"A"
"u"	"u"
"B"	"B"
"B"	"i"
"i"	"B"
"i"	"i"

proj_OI(L_A)

Object	Interpretant
A	"A"
A	"i"
B	"B"
B	"u"

proj_OI(L_B)

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

proj_XY(L₀)
X	Y
0	0
0	1
1	0
1	1

proj_XZ(L₀)
X	Z
0	0
0	1
1	1
1	0

proj_YZ(L₀)
Y	Z
0	0
1	1
0	1
1	0

proj_XY(L₁)
X	Y
0	0
0	1
1	0
1	1

proj_XZ(L₁)
X	Z
0	1
0	0
1	0
1	1

proj_YZ(L₁)
Y	Z
0	1
1	0
0	0
1	1

proj_XY(L₀) = proj_XY(L₁)

proj_XZ(L₀) = proj_XZ(L₁)

proj_YZ(L₀) = proj_YZ(L₁)

L_A = 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"

L_B = 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"

proj_XY(L_A)
Object	Sign
A	"A"
A	"i"
B	"B"
B	"u"

proj_XZ(L_A)
Object	Interpretant
A	"A"
A	"i"
B	"B"
B	"u"

proj_YZ(L_A)
Sign	Interpretant
"A"	"A"
"A"	"i"
"i"	"A"
"i"	"i"
"B"	"B"
"B"	"u"
"u"	"B"
"u"	"u"

proj_XY(L_B)
Object	Sign
A	"A"
A	"u"
B	"B"
B	"i"

proj_XZ(L_B)
Object	Interpretant
A	"A"
A	"u"
B	"B"
B	"i"

proj_YZ(L_B)
Sign	Interpretant
"A"	"A"
"A"	"u"
"u"	"A"
"u"	"u"
"B"	"B"
"B"	"i"
"i"	"B"
"i"	"i"

proj_XY(L_A) ≠ proj_XY(L_B)

proj_XZ(L_A) ≠ proj_XZ(L_B)

proj_YZ(L_A) ≠ proj_YZ(L_B)

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

proj_XY(L₀)

X	Y
0	0
0	1
1	0
1	1

proj_XZ(L₀)

X	Z
0	0
0	1
1	1
1	0

proj_YZ(L₀)

Y	Z
0	0
1	1
0	1
1	0

proj_XY(L₁)

X	Y
0	0
0	1
1	0
1	1

proj_XZ(L₁)

X	Z
0	1
0	0
1	0
1	1

proj_YZ(L₁)

Y	Z
0	1
1	0
0	0
1	1

proj_XY(L₀) = proj_XY(L₁)

proj_XZ(L₀) = proj_XZ(L₁)

proj_YZ(L₀) = proj_YZ(L₁)

L_A = 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"

L_B = 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"

proj_XY(L_A)

Object	Sign
A	"A"
A	"i"
B	"B"
B	"u"

proj_XZ(L_A)

Object	Interpretant
A	"A"
A	"i"
B	"B"
B	"u"

proj_YZ(L_A)

Sign	Interpretant
"A"	"A"
"A"	"i"
"i"	"A"
"i"	"i"
"B"	"B"
"B"	"u"
"u"	"B"
"u"	"u"

proj_XY(L_B)

Object	Sign
A	"A"
A	"u"
B	"B"
B	"i"

proj_XZ(L_B)

Object	Interpretant
A	"A"
A	"u"
B	"B"
B	"i"

proj_YZ(L_B)

Sign	Interpretant
"A"	"A"
"A"	"u"
"u"	"A"
"u"	"u"
"B"	"B"
"B"	"i"
"i"	"B"
"i"	"i"

proj_XY(L_A) ≠ proj_XY(L_B)

proj_XZ(L_A) ≠ proj_XZ(L_B)

proj_YZ(L_A) ≠ proj_YZ(L_B)

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**
\(\text{Mnemonic}\)	\(\text{Category}\)	\(\text{Classical Form}\)	\(\text{Alternate Form}\)	\(\text{Symmetric Form}\)	\(\text{Operator}\)
\(\text{E}\!\) \(\text{Exclusive}\)	\(\text{Universal}\) \(\text{Negative}\)	\(\text{All}\ u\ \text{is}\ (v)\)		\(\text{No}\ u\ \text{is}\ v \)	\((\ell_{11})\)
\(\text{A}\!\) \(\text{Absolute}\)	\(\text{Universal}\) \(\text{Affirmative}\)	\(\text{All}\ u\ \text{is}\ v \)		\(\text{No}\ u\ \text{is}\ (v)\)	\((\ell_{10})\)
		\(\text{All}\ v\ \text{is}\ u \)	\(\text{No}\ v\ \text{is}\ (u)\)	\(\text{No}\ (u)\ \text{is}\ v \)	\((\ell_{01})\)
		\(\text{All}\ (v)\ \text{is}\ u \)	\(\text{No}\ (v)\ \text{is}\ (u)\)	\(\text{No}\ (u)\ \text{is}\ (v)\)	\((\ell_{00})\)
		\(\text{Some}\ (u)\ \text{is}\ (v)\)		\(\text{Some}\ (u)\ \text{is}\ (v)\)	\(\ell_{00}\!\)
		\(\text{Some}\ (u)\ \text{is}\ v\)		\(\text{Some}\ (u)\ \text{is}\ v\)	\(\ell_{01}\!\)
\(\text{O}\!\) \(\text{Obtrusive}\)	\(\text{Particular}\) \(\text{Negative}\)	\(\text{Some}\ u\ \text{is}\ (v)\)		\(\text{Some}\ u\ \text{is}\ (v)\)	\(\ell_{10}\!\)
\(\text{I}\!\) \(\text{Indefinite}\)	\(\text{Particular}\) \(\text{Affirmative}\)	\(\text{Some}\ u\ \text{is}\ v\)		\(\text{Some}\ u\ \text{is}\ v\)	\(\ell_{11}\!\)