Difference between revisions of "User:Jon Awbrey/TABLE"
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{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" | ||
| − | |+ '''Table | + | |+ '''Table A1. Propositional Forms on Two Variables''' |
|- style="background:#f0f0ff" | |- style="background:#f0f0ff" | ||
! width="15%" | L<sub>1</sub> | ! width="15%" | L<sub>1</sub> | ||
| Line 477: | Line 477: | ||
<br> | <br> | ||
| − | {| align="center" border="1" cellpadding=" | + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" |
| − | |+ '''Table | + | |+ '''Table A2. Propositional Forms on Two Variables''' |
| + | |- style="background:#f0f0ff" | ||
| + | ! width="15%" | L<sub>1</sub> | ||
| + | ! width="15%" | L<sub>2</sub> | ||
| + | ! width="15%" | L<sub>3</sub> | ||
| + | ! width="15%" | L<sub>4</sub> | ||
| + | ! width="25%" | L<sub>5</sub> | ||
| + | ! width="15%" | L<sub>6</sub> | ||
|- style="background:#f0f0ff" | |- style="background:#f0f0ff" | ||
| | | | ||
| − | | align="right" | | + | | align="right" | x : |
| 1 1 0 0 | | 1 1 0 0 | ||
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| Line 488: | Line 495: | ||
|- style="background:#f0f0ff" | |- style="background:#f0f0ff" | ||
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| − | | align="right" | | + | | align="right" | y : |
| 1 0 1 0 | | 1 0 1 0 | ||
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| Line 495: | Line 502: | ||
|- | |- | ||
| f<sub>0</sub> | | f<sub>0</sub> | ||
| − | | | + | | f<sub>0000</sub> |
| 0 0 0 0 | | 0 0 0 0 | ||
| ( ) | | ( ) | ||
| − | | | + | | false |
| 0 | | 0 | ||
|- | |- | ||
| | | | ||
| − | {| | + | {| align="center" |
| | | | ||
| − | + | <p>f<sub>1</sub></p> | |
| − | + | <p>f<sub>2</sub></p> | |
| − | + | <p>f<sub>4</sub></p> | |
| − | + | <p>f<sub>8</sub></p> | |
|} | |} | ||
| | | | ||
| − | {| | + | {| align="center" |
| | | | ||
| − | + | <p>f<sub>0001</sub></p> | |
| − | + | <p>f<sub>0010</sub></p> | |
| − | + | <p>f<sub>0100</sub></p> | |
| − | + | <p>f<sub>1000</sub></p> | |
|} | |} | ||
| | | | ||
| − | {| | + | {| align="center" |
| | | | ||
| − | 0 0 0 1< | + | <p>0 0 0 1</p> |
| − | 0 0 1 0< | + | <p>0 0 1 0</p> |
| − | 0 1 0 0< | + | <p>0 1 0 0</p> |
| − | 1 0 0 0 | + | <p>1 0 0 0</p> |
|} | |} | ||
| | | | ||
| − | {| | + | {| align="center" |
| | | | ||
| − | ( | + | <p>(x)(y)</p> |
| − | ( | + | <p>(x) y </p> |
| − | + | <p> x (y)</p> | |
| − | + | <p> x y </p> | |
|} | |} | ||
| | | | ||
| − | {| | + | {| align="center" |
| | | | ||
| − | + | <p>neither x nor y</p> | |
| − | + | <p>not x but y</p> | |
| − | + | <p>x but not y</p> | |
| − | + | <p>x and y</p> | |
|} | |} | ||
| | | | ||
| − | {| | + | {| align="center" |
| | | | ||
| − | ¬ | + | <p>¬x ∧ ¬y</p> |
| − | ¬ | + | <p>¬x ∧ y</p> |
| − | + | <p>x ∧ ¬y</p> | |
| − | + | <p>x ∧ y</p> | |
|} | |} | ||
|- | |- | ||
| | | | ||
| − | {| | + | {| align="center" |
| | | | ||
| − | f<sub> | + | <p>f<sub>3</sub></p> |
| − | f<sub> | + | <p>f<sub>12</sub></p> |
|} | |} | ||
| | | | ||
| − | {| | + | {| align="center" |
| | | | ||
| − | + | <p>f<sub>0011</sub></p> | |
| − | + | <p>f<sub>1100</sub></p> | |
|} | |} | ||
| | | | ||
| − | {| | + | {| align="center" |
| | | | ||
| − | 0 0 1 1< | + | <p>0 0 1 1</p> |
| − | 1 1 0 0 | + | <p>1 1 0 0</p> |
|} | |} | ||
| | | | ||
| − | {| | + | {| align="center" |
| | | | ||
| − | ( | + | <p>(x)</p> |
| − | + | <p> x </p> | |
|} | |} | ||
| | | | ||
| − | {| | + | {| align="center" |
| | | | ||
| − | + | <p>not x</p> | |
| − | + | <p>x</p> | |
|} | |} | ||
| | | | ||
| − | {| | + | {| align="center" |
| | | | ||
| − | ¬ | + | <p>¬x</p> |
| − | + | <p>x</p> | |
|} | |} | ||
|- | |- | ||
| | | | ||
| − | {| | + | {| align="center" |
| | | | ||
| − | + | <p>f<sub>6</sub></p> | |
| − | + | <p>f<sub>9</sub></p> | |
|} | |} | ||
| | | | ||
| − | {| | + | {| align="center" |
| | | | ||
| − | + | <p>f<sub>0110</sub></p> | |
| − | + | <p>f<sub>1001</sub></p> | |
|} | |} | ||
| | | | ||
| − | {| | + | {| align="center" |
| | | | ||
| − | 0 1 1 0< | + | <p>0 1 1 0</p> |
| − | 1 0 0 1 | + | <p>1 0 0 1</p> |
|} | |} | ||
| | | | ||
| − | {| | + | {| align="center" |
| | | | ||
| − | ( | + | <p> (x, y) </p> |
| − | (( | + | <p>((x, y))</p> |
|} | |} | ||
| | | | ||
| − | {| | + | {| align="center" |
| | | | ||
| − | + | <p>x not equal to y</p> | |
| − | + | <p>x equal to y</p> | |
|} | |} | ||
| | | | ||
| − | {| | + | {| align="center" |
| | | | ||
| − | + | <p>x ≠ y</p> | |
| − | + | <p>x = y</p> | |
|} | |} | ||
|- | |- | ||
| | | | ||
| − | {| | + | {| align="center" |
| | | | ||
| − | + | <p>f<sub>5</sub></p> | |
| − | + | <p>f<sub>10</sub></p> | |
|} | |} | ||
| | | | ||
| − | {| | + | {| align="center" |
| | | | ||
| − | + | <p>f<sub>0101</sub></p> | |
| − | + | <p>f<sub>1010</sub></p> | |
|} | |} | ||
| | | | ||
| − | {| | + | {| align="center" |
| | | | ||
| − | 0 1 0 1< | + | <p>0 1 0 1</p> |
| − | 1 0 1 0 | + | <p>1 0 1 0</p> |
|} | |} | ||
| | | | ||
| − | {| | + | {| align="center" |
| | | | ||
| − | ( | + | <p>(y)</p> |
| − | + | <p> y </p> | |
|} | |} | ||
| | | | ||
| − | {| | + | {| align="center" |
| | | | ||
| − | + | <p>not y</p> | |
| − | + | <p>y</p> | |
|} | |} | ||
| | | | ||
| − | {| | + | {| align="center" |
| | | | ||
| − | ¬ | + | <p>¬y</p> |
| − | + | <p>y</p> | |
|} | |} | ||
|- | |- | ||
| | | | ||
| − | {| | + | {| align="center" |
| | | | ||
| − | + | <p>f<sub>7</sub></p> | |
| − | + | <p>f<sub>11</sub></p> | |
| − | + | <p>f<sub>13</sub></p> | |
| − | + | <p>f<sub>14</sub></p> | |
|} | |} | ||
| | | | ||
| − | {| | + | {| align="center" |
| | | | ||
| − | + | <p>f<sub>0111</sub></p> | |
| − | + | <p>f<sub>1011</sub></p> | |
| − | + | <p>f<sub>1101</sub></p> | |
| − | + | <p>f<sub>1110</sub></p> | |
|} | |} | ||
| | | | ||
| − | {| | + | {| align="center" |
| | | | ||
| − | 0 1 1 1< | + | <p>0 1 1 1</p> |
| − | 1 0 1 1< | + | <p>1 0 1 1</p> |
| − | 1 1 0 1< | + | <p>1 1 0 1</p> |
| − | 1 1 1 0 | + | <p>1 1 1 0</p> |
|} | |} | ||
| | | | ||
| − | {| | + | {| align="center" |
| | | | ||
| − | ( | + | <p>(x y)</p> |
| − | ( | + | <p>(x (y))</p> |
| − | (( | + | <p>((x) y)</p> |
| − | (( | + | <p>((x)(y))</p> |
|} | |} | ||
| | | | ||
| − | {| | + | {| align="center" |
| | | | ||
| − | + | <p>not both x and y</p> | |
| − | + | <p>not x without y</p> | |
| − | + | <p>not y without x</p> | |
| − | + | <p>x or y</p> | |
|} | |} | ||
| | | | ||
| − | {| | + | {| align="center" |
| | | | ||
| − | ¬ | + | <p>¬x ∨ ¬y</p> |
| − | + | <p>x ⇒ y</p> | |
| − | + | <p>x ⇐ y</p> | |
| − | + | <p>x ∨ y</p> | |
|} | |} | ||
|- | |- | ||
| − | | f<sub> | + | | f<sub>15</sub> |
| − | | | + | | f<sub>1111</sub> |
| 1 1 1 1 | | 1 1 1 1 | ||
| (( )) | | (( )) | ||
| − | | | + | | true |
| 1 | | 1 | ||
|} | |} | ||
Revision as of 21:52, 29 May 2009
Differential Logic
Ascii Tables
Table A1. Propositional Forms On Two Variables o---------o---------o---------o----------o------------------o----------o | L_1 | L_2 | L_3 | L_4 | L_5 | L_6 | | | | | | | | | Decimal | Binary | Vector | Cactus | English | Ordinary | o---------o---------o---------o----------o------------------o----------o | | x : 1 1 0 0 | | | | | | y : 1 0 1 0 | | | | o---------o---------o---------o----------o------------------o----------o | | | | | | | | f_0 | f_0000 | 0 0 0 0 | () | false | 0 | | | | | | | | | f_1 | f_0001 | 0 0 0 1 | (x)(y) | neither x nor y | ~x & ~y | | | | | | | | | f_2 | f_0010 | 0 0 1 0 | (x) y | y and not x | ~x & y | | | | | | | | | f_3 | f_0011 | 0 0 1 1 | (x) | not x | ~x | | | | | | | | | f_4 | f_0100 | 0 1 0 0 | x (y) | x and not y | x & ~y | | | | | | | | | f_5 | f_0101 | 0 1 0 1 | (y) | not y | ~y | | | | | | | | | f_6 | f_0110 | 0 1 1 0 | (x, y) | x not equal to y | x + y | | | | | | | | | f_7 | f_0111 | 0 1 1 1 | (x y) | not both x and y | ~x v ~y | | | | | | | | | f_8 | f_1000 | 1 0 0 0 | x y | x and y | x & y | | | | | | | | | f_9 | f_1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y | | | | | | | | | f_10 | f_1010 | 1 0 1 0 | y | y | y | | | | | | | | | f_11 | f_1011 | 1 0 1 1 | (x (y)) | not x without y | x => y | | | | | | | | | f_12 | f_1100 | 1 1 0 0 | x | x | x | | | | | | | | | f_13 | f_1101 | 1 1 0 1 | ((x) y) | not y without x | x <= y | | | | | | | | | f_14 | f_1110 | 1 1 1 0 | ((x)(y)) | x or y | x v y | | | | | | | | | f_15 | f_1111 | 1 1 1 1 | (()) | true | 1 | | | | | | | | o---------o---------o---------o----------o------------------o----------o
Table A2. Propositional Forms On Two Variables o---------o---------o---------o----------o------------------o----------o | L_1 | L_2 | L_3 | L_4 | L_5 | L_6 | | | | | | | | | Decimal | Binary | Vector | Cactus | English | Ordinary | o---------o---------o---------o----------o------------------o----------o | | x : 1 1 0 0 | | | | | | y : 1 0 1 0 | | | | o---------o---------o---------o----------o------------------o----------o | | | | | | | | f_0 | f_0000 | 0 0 0 0 | () | false | 0 | | | | | | | | o---------o---------o---------o----------o------------------o----------o | | | | | | | | f_1 | f_0001 | 0 0 0 1 | (x)(y) | neither x nor y | ~x & ~y | | | | | | | | | f_2 | f_0010 | 0 0 1 0 | (x) y | y and not x | ~x & y | | | | | | | | | f_4 | f_0100 | 0 1 0 0 | x (y) | x and not y | x & ~y | | | | | | | | | f_8 | f_1000 | 1 0 0 0 | x y | x and y | x & y | | | | | | | | o---------o---------o---------o----------o------------------o----------o | | | | | | | | f_3 | f_0011 | 0 0 1 1 | (x) | not x | ~x | | | | | | | | | f_12 | f_1100 | 1 1 0 0 | x | x | x | | | | | | | | o---------o---------o---------o----------o------------------o----------o | | | | | | | | f_6 | f_0110 | 0 1 1 0 | (x, y) | x not equal to y | x + y | | | | | | | | | f_9 | f_1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y | | | | | | | | o---------o---------o---------o----------o------------------o----------o | | | | | | | | f_5 | f_0101 | 0 1 0 1 | (y) | not y | ~y | | | | | | | | | f_10 | f_1010 | 1 0 1 0 | y | y | y | | | | | | | | o---------o---------o---------o----------o------------------o----------o | | | | | | | | f_7 | f_0111 | 0 1 1 1 | (x y) | not both x and y | ~x v ~y | | | | | | | | | f_11 | f_1011 | 1 0 1 1 | (x (y)) | not x without y | x => y | | | | | | | | | f_13 | f_1101 | 1 1 0 1 | ((x) y) | not y without x | x <= y | | | | | | | | | f_14 | f_1110 | 1 1 1 0 | ((x)(y)) | x or y | x v y | | | | | | | | o---------o---------o---------o----------o------------------o----------o | | | | | | | | f_15 | f_1111 | 1 1 1 1 | (()) | true | 1 | | | | | | | | o---------o---------o---------o----------o------------------o----------o
Table A3. Ef Expanded Over Differential Features {dx, dy}
o------o------------o------------o------------o------------o------------o
| | | | | | |
| | f | T_11 f | T_10 f | T_01 f | T_00 f |
| | | | | | |
| | | Ef| dx dy | Ef| dx(dy) | Ef| (dx)dy | Ef|(dx)(dy)|
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_0 | () | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_1 | (x)(y) | x y | x (y) | (x) y | (x)(y) |
| | | | | | |
| f_2 | (x) y | x (y) | x y | (x)(y) | (x) y |
| | | | | | |
| f_4 | x (y) | (x) y | (x)(y) | x y | x (y) |
| | | | | | |
| f_8 | x y | (x)(y) | (x) y | x (y) | x y |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_3 | (x) | x | x | (x) | (x) |
| | | | | | |
| f_12 | x | (x) | (x) | x | x |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_6 | (x, y) | (x, y) | ((x, y)) | ((x, y)) | (x, y) |
| | | | | | |
| f_9 | ((x, y)) | ((x, y)) | (x, y) | (x, y) | ((x, y)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_5 | (y) | y | (y) | y | (y) |
| | | | | | |
| f_10 | y | (y) | y | (y) | y |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_7 | (x y) | ((x)(y)) | ((x) y) | (x (y)) | (x y) |
| | | | | | |
| f_11 | (x (y)) | ((x) y) | ((x)(y)) | (x y) | (x (y)) |
| | | | | | |
| f_13 | ((x) y) | (x (y)) | (x y) | ((x)(y)) | ((x) y) |
| | | | | | |
| f_14 | ((x)(y)) | (x y) | (x (y)) | ((x) y) | ((x)(y)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_15 | (()) | (()) | (()) | (()) | (()) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | |
| Fixed Point Total | 4 | 4 | 4 | 16 |
| | | | | |
o-------------------o------------o------------o------------o------------o
Table A4. Df Expanded Over Differential Features {dx, dy}
o------o------------o------------o------------o------------o------------o
| | | | | | |
| | f | Df| dx dy | Df| dx(dy) | Df| (dx)dy | Df|(dx)(dy)|
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_0 | () | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_1 | (x)(y) | ((x, y)) | (y) | (x) | () |
| | | | | | |
| f_2 | (x) y | (x, y) | y | (x) | () |
| | | | | | |
| f_4 | x (y) | (x, y) | (y) | x | () |
| | | | | | |
| f_8 | x y | ((x, y)) | y | x | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_3 | (x) | (()) | (()) | () | () |
| | | | | | |
| f_12 | x | (()) | (()) | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_6 | (x, y) | () | (()) | (()) | () |
| | | | | | |
| f_9 | ((x, y)) | () | (()) | (()) | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_5 | (y) | (()) | () | (()) | () |
| | | | | | |
| f_10 | y | (()) | () | (()) | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_7 | (x y) | ((x, y)) | y | x | () |
| | | | | | |
| f_11 | (x (y)) | (x, y) | (y) | x | () |
| | | | | | |
| f_13 | ((x) y) | (x, y) | y | (x) | () |
| | | | | | |
| f_14 | ((x)(y)) | ((x, y)) | (y) | (x) | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_15 | (()) | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
Table A5. Ef Expanded Over Ordinary Features {x, y}
o------o------------o------------o------------o------------o------------o
| | | | | | |
| | f | Ef | xy | Ef | x(y) | Ef | (x)y | Ef | (x)(y)|
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_0 | () | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_1 | (x)(y) | dx dy | dx (dy) | (dx) dy | (dx)(dy) |
| | | | | | |
| f_2 | (x) y | dx (dy) | dx dy | (dx)(dy) | (dx) dy |
| | | | | | |
| f_4 | x (y) | (dx) dy | (dx)(dy) | dx dy | dx (dy) |
| | | | | | |
| f_8 | x y | (dx)(dy) | (dx) dy | dx (dy) | dx dy |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_3 | (x) | dx | dx | (dx) | (dx) |
| | | | | | |
| f_12 | x | (dx) | (dx) | dx | dx |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_6 | (x, y) | (dx, dy) | ((dx, dy)) | ((dx, dy)) | (dx, dy) |
| | | | | | |
| f_9 | ((x, y)) | ((dx, dy)) | (dx, dy) | (dx, dy) | ((dx, dy)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_5 | (y) | dy | (dy) | dy | (dy) |
| | | | | | |
| f_10 | y | (dy) | dy | (dy) | dy |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_7 | (x y) | ((dx)(dy)) | ((dx) dy) | (dx (dy)) | (dx dy) |
| | | | | | |
| f_11 | (x (y)) | ((dx) dy) | ((dx)(dy)) | (dx dy) | (dx (dy)) |
| | | | | | |
| f_13 | ((x) y) | (dx (dy)) | (dx dy) | ((dx)(dy)) | ((dx) dy) |
| | | | | | |
| f_14 | ((x)(y)) | (dx dy) | (dx (dy)) | ((dx) dy) | ((dx)(dy)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_15 | (()) | (()) | (()) | (()) | (()) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
Table A6. Df Expanded Over Ordinary Features {x, y}
o------o------------o------------o------------o------------o------------o
| | | | | | |
| | f | Df | xy | Df | x(y) | Df | (x)y | Df | (x)(y)|
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_0 | () | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_1 | (x)(y) | dx dy | dx (dy) | (dx) dy | ((dx)(dy)) |
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| f_2 | (x) y | dx (dy) | dx dy | ((dx)(dy)) | (dx) dy |
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| f_4 | x (y) | (dx) dy | ((dx)(dy)) | dx dy | dx (dy) |
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| f_8 | x y | ((dx)(dy)) | (dx) dy | dx (dy) | dx dy |
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o------o------------o------------o------------o------------o------------o
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| f_3 | (x) | dx | dx | dx | dx |
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| f_12 | x | dx | dx | dx | dx |
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o------o------------o------------o------------o------------o------------o
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| f_6 | (x, y) | (dx, dy) | (dx, dy) | (dx, dy) | (dx, dy) |
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| f_9 | ((x, y)) | (dx, dy) | (dx, dy) | (dx, dy) | (dx, dy) |
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o------o------------o------------o------------o------------o------------o
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| f_5 | (y) | dy | dy | dy | dy |
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| f_10 | y | dy | dy | dy | dy |
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o------o------------o------------o------------o------------o------------o
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| f_7 | (x y) | ((dx)(dy)) | (dx) dy | dx (dy) | dx dy |
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| 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)) |
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o------o------------o------------o------------o------------o------------o
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| f_15 | (()) | () | () | () | () |
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o------o------------o------------o------------o------------o------------o
Wiki Tables
New Versions
| L1 | L2 | L3 | L4 | L5 | L6 |
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| 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 |
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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
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Wiki TeX Tables
|
\(\mathcal{L}_1\) \(\text{Decimal}\) |
\(\mathcal{L}_2\) \(\text{Binary}\) |
\(\mathcal{L}_3\) \(\text{Vector}\) |
\(\mathcal{L}_4\) \(\text{Cactus}\) |
\(\mathcal{L}_5\) \(\text{English}\) |
\(\mathcal{L}_6\) \(\text{Ordinary}\) |
| \(x\colon\!\) | \(1~1~0~0\!\) | ||||
| \(y\colon\!\) | \(1~0~1~0\!\) | ||||
| \(f_{0}\!\) | \(f_{0000}\!\) | \(0~0~0~0\!\) | \((~)\!\) | \(\text{false}\!\) | \(0\!\) |
| \(f_{1}\!\) | \(f_{0001}\!\) | \(0~0~0~1\!\) | \((x)(y)\!\) | \(\text{neither}~ x ~\text{nor}~ y\!\) | \(\lnot x \land \lnot y\!\) |
| \(f_{2}\!\) | \(f_{0010}\!\) | \(0~0~1~0\!\) | \((x)~y\!\) | \(y ~\text{without}~ x\!\) | \(\lnot x \land y\!\) |
| \(f_{3}\!\) | \(f_{0011}\!\) | \(0~0~1~1\!\) | \((x)\!\) | \(\text{not}~ x\!\) | \(\lnot x\!\) |
| \(f_{4}\!\) | \(f_{0100}\!\) | \(0~1~0~0\!\) | \(x~(y)\!\) | \(x ~\text{without}~ y\!\) | \(x \land \lnot y\!\) |
| \(f_{5}\!\) | \(f_{0101}\!\) | \(0~1~0~1\!\) | \((y)\!\) | \(\text{not}~ y\!\) | \(\lnot y\!\) |
| \(f_{6}\!\) | \(f_{0110}\!\) | \(0~1~1~0\!\) | \((x,~y)\!\) | \(x ~\text{not equal to}~ y\!\) | \(x \ne y\!\) |
| \(f_{7}\!\) | \(f_{0111}\!\) | \(0~1~1~1\!\) | \((x~y)\!\) | \(\text{not both}~ x ~\text{and}~ y\!\) | \(\lnot x \lor \lnot y\!\) |
| \(f_{8}\!\) | \(f_{1000}\!\) | \(1~0~0~0\!\) | \(x~y\!\) | \(x ~\text{and}~ y\!\) | \(x \land y\!\) |
| \(f_{9}\!\) | \(f_{1001}\!\) | \(1~0~0~1\!\) | \(((x,~y))\!\) | \(x ~\text{equal to}~ y\!\) | \(x = y\!\) |
| \(f_{10}\!\) | \(f_{1010}\!\) | \(1~0~1~0\!\) | \(y\!\) | \(y\!\) | \(y\!\) |
| \(f_{11}\!\) | \(f_{1011}\!\) | \(1~0~1~1\!\) | \((x~(y))\!\) | \(\text{not}~ x ~\text{without}~ y\!\) | \(x \Rightarrow y\!\) |
| \(f_{12}\!\) | \(f_{1100}\!\) | \(1~1~0~0\!\) | \(x\!\) | \(x\!\) | \(x\!\) |
| \(f_{13}\!\) | \(f_{1101}\!\) | \(1~1~0~1\!\) | \(((x)~y)\!\) | \(\text{not}~ y ~\text{without}~ x\!\) | \(x \Leftarrow y\!\) |
| \(f_{14}\!\) | \(f_{1110}\!\) | \(1~1~1~0\!\) | \(((x)(y))\!\) | \(x ~\text{or}~ y\!\) | \(x \lor y\!\) |
| \(f_{15}\!\) | \(f_{1111}\!\) | \(1~1~1~1\!\) | \(((~))\!\) | \(\text{true}\!\) | \(1\!\) |
TeX Tables
\tableofcontents
\subsection{Table A1. Propositional Forms on Two Variables}
Table A1 lists equivalent expressions for the Boolean functions of two variables in a number of different notational systems.
\begin{quote}\begin{tabular}{|c|c|c|c|c|c|c|}
\multicolumn{7}{c}{\textbf{Table A1. Propositional Forms on Two Variables}} \\
\hline
$\mathcal{L}_1$ &
$\mathcal{L}_2$ &&
$\mathcal{L}_3$ &
$\mathcal{L}_4$ &
$\mathcal{L}_5$ &
$\mathcal{L}_6$ \\
\hline
& & $x =$ & 1 1 0 0 & & & \\
& & $y =$ & 1 0 1 0 & & & \\
\hline
$f_{0}$ &
$f_{0000}$ &&
0 0 0 0 &
$(~)$ &
$\operatorname{false}$ &
$0$ \\
$f_{1}$ &
$f_{0001}$ &&
0 0 0 1 &
$(x)(y)$ &
$\operatorname{neither}\ x\ \operatorname{nor}\ y$ &
$\lnot x \land \lnot y$ \\
$f_{2}$ &
$f_{0010}$ &&
0 0 1 0 &
$(x)\ y$ &
$y\ \operatorname{without}\ x$ &
$\lnot x \land y$ \\
$f_{3}$ &
$f_{0011}$ &&
0 0 1 1 &
$(x)$ &
$\operatorname{not}\ x$ &
$\lnot x$ \\
$f_{4}$ &
$f_{0100}$ &&
0 1 0 0 &
$x\ (y)$ &
$x\ \operatorname{without}\ y$ &
$x \land \lnot y$ \\
$f_{5}$ &
$f_{0101}$ &&
0 1 0 1 &
$(y)$ &
$\operatorname{not}\ y$ &
$\lnot y$ \\
$f_{6}$ &
$f_{0110}$ &&
0 1 1 0 &
$(x,\ y)$ &
$x\ \operatorname{not~equal~to}\ y$ &
$x \ne y$ \\
$f_{7}$ &
$f_{0111}$ &&
0 1 1 1 &
$(x\ y)$ &
$\operatorname{not~both}\ x\ \operatorname{and}\ y$ &
$\lnot x \lor \lnot y$ \\
\hline
$f_{8}$ &
$f_{1000}$ &&
1 0 0 0 &
$x\ y$ &
$x\ \operatorname{and}\ y$ &
$x \land y$ \\
$f_{9}$ &
$f_{1001}$ &&
1 0 0 1 &
$((x,\ y))$ &
$x\ \operatorname{equal~to}\ y$ &
$x = y$ \\
$f_{10}$ &
$f_{1010}$ &&
1 0 1 0 &
$y$ &
$y$ &
$y$ \\
$f_{11}$ &
$f_{1011}$ &&
1 0 1 1 &
$(x\ (y))$ &
$\operatorname{not}\ x\ \operatorname{without}\ y$ &
$x \Rightarrow y$ \\
$f_{12}$ &
$f_{1100}$ &&
1 1 0 0 &
$x$ &
$x$ &
$x$ \\
$f_{13}$ &
$f_{1101}$ &&
1 1 0 1 &
$((x)\ y)$ &
$\operatorname{not}\ y\ \operatorname{without}\ x$ &
$x \Leftarrow y$ \\
$f_{14}$ &
$f_{1110}$ &&
1 1 1 0 &
$((x)(y))$ &
$x\ \operatorname{or}\ y$ &
$x \lor y$ \\
$f_{15}$ &
$f_{1111}$ &&
1 1 1 1 &
$((~))$ &
$\operatorname{true}$ &
$1$ \\
\hline
\end{tabular}\end{quote}
\subsection{Table A2. Propositional Forms on Two Variables}
Table A2 lists the sixteen Boolean functions of two variables in a different order, grouping them by structural similarity into seven natural classes.
\begin{quote}\begin{tabular}{|c|c|c|c|c|c|c|}
\multicolumn{7}{c}{\textbf{Table A2. Propositional Forms on Two Variables}} \\
\hline
$\mathcal{L}_1$ &
$\mathcal{L}_2$ &&
$\mathcal{L}_3$ &
$\mathcal{L}_4$ &
$\mathcal{L}_5$ &
$\mathcal{L}_6$ \\
\hline
& & $x =$ & 1 1 0 0 & & & \\
& & $y =$ & 1 0 1 0 & & & \\
\hline
$f_{0}$ &
$f_{0000}$ &&
0 0 0 0 &
$(~)$ &
$\operatorname{false}$ &
$0$ \\
\hline
$f_{1}$ &
$f_{0001}$ &&
0 0 0 1 &
$(x)(y)$ &
$\operatorname{neither}\ x\ \operatorname{nor}\ y$ &
$\lnot x \land \lnot y$ \\
$f_{2}$ &
$f_{0010}$ &&
0 0 1 0 &
$(x)\ y$ &
$y\ \operatorname{without}\ x$ &
$\lnot x \land y$ \\
$f_{4}$ &
$f_{0100}$ &&
0 1 0 0 &
$x\ (y)$ &
$x\ \operatorname{without}\ y$ &
$x \land \lnot y$ \\
$f_{8}$ &
$f_{1000}$ &&
1 0 0 0 &
$x\ y$ &
$x\ \operatorname{and}\ y$ &
$x \land y$ \\
\hline
$f_{3}$ &
$f_{0011}$ &&
0 0 1 1 &
$(x)$ &
$\operatorname{not}\ x$ &
$\lnot x$ \\
$f_{12}$ &
$f_{1100}$ &&
1 1 0 0 &
$x$ &
$x$ &
$x$ \\
\hline
$f_{6}$ &
$f_{0110}$ &&
0 1 1 0 &
$(x,\ y)$ &
$x\ \operatorname{not~equal~to}\ y$ &
$x \ne y$ \\
$f_{9}$ &
$f_{1001}$ &&
1 0 0 1 &
$((x,\ y))$ &
$x\ \operatorname{equal~to}\ y$ &
$x = y$ \\
\hline
$f_{5}$ &
$f_{0101}$ &&
0 1 0 1 &
$(y)$ &
$\operatorname{not}\ y$ &
$\lnot y$ \\
$f_{10}$ &
$f_{1010}$ &&
1 0 1 0 &
$y$ &
$y$ &
$y$ \\
\hline
$f_{7}$ &
$f_{0111}$ &&
0 1 1 1 &
$(x\ y)$ &
$\operatorname{not~both}\ x\ \operatorname{and}\ y$ &
$\lnot x \lor \lnot y$ \\
$f_{11}$ &
$f_{1011}$ &&
1 0 1 1 &
$(x\ (y))$ &
$\operatorname{not}\ x\ \operatorname{without}\ y$ &
$x \Rightarrow y$ \\
$f_{13}$ &
$f_{1101}$ &&
1 1 0 1 &
$((x)\ y)$ &
$\operatorname{not}\ y\ \operatorname{without}\ x$ &
$x \Leftarrow y$ \\
$f_{14}$ &
$f_{1110}$ &&
1 1 1 0 &
$((x)(y))$ &
$x\ \operatorname{or}\ y$ &
$x \lor y$ \\
\hline
$f_{15}$ &
$f_{1111}$ &&
1 1 1 1 &
$((~))$ &
$\operatorname{true}$ &
$1$ \\
\hline
\end{tabular}\end{quote}
\subsection{Table A3. $\operatorname{E}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}
\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
\multicolumn{6}{c}{\textbf{Table A3. $\operatorname{E}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}} \\
\hline
& &
$\operatorname{T}_{11}$ &
$\operatorname{T}_{10}$ &
$\operatorname{T}_{01}$ &
$\operatorname{T}_{00}$ \\
& $f$ &
$\operatorname{E}f|_{\operatorname{d}x\ \operatorname{d}y}$ &
$\operatorname{E}f|_{\operatorname{d}x (\operatorname{d}y)}$ &
$\operatorname{E}f|_{(\operatorname{d}x) \operatorname{d}y}$ &
$\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}$ \\
\hline
$f_{0}$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ \\
\hline
$f_{1}$ & $(x)(y)$ & $x\ y$ & $x\ (y)$ & $(x)\ y$ & $(x)(y)$ \\
$f_{2}$ & $(x)\ y$ & $x\ (y)$ & $x\ y$ & $(x)(y)$ & $(x)\ y$ \\
$f_{4}$ & $x\ (y)$ & $(x)\ y$ & $(x)(y)$ & $x\ y$ & $x\ (y)$ \\
$f_{8}$ & $x\ y$ & $(x)(y)$ & $(x)\ y$ & $x\ (y)$ & $x\ y$ \\
\hline
$f_{3}$ & $(x)$ & $x$ & $x$ & $(x)$ & $(x)$ \\
$f_{12}$ & $x$ & $(x)$ & $(x)$ & $x$ & $x$ \\
\hline
$f_{6}$ & $(x,\ y)$ & $(x,\ y)$ & $((x,\ y))$ & $((x,\ y))$ & $(x,\ y)$ \\
$f_{9}$ & $((x,\ y))$ & $((x,\ y))$ & $(x,\ y)$ & $(x,\ y)$ & $((x,\ y))$ \\
\hline
$f_{5}$ & $(y)$ & $y$ & $(y)$ & $y$ & $(y)$ \\
$f_{10}$ & $y$ & $(y)$ & $y$ & $(y)$ & $y$ \\
\hline
$f_{7}$ & $(x\ y)$ & $((x)(y))$ & $((x)\ y)$ & $(x\ (y))$ & $(x\ y)$ \\
$f_{11}$ & $(x\ (y))$ & $((x)\ y)$ & $((x)(y))$ & $(x\ y)$ & $(x\ (y))$ \\
$f_{13}$ & $((x)\ y)$ & $(x\ (y))$ & $(x\ y)$ & $((x)(y))$ & $((x)\ y)$ \\
$f_{14}$ & $((x)(y))$ & $(x\ y)$ & $(x\ (y))$ & $((x)\ y)$ & $((x)(y))$ \\
\hline
$f_{15}$ & $((~))$ & $((~))$ & $((~))$ & $((~))$ & $((~))$ \\
\hline
\multicolumn{2}{|c||}{\PMlinkname{Fixed Point}{FixedPoint} Total:} & 4 & 4 & 4 & 16 \\
\hline
\end{tabular}\end{quote}
\subsection{Table A4. $\operatorname{D}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}
\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
\multicolumn{6}{c}{\textbf{Table A4. $\operatorname{D}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}} \\
\hline
& $f$ &
$\operatorname{D}f|_{\operatorname{d}x\ \operatorname{d}y}$ &
$\operatorname{D}f|_{\operatorname{d}x (\operatorname{d}y)}$ &
$\operatorname{D}f|_{(\operatorname{d}x) \operatorname{d}y}$ &
$\operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)}$ \\
\hline
$f_{0}$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ \\
\hline
$f_{1}$ & $(x)(y)$ & $((x,\ y))$ & $(y)$ & $(x)$ & $(~)$ \\
$f_{2}$ & $(x)\ y$ & $(x,\ y)$ & $y$ & $(x)$ & $(~)$ \\
$f_{4}$ & $x\ (y)$ & $(x,\ y)$ & $(y)$ & $x$ & $(~)$ \\
$f_{8}$ & $x\ y$ & $((x,\ y))$ & $y$ & $x$ & $(~)$ \\
\hline
$f_{3}$ & $(x)$ & $((~))$ & $((~))$ & $(~)$ & $(~)$ \\
$f_{12}$ & $x$ & $((~))$ & $((~))$ & $(~)$ & $(~)$ \\
\hline
$f_{6}$ & $(x,\ y)$ & $(~)$ & $((~))$ & $((~))$ & $(~)$ \\
$f_{9}$ & $((x,\ y))$ & $(~)$ & $((~))$ & $((~))$ & $(~)$ \\
\hline
$f_{5}$ & $(y)$ & $((~))$ & $(~)$ & $((~))$ & $(~)$ \\
$f_{10}$ & $y$ & $((~))$ & $(~)$ & $((~))$ & $(~)$ \\
\hline
$f_{7}$ & $(x\ y)$ & $((x,\ y))$ & $y$ & $x$ & $(~)$ \\
$f_{11}$ & $(x\ (y))$ & $(x,\ y)$ & $(y)$ & $x$ & $(~)$ \\
$f_{13}$ & $((x)\ y)$ & $(x,\ y)$ & $y$ & $(x)$ & $(~)$ \\
$f_{14}$ & $((x)(y))$ & $((x,\ y))$ & $(y)$ & $(x)$ & $(~)$ \\
\hline
$f_{15}$ & $((~))$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ \\
\hline
\end{tabular}\end{quote}
\subsection{Table A5. $\operatorname{E}f$ Expanded Over Ordinary Features $\{ x, y \}$}
\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
\multicolumn{6}{c}{\textbf{Table A5. $\operatorname{E}f$ Expanded Over Ordinary Features $\{ x, y \}$}} \\
\hline
& $f$ &
$\operatorname{E}f|_{x\ y}$ &
$\operatorname{E}f|_{x (y)}$ &
$\operatorname{E}f|_{(x) y}$ &
$\operatorname{E}f|_{(x)(y)}$ \\
\hline
$f_{0}$ &
$(~)$ &
$(~)$ &
$(~)$ &
$(~)$ &
$(~)$ \\
\hline
$f_{1}$ &
$(x)(y)$ &
$\operatorname{d}x\ \operatorname{d}y$ &
$\operatorname{d}x\ (\operatorname{d}y)$ &
$(\operatorname{d}x)\ \operatorname{d}y$ &
$(\operatorname{d}x)(\operatorname{d}y)$ \\
$f_{2}$ &
$(x)\ y$ &
$\operatorname{d}x\ (\operatorname{d}y)$ &
$\operatorname{d}x\ \operatorname{d}y$ &
$(\operatorname{d}x)(\operatorname{d}y)$ &
$(\operatorname{d}x)\ \operatorname{d}y$ \\
$f_{4}$ &
$x\ (y)$ &
$(\operatorname{d}x)\ \operatorname{d}y$ &
$(\operatorname{d}x)(\operatorname{d}y)$ &
$\operatorname{d}x\ \operatorname{d}y$ &
$\operatorname{d}x\ (\operatorname{d}y)$ \\
$f_{8}$ &
$x\ y$ &
$(\operatorname{d}x)(\operatorname{d}y)$ &
$(\operatorname{d}x)\ \operatorname{d}y$ &
$\operatorname{d}x\ (\operatorname{d}y)$ &
$\operatorname{d}x\ \operatorname{d}y$ \\
\hline
$f_{3}$ &
$(x)$ &
$\operatorname{d}x$ &
$\operatorname{d}x$ &
$(\operatorname{d}x)$ &
$(\operatorname{d}x)$ \\
$f_{12}$ &
$x$ &
$(\operatorname{d}x)$ &
$(\operatorname{d}x)$ &
$\operatorname{d}x$ &
$\operatorname{d}x$ \\
\hline
$f_{6}$ &
$(x,\ y)$ &
$(\operatorname{d}x,\ \operatorname{d}y)$ &
$((\operatorname{d}x,\ \operatorname{d}y))$ &
$((\operatorname{d}x,\ \operatorname{d}y))$ &
$(\operatorname{d}x,\ \operatorname{d}y)$ \\
$f_{9}$ &
$((x,\ y))$ &
$((\operatorname{d}x,\ \operatorname{d}y))$ &
$(\operatorname{d}x,\ \operatorname{d}y)$ &
$(\operatorname{d}x,\ \operatorname{d}y)$ &
$((\operatorname{d}x,\ \operatorname{d}y))$ \\
\hline
$f_{5}$ &
$(y)$ &
$\operatorname{d}y$ &
$(\operatorname{d}y)$ &
$\operatorname{d}y$ &
$(\operatorname{d}y)$ \\
$f_{10}$ &
$y$ &
$(\operatorname{d}y)$ &
$\operatorname{d}y$ &
$(\operatorname{d}y)$ &
$\operatorname{d}y$ \\
\hline
$f_{7}$ &
$(x\ y)$ &
$((\operatorname{d}x)(\operatorname{d}y))$ &
$((\operatorname{d}x)\ \operatorname{d}y)$ &
$(\operatorname{d}x\ (\operatorname{d}y))$ &
$(\operatorname{d}x\ \operatorname{d}y)$ \\
$f_{11}$ &
$(x\ (y))$ &
$((\operatorname{d}x)\ \operatorname{d}y)$ &
$((\operatorname{d}x)(\operatorname{d}y))$ &
$(\operatorname{d}x\ \operatorname{d}y)$ &
$(\operatorname{d}x\ (\operatorname{d}y))$ \\
$f_{13}$ &
$((x)\ y)$ &
$(\operatorname{d}x\ (\operatorname{d}y))$ &
$(\operatorname{d}x\ \operatorname{d}y)$ &
$((\operatorname{d}x)(\operatorname{d}y))$ &
$((\operatorname{d}x)\ \operatorname{d}y)$ \\
$f_{14}$ &
$((x)(y))$ &
$(\operatorname{d}x\ \operatorname{d}y)$ &
$(\operatorname{d}x\ (\operatorname{d}y))$ &
$((\operatorname{d}x)\ \operatorname{d}y)$ &
$((\operatorname{d}x)(\operatorname{d}y))$ \\
\hline
$f_{15}$ &
$((~))$ &
$((~))$ &
$((~))$ &
$((~))$ &
$((~))$ \\
\hline
\end{tabular}\end{quote}
\subsection{Table A6. $\operatorname{D}f$ Expanded Over Ordinary Features $\{ x, y \}$}
\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
\multicolumn{6}{c}{\textbf{Table A6. $\operatorname{D}f$ Expanded Over Ordinary Features $\{ x, y \}$}} \\
\hline
& $f$ &
$\operatorname{D}f|_{x\ y}$ &
$\operatorname{D}f|_{x (y)}$ &
$\operatorname{D}f|_{(x) y}$ &
$\operatorname{D}f|_{(x)(y)}$ \\
\hline
$f_{0}$ &
$(~)$ &
$(~)$ &
$(~)$ &
$(~)$ &
$(~)$ \\
\hline
$f_{1}$ &
$(x)(y)$ &
$\operatorname{d}x\ \operatorname{d}y$ &
$\operatorname{d}x\ (\operatorname{d}y)$ &
$(\operatorname{d}x)\ \operatorname{d}y$ &
$((\operatorname{d}x)(\operatorname{d}y))$ \\
$f_{2}$ &
$(x)\ y$ &
$\operatorname{d}x\ (\operatorname{d}y)$ &
$\operatorname{d}x\ \operatorname{d}y$ &
$((\operatorname{d}x)(\operatorname{d}y))$ &
$(\operatorname{d}x)\ \operatorname{d}y$ \\
$f_{4}$ &
$x\ (y)$ &
$(\operatorname{d}x)\ \operatorname{d}y$ &
$((\operatorname{d}x)(\operatorname{d}y))$ &
$\operatorname{d}x\ \operatorname{d}y$ &
$\operatorname{d}x\ (\operatorname{d}y)$ \\
$f_{8}$ &
$x\ y$ &
$((\operatorname{d}x)(\operatorname{d}y))$ &
$(\operatorname{d}x)\ \operatorname{d}y$ &
$\operatorname{d}x\ (\operatorname{d}y)$ &
$\operatorname{d}x\ \operatorname{d}y$ \\
\hline
$f_{3}$ &
$(x)$ &
$\operatorname{d}x$ &
$\operatorname{d}x$ &
$\operatorname{d}x$ &
$\operatorname{d}x$ \\
$f_{12}$ &
$x$ &
$\operatorname{d}x$ &
$\operatorname{d}x$ &
$\operatorname{d}x$ &
$\operatorname{d}x$ \\
\hline
$f_{6}$ &
$(x,\ y)$ &
$(\operatorname{d}x,\ \operatorname{d}y)$ &
$(\operatorname{d}x,\ \operatorname{d}y)$ &
$(\operatorname{d}x,\ \operatorname{d}y)$ &
$(\operatorname{d}x,\ \operatorname{d}y)$ \\
$f_{9}$ &
$((x,\ y))$ &
$(\operatorname{d}x,\ \operatorname{d}y)$ &
$(\operatorname{d}x,\ \operatorname{d}y)$ &
$(\operatorname{d}x,\ \operatorname{d}y)$ &
$(\operatorname{d}x,\ \operatorname{d}y)$ \\
\hline
$f_{5}$ &
$(y)$ &
$\operatorname{d}y$ &
$\operatorname{d}y$ &
$\operatorname{d}y$ &
$\operatorname{d}y$ \\
$f_{10}$ &
$y$ &
$\operatorname{d}y$ &
$\operatorname{d}y$ &
$\operatorname{d}y$ &
$\operatorname{d}y$ \\
\hline
$f_{7}$ &
$(x\ y)$ &
$((\operatorname{d}x)(\operatorname{d}y))$ &
$(\operatorname{d}x)\ \operatorname{d}y$ &
$\operatorname{d}x\ (\operatorname{d}y)$ &
$\operatorname{d}x\ \operatorname{d}y$ \\
$f_{11}$ &
$(x\ (y))$ &
$(\operatorname{d}x)\ \operatorname{d}y$ &
$((\operatorname{d}x)(\operatorname{d}y))$ &
$\operatorname{d}x\ \operatorname{d}y$ &
$\operatorname{d}x\ (\operatorname{d}y)$ \\
$f_{13}$ &
$((x)\ y)$ &
$\operatorname{d}x\ (\operatorname{d}y)$ &
$\operatorname{d}x\ \operatorname{d}y$ &
$((\operatorname{d}x)(\operatorname{d}y))$ &
$(\operatorname{d}x)\ \operatorname{d}y$ \\
$f_{14}$ &
$((x)(y))$ &
$\operatorname{d}x\ \operatorname{d}y$ &
$\operatorname{d}x\ (\operatorname{d}y)$ &
$(\operatorname{d}x)\ \operatorname{d}y$ &
$((\operatorname{d}x)(\operatorname{d}y))$ \\
\hline
$f_{15}$ &
$((~))$ &
$(~)$ &
$(~)$ &
$(~)$ &
$(~)$ \\
\hline
\end{tabular}\end{quote}
Inquiry Driven Systems
Table 1. Sign Relation of Interpreter A
Table 1. Sign Relation of Interpreter A o---------------o---------------o---------------o | Object | Sign | Interpretant | o---------------o---------------o---------------o | A | "A" | "A" | | A | "A" | "i" | | A | "i" | "A" | | A | "i" | "i" | | B | "B" | "B" | | B | "B" | "u" | | B | "u" | "B" | | B | "u" | "u" | o---------------o---------------o---------------o
| Object | Sign | Interpretant |
|---|---|---|
| A | "A" | "A" |
| A | "A" | "i" |
| A | "i" | "A" |
| A | "i" | "i" |
| B | "B" | "B" |
| B | "B" | "u" |
| B | "u" | "B" |
| B | "u" | "u" |
Table 2. Sign Relation of Interpreter B
Table 2. Sign Relation of Interpreter B o---------------o---------------o---------------o | Object | Sign | Interpretant | o---------------o---------------o---------------o | A | "A" | "A" | | A | "A" | "u" | | A | "u" | "A" | | A | "u" | "u" | | B | "B" | "B" | | B | "B" | "i" | | B | "i" | "B" | | B | "i" | "i" | o---------------o---------------o---------------o
| Object | Sign | Interpretant |
|---|---|---|
| A | "A" | "A" |
| A | "A" | "u" |
| A | "u" | "A" |
| A | "u" | "u" |
| B | "B" | "B" |
| B | "B" | "i" |
| B | "i" | "B" |
| B | "i" | "i" |
Table 3. Semiotic Partition of Interpreter A
Table 3. A's Semiotic Partition o-------------------------------o | "A" "i" | o-------------------------------o | "u" "B" | o-------------------------------o
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Table 4. Semiotic Partition of Interpreter B
Table 4. B's Semiotic Partition o---------------o---------------o | "A" | "i" | | | | | "u" | "B" | o---------------o---------------o
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Table 5. Alignments of Capacities
Table 5. Alignments of Capacities o-------------------o-----------------------------o | Formal | Formative | o-------------------o-----------------------------o | Objective | Instrumental | | Passive | Active | o-------------------o--------------o--------------o | Afforded | Possessed | Exercised | o-------------------o--------------o--------------o
Table 6. Alignments of Capacities in Aristotle
Table 6. Alignments of Capacities in Aristotle o-------------------o-----------------------------o | Matter | Form | o-------------------o-----------------------------o | Potentiality | Actuality | | Receptivity | Possession | Exercise | | Life | Sleep | Waking | | Wax | Impression | | Axe | Edge | Cutting | | Eye | Vision | Seeing | | Body | Soul | o-------------------o-----------------------------o | Ship? | Sailor? | o-------------------o-----------------------------o
Table 7. Synthesis of Alignments
Table 7. Synthesis of Alignments o-------------------o-----------------------------o | Formal | Formative | o-------------------o-----------------------------o | Objective | Instrumental | | Passive | Active | | Afforded | Possessed | Exercised | | To Hold | To Have | To Use | | Receptivity | Possession | Exercise | | Potentiality | Actuality | | Matter | Form | o-------------------o-----------------------------o
Table 8. Boolean Product
Table 8. Boolean Product o---------o---------o---------o | %*% % %0% | %1% | o=========o=========o=========o | %0% % %0% | %0% | o---------o---------o---------o | %1% % %0% | %1% | o---------o---------o---------o
Table 9. Boolean Sum
Table 9. Boolean Sum o---------o---------o---------o | %+% % %0% | %1% | o=========o=========o=========o | %0% % %0% | %1% | o---------o---------o---------o | %1% % %1% | %0% | o---------o---------o---------o
Logical Tables
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Higher Order Propositions
| \ x | 1 0 | F | m | m | m | m | m | m | m | m | m | m | m | m | m | m | m | m |
| F \ | 00 | 01 | 02 | 03 | 04 | 05 | 06 | 07 | 08 | 09 | 10 | 11 | 12 | 13 | 14 | 15 | ||
| F0 | 0 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| F1 | 0 1 | (x) | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| F2 | 1 0 | x | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| F3 | 1 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| Measure | Happening | Exactness | Existence | Linearity | Uniformity | Information |
| m0 | nothing happens | |||||
| m1 | just false | nothing exists | ||||
| m2 | just not x | |||||
| m3 | nothing is x | |||||
| m4 | just x | |||||
| m5 | everything is x | F is linear | ||||
| m6 | F is not uniform | F is informed | ||||
| m7 | not just true | |||||
| m8 | just true | |||||
| m9 | F is uniform | F is not informed | ||||
| m10 | something is not x | F is not linear | ||||
| m11 | not just x | |||||
| m12 | something is x | |||||
| m13 | not just not x | |||||
| m14 | not just false | something exists | ||||
| m15 | anything happens |
| x : | 1100 | f | m | m | m | m | m | m | m | m | m | m | m | m | m | m | m | m | m | m | m | m | m | m | m | m |
| y : | 1010 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
| f0 | 0000 | ( ) | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| f1 | 0001 | (x)(y) | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | ||
| f2 | 0010 | (x) y | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | ||||
| f3 | 0011 | (x) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||||||||
| f4 | 0100 | x (y) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||||||||||
| f5 | 0101 | (y) | ||||||||||||||||||||||||
| f6 | 0110 | (x, y) | ||||||||||||||||||||||||
| f7 | 0111 | (x y) | ||||||||||||||||||||||||
| f8 | 1000 | x y | ||||||||||||||||||||||||
| f9 | 1001 | ((x, y)) | ||||||||||||||||||||||||
| f10 | 1010 | y | ||||||||||||||||||||||||
| f11 | 1011 | (x (y)) | ||||||||||||||||||||||||
| f12 | 1100 | x | ||||||||||||||||||||||||
| f13 | 1101 | ((x) y) | ||||||||||||||||||||||||
| f14 | 1110 | ((x)(y)) | ||||||||||||||||||||||||
| f15 | 1111 | (( )) |
| x : | 1100 | f | α | α | α | α | α | α | α | α | α | α | α | α | α | α | α | α |
| y : | 1010 | 15 | 14 | 13 | 12 | 11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 | |
| f0 | 0000 | ( ) | 1 | |||||||||||||||
| f1 | 0001 | (x)(y) | 1 | 1 | ||||||||||||||
| f2 | 0010 | (x) y | 1 | 1 | ||||||||||||||
| f3 | 0011 | (x) | 1 | 1 | 1 | 1 | ||||||||||||
| f4 | 0100 | x (y) | 1 | 1 | ||||||||||||||
| f5 | 0101 | (y) | 1 | 1 | 1 | 1 | ||||||||||||
| f6 | 0110 | (x, y) | 1 | 1 | 1 | 1 | ||||||||||||
| f7 | 0111 | (x y) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
| f8 | 1000 | x y | 1 | 1 | ||||||||||||||
| f9 | 1001 | ((x, y)) | 1 | 1 | 1 | 1 | ||||||||||||
| f10 | 1010 | y | 1 | 1 | 1 | 1 | ||||||||||||
| f11 | 1011 | (x (y)) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
| f12 | 1100 | x | 1 | 1 | 1 | 1 | ||||||||||||
| f13 | 1101 | ((x) y) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
| f14 | 1110 | ((x)(y)) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
| f15 | 1111 | (( )) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| x : | 1100 | f | β | β | β | β | β | β | β | β | β | β | β | β | β | β | β | β |
| y : | 1010 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |
| f0 | 0000 | ( ) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| f1 | 0001 | (x)(y) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
| f2 | 0010 | (x) y | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
| f3 | 0011 | (x) | 1 | 1 | 1 | 1 | ||||||||||||
| f4 | 0100 | x (y) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
| f5 | 0101 | (y) | 1 | 1 | 1 | 1 | ||||||||||||
| f6 | 0110 | (x, y) | 1 | 1 | 1 | 1 | ||||||||||||
| f7 | 0111 | (x y) | 1 | 1 | ||||||||||||||
| f8 | 1000 | x y | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
| f9 | 1001 | ((x, y)) | 1 | 1 | 1 | 1 | ||||||||||||
| f10 | 1010 | y | 1 | 1 | 1 | 1 | ||||||||||||
| f11 | 1011 | (x (y)) | 1 | 1 | ||||||||||||||
| f12 | 1100 | x | 1 | 1 | 1 | 1 | ||||||||||||
| f13 | 1101 | ((x) y) | 1 | 1 | ||||||||||||||
| f14 | 1110 | ((x)(y)) | 1 | 1 | ||||||||||||||
| f15 | 1111 | (( )) | 1 |
| A | Universal Affirmative | All | x | is | y | Indicator of " x (y)" = 0 |
| E | Universal Negative | All | x | is | (y) | Indicator of " x y " = 0 |
| I | Particular Affirmative | Some | x | is | y | Indicator of " x y " = 1 |
| O | Particular Negative | Some | x | is | (y) | Indicator of " x (y)" = 1 |
| Mnemonic | Category | Classical Form | Alternate Form | Symmetric Form | Operator |
| E Exclusive |
Universal Negative |
All x is (y) | No x is y | (L11) | |
| A Absolute |
Universal Affirmative |
All x is y | No x is (y) | (L10) | |
| All y is x | No y is (x) | No (x) is y | (L01) | ||
| All (y) is x | No (y) is (x) | No (x) is (y) | (L00) | ||
| Some (x) is (y) | Some (x) is (y) | L00 | |||
| Some (x) is y | Some (x) is y | L01 | |||
| O Obtrusive |
Particular Negative |
Some x is (y) | Some x is (y) | L10 | |
| I Indefinite |
Particular Affirmative |
Some x is y | Some x is y | L11 |
| x : | 1100 | f | (L11) | (L10) | (L01) | (L00) | L00 | L01 | L10 | L11 |
| y : | 1010 | no x is y |
no x is (y) |
no (x) is y |
no (x) is (y) |
some (x) is (y) |
some (x) is y |
some x is (y) |
some x is y | |
| f0 | 0000 | ( ) | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
| f1 | 0001 | (x)(y) | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 |
| f2 | 0010 | (x) y | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 |
| f3 | 0011 | (x) | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 |
| f4 | 0100 | x (y) | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 |
| f5 | 0101 | (y) | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |
| f6 | 0110 | (x, y) | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 |
| f7 | 0111 | (x y) | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 |
| f8 | 1000 | x y | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 |
| f9 | 1001 | ((x, y)) | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 |
| f10 | 1010 | y | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| f11 | 1011 | (x (y)) | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 |
| f12 | 1100 | x | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| f13 | 1101 | ((x) y) | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 |
| f14 | 1110 | ((x)(y)) | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 1 |
| f15 | 1111 | (( )) | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
Table 7. Higher Order Propositions (n = 1) o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o | \ x | 1 0 | F |m |m |m |m |m |m |m |m |m |m |m |m |m |m |m |m | | F \ | | |00|01|02|03|04|05|06|07|08|09|10|11|12|13|14|15 | o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o | | | | | | F_0 | 0 0 | 0 | 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 | | | | | | | F_1 | 0 1 | (x) | 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 | | | | | | | F_2 | 1 0 | x | 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 | | | | | | | F_3 | 1 1 | 1 | 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 | | | | | | o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o
Table 8. Interpretive Categories for Higher Order Propositions (n = 1) o-------o----------o------------o------------o----------o----------o-----------o |Measure| Happening| Exactness | Existence | Linearity|Uniformity|Information| o-------o----------o------------o------------o----------o----------o-----------o | m_0 | nothing | | | | | | | | happens | | | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_1 | | | nothing | | | | | | | just false | exists | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_2 | | | | | | | | | | just not x | | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_3 | | | nothing | | | | | | | | is x | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_4 | | | | | | | | | | just x | | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_5 | | | everything | F is | | | | | | | is x | linear | | | o-------o----------o------------o------------o----------o----------o-----------o | m_6 | | | | | F is not | F is | | | | | | | uniform | informed | o-------o----------o------------o------------o----------o----------o-----------o | m_7 | | not | | | | | | | | just true | | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_8 | | | | | | | | | | just true | | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_9 | | | | | F is | F is not | | | | | | | uniform | informed | o-------o----------o------------o------------o----------o----------o-----------o | m_10 | | | something | F is not | | | | | | | is not x | linear | | | o-------o----------o------------o------------o----------o----------o-----------o | m_11 | | not | | | | | | | | just x | | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_12 | | | something | | | | | | | | is x | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_13 | | not | | | | | | | | just not x | | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_14 | | not | something | | | | | | | just false | exists | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_15 | anything | | | | | | | | happens | | | | | | o-------o----------o------------o------------o----------o----------o-----------o
Table 9. Higher Order Propositions (n = 2) o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o | | x | 1100 | f |m|m|m|m|m|m|m|m|m|m|m|m|m|m|m|m|.| | | y | 1010 | |0|0|0|0|0|0|0|0|0|0|1|1|1|1|1|1|.| | f \ | | |0|1|2|3|4|5|6|7|8|9|0|1|2|3|4|5|.| o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o | | | | | | f_0 | 0000 | () |0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 | | | | | | | f_1 | 0001 | (x)(y) | 1 1 0 0 1 1 0 0 1 1 0 0 1 1 | | | | | | | f_2 | 0010 | (x) y | 1 1 1 1 0 0 0 0 1 1 1 1 | | | | | | | f_3 | 0011 | (x) | 1 1 1 1 1 1 1 1 | | | | | | | f_4 | 0100 | x (y) | | | | | | | | f_5 | 0101 | (y) | | | | | | | | f_6 | 0110 | (x, y) | | | | | | | | f_7 | 0111 | (x y) | | | | | | | o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o | | | | | | f_8 | 1000 | x y | | | | | | | | f_9 | 1001 | ((x, y)) | | | | | | | | f_10 | 1010 | y | | | | | | | | f_11 | 1011 | (x (y)) | | | | | | | | f_12 | 1100 | x | | | | | | | | f_13 | 1101 | ((x) y) | | | | | | | | f_14 | 1110 | ((x)(y)) | | | | | | | | f_15 | 1111 | (()) | | | | | | | o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
Table 10. Qualifiers of Implication Ordering: !a!_i f = !Y!(f_i => f) o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o | | x | 1100 | f |a |a |a |a |a |a |a |a |a |a |a |a |a |a |a |a | | | y | 1010 | |1 |1 |1 |1 |1 |1 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 | | f \ | | |5 |4 |3 |2 |1 |0 |9 |8 |7 |6 |5 |4 |3 |2 |1 |0 | o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o | | | | | | f_0 | 0000 | () | 1 | | | | | | | f_1 | 0001 | (x)(y) | 1 1 | | | | | | | f_2 | 0010 | (x) y | 1 1 | | | | | | | f_3 | 0011 | (x) | 1 1 1 1 | | | | | | | f_4 | 0100 | x (y) | 1 1 | | | | | | | f_5 | 0101 | (y) | 1 1 1 1 | | | | | | | f_6 | 0110 | (x, y) | 1 1 1 1 | | | | | | | f_7 | 0111 | (x y) | 1 1 1 1 1 1 1 1 | | | | | | | f_8 | 1000 | x y | 1 1 | | | | | | | f_9 | 1001 | ((x, y)) | 1 1 1 1 | | | | | | | f_10 | 1010 | y | 1 1 1 1 | | | | | | | f_11 | 1011 | (x (y)) | 1 1 1 1 1 1 1 1 | | | | | | | f_12 | 1100 | x | 1 1 1 1 | | | | | | | f_13 | 1101 | ((x) y) | 1 1 1 1 1 1 1 1 | | | | | | | f_14 | 1110 | ((x)(y)) | 1 1 1 1 1 1 1 1 | | | | | | | f_15 | 1111 | (()) |1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 | | | | | | o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
Table 11. Qualifiers of Implication Ordering: !b!_i f = !Y!(f => f_i) o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o | | x | 1100 | f |b |b |b |b |b |b |b |b |b |b |b |b |b |b |b |b | | | y | 1010 | |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |1 |1 |1 |1 |1 |1 | | f \ | | |0 |1 |2 |3 |4 |5 |6 |7 |8 |9 |0 |1 |2 |3 |4 |5 | o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o | | | | | | f_0 | 0000 | () |1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 | | | | | | | f_1 | 0001 | (x)(y) | 1 1 1 1 1 1 1 1 | | | | | | | f_2 | 0010 | (x) y | 1 1 1 1 1 1 1 1 | | | | | | | f_3 | 0011 | (x) | 1 1 1 1 | | | | | | | f_4 | 0100 | x (y) | 1 1 1 1 1 1 1 1 | | | | | | | f_5 | 0101 | (y) | 1 1 1 1 | | | | | | | f_6 | 0110 | (x, y) | 1 1 1 1 | | | | | | | f_7 | 0111 | (x y) | 1 1 | | | | | | | f_8 | 1000 | x y | 1 1 1 1 1 1 1 1 | | | | | | | f_9 | 1001 | ((x, y)) | 1 1 1 1 | | | | | | | f_10 | 1010 | y | 1 1 1 1 | | | | | | | f_11 | 1011 | (x (y)) | 1 1 | | | | | | | f_12 | 1100 | x | 1 1 1 1 | | | | | | | f_13 | 1101 | ((x) y) | 1 1 | | | | | | | f_14 | 1110 | ((x)(y)) | 1 1 | | | | | | | f_15 | 1111 | (()) | 1 | | | | | | o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
Table 13. Syllogistic Premisses as Higher Order Indicator Functions o---o------------------------o-----------------o---------------------------o | | | | | | A | Universal Affirmative | All x is y | Indicator of " x (y)" = 0 | | | | | | | E | Universal Negative | All x is (y) | Indicator of " x y " = 0 | | | | | | | I | Particular Affirmative | Some x is y | Indicator of " x y " = 1 | | | | | | | O | Particular Negative | Some x is (y) | Indicator of " x (y)" = 1 | | | | | | o---o------------------------o-----------------o---------------------------o
Table 14. Relation of Quantifiers to Higher Order Propositions o------------o------------o-----------o-----------o-----------o-----------o | Mnemonic | Category | Classical | Alternate | Symmetric | Operator | | | | Form | Form | Form | | o============o============o===========o===========o===========o===========o | E | Universal | All x | | No x | (L_11) | | Exclusive | Negative | is (y) | | is y | | o------------o------------o-----------o-----------o-----------o-----------o | A | Universal | All x | | No x | (L_10) | | Absolute | Affrmtve | is y | | is (y) | | o------------o------------o-----------o-----------o-----------o-----------o | | | All y | No y | No (x) | (L_01) | | | | is x | is (x) | is y | | o------------o------------o-----------o-----------o-----------o-----------o | | | All (y) | No (y) | No (x) | (L_00) | | | | is x | is (x) | is (y) | | o------------o------------o-----------o-----------o-----------o-----------o | | | Some (x) | | Some (x) | L_00 | | | | is (y) | | is (y) | | o------------o------------o-----------o-----------o-----------o-----------o | | | Some (x) | | Some (x) | L_01 | | | | is y | | is y | | o------------o------------o-----------o-----------o-----------o-----------o | O | Particular | Some x | | Some x | L_10 | | Obtrusive | Negative | is (y) | | is (y) | | o------------o------------o-----------o-----------o-----------o-----------o | I | Particular | Some x | | Some x | L_11 | | Indefinite | Affrmtve | is y | | is y | | o------------o------------o-----------o-----------o-----------o-----------o
Table 15. Simple Qualifiers of Propositions (n = 2) o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o | | x | 1100 | f |(L11)|(L10)|(L01)|(L00)| L00 | L01 | L10 | L11 | | | y | 1010 | |no x|no x|no ~x|no ~x|sm ~x|sm ~x|sm x|sm x| | f \ | | |is y|is ~y|is y|is ~y|is ~y|is y|is ~y|is y| o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o | | | | | | f_0 | 0000 | () | 1 1 1 1 0 0 0 0 | | | | | | | f_1 | 0001 | (x)(y) | 1 1 1 0 1 0 0 0 | | | | | | | f_2 | 0010 | (x) y | 1 1 0 1 0 1 0 0 | | | | | | | f_3 | 0011 | (x) | 1 1 0 0 1 1 0 0 | | | | | | | f_4 | 0100 | x (y) | 1 0 1 1 0 0 1 0 | | | | | | | f_5 | 0101 | (y) | 1 0 1 0 1 0 1 0 | | | | | | | f_6 | 0110 | (x, y) | 1 0 0 1 0 1 1 0 | | | | | | | f_7 | 0111 | (x y) | 1 0 0 0 1 1 1 0 | | | | | | | f_8 | 1000 | x y | 0 1 1 1 0 0 0 1 | | | | | | | f_9 | 1001 | ((x, y)) | 0 1 1 0 1 0 0 1 | | | | | | | f_10 | 1010 | y | 0 1 0 1 0 1 0 1 | | | | | | | f_11 | 1011 | (x (y)) | 0 1 0 0 1 1 0 1 | | | | | | | f_12 | 1100 | x | 0 0 1 1 0 0 1 1 | | | | | | | f_13 | 1101 | ((x) y) | 0 0 1 0 1 0 1 1 | | | | | | | f_14 | 1110 | ((x)(y)) | 0 0 0 1 0 1 1 1 | | | | | | | f_15 | 1111 | (()) | 0 0 0 0 1 1 1 1 | | | | | | o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o
Zeroth Order Logic
| L1 | L2 | L3 | L4 | L5 | L6 |
|---|---|---|---|---|---|
| x : | 1 1 0 0 | ||||
| y : | 1 0 1 0 | ||||
| f0 | f0000 | 0 0 0 0 | ( ) | false | 0 |
| f1 | f0001 | 0 0 0 1 | (x)(y) | neither x nor y | ¬x ∧ ¬y |
| f2 | f0010 | 0 0 1 0 | (x) y | y and not x | ¬x ∧ y |
| f3 | f0011 | 0 0 1 1 | (x) | not x | ¬x |
| f4 | f0100 | 0 1 0 0 | x (y) | x and not y | x ∧ ¬y |
| f5 | f0101 | 0 1 0 1 | (y) | not y | ¬y |
| f6 | f0110 | 0 1 1 0 | (x, y) | x not equal to y | x ≠ y |
| f7 | f0111 | 0 1 1 1 | (x y) | not both x and y | ¬x ∨ ¬y |
| f8 | f1000 | 1 0 0 0 | x y | x and y | x ∧ y |
| f9 | f1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y |
| f10 | f1010 | 1 0 1 0 | y | y | y |
| f11 | f1011 | 1 0 1 1 | (x (y)) | not x without y | x → y |
| f12 | f1100 | 1 1 0 0 | x | x | x |
| f13 | f1101 | 1 1 0 1 | ((x) y) | not y without x | x ← y |
| f14 | f1110 | 1 1 1 0 | ((x)(y)) | x or y | x ∨ y |
| f15 | f1111 | 1 1 1 1 | (( )) | true | 1 |
| L1 | L2 | L3 | L4 | L5 | L6 |
|---|---|---|---|---|---|
| x : | 1 1 0 0 | ||||
| y : | 1 0 1 0 | ||||
| f0 | f0000 | 0 0 0 0 | ( ) | false | 0 |
| f1 | f0001 | 0 0 0 1 | (x)(y) | neither x nor y | ¬x ∧ ¬y |
| f2 | f0010 | 0 0 1 0 | (x) y | y and not x | ¬x ∧ y |
| f3 | f0011 | 0 0 1 1 | (x) | not x | ¬x |
| f4 | f0100 | 0 1 0 0 | x (y) | x and not y | x ∧ ¬y |
| f5 | f0101 | 0 1 0 1 | (y) | not y | ¬y |
| f6 | f0110 | 0 1 1 0 | (x, y) | x not equal to y | x ≠ y |
| f7 | f0111 | 0 1 1 1 | (x y) | not both x and y | ¬x ∨ ¬y |
| f8 | f1000 | 1 0 0 0 | x y | x and y | x ∧ y |
| f9 | f1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y |
| f10 | f1010 | 1 0 1 0 | y | y | y |
| f11 | f1011 | 1 0 1 1 | (x (y)) | not x without y | x → y |
| f12 | f1100 | 1 1 0 0 | x | x | x |
| f13 | f1101 | 1 1 0 1 | ((x) y) | not y without x | x ← y |
| f14 | f1110 | 1 1 1 0 | ((x)(y)) | x or y | x ∨ y |
| f15 | f1111 | 1 1 1 1 | (( )) | true | 1 |
Template Draft
| L1 | L2 | L3 | L4 | L5 | L6 | Name |
|---|---|---|---|---|---|---|
| x : | 1 1 0 0 | |||||
| y : | 1 0 1 0 | |||||
| f0 | f0000 | 0 0 0 0 | ( ) | false | 0 | Falsity |
| f1 | f0001 | 0 0 0 1 | (x)(y) | neither x nor y | ¬x ∧ ¬y | NNOR |
| f2 | f0010 | 0 0 1 0 | (x) y | y and not x | ¬x ∧ y | Insuccede |
| f3 | f0011 | 0 0 1 1 | (x) | not x | ¬x | Not One |
| f4 | f0100 | 0 1 0 0 | x (y) | x and not y | x ∧ ¬y | Imprecede |
| f5 | f0101 | 0 1 0 1 | (y) | not y | ¬y | Not Two |
| f6 | f0110 | 0 1 1 0 | (x, y) | x not equal to y | x ≠ y | Inequality |
| f7 | f0111 | 0 1 1 1 | (x y) | not both x and y | ¬x ∨ ¬y | NAND |
| f8 | f1000 | 1 0 0 0 | x y | x and y | x ∧ y | Conjunction |
| f9 | f1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y | Equality |
| f10 | f1010 | 1 0 1 0 | y | y | y | Two |
| f11 | f1011 | 1 0 1 1 | (x (y)) | not x without y | x → y | Implication |
| f12 | f1100 | 1 1 0 0 | x | x | x | One |
| f13 | f1101 | 1 1 0 1 | ((x) y) | not y without x | x ← y | Involution |
| f14 | f1110 | 1 1 1 0 | ((x)(y)) | x or y | x ∨ y | Disjunction |
| f15 | f1111 | 1 1 1 1 | (( )) | true | 1 | Tautology |
Truth Tables
Logical negation
Logical negation is an operation on one logical value, typically the value of a proposition, that produces a value of true when its operand is false and a value of false when its operand is true.
The truth table of NOT p (also written as ~p or ¬p) is as follows:
| p | ¬p |
|---|---|
| F | T |
| T | F |
The logical negation of a proposition p is notated in different ways in various contexts of discussion and fields of application. Among these variants are the following:
| Notation | Vocalization |
|---|---|
| \(\bar{p}\) | bar p |
| \(p'\!\) | p prime, p complement |
| \(!p\!\) | bang p |
No matter how it is notated or symbolized, the logical negation ¬p is read as "it is not the case that p", or usually more simply as "not p".
- Within a system of classical logic, double negation, that is, the negation of the negation of a proposition p, is logically equivalent to the initial proposition p. Expressed in symbolic terms, ¬(¬p) ⇔ p.
- Within a system of intuitionistic logic, however, ¬¬p is a weaker statement than p. On the other hand, the logical equivalence ¬¬¬p ⇔ ¬p remains valid.
Logical negation can be defined in terms of other logical operations. For example, ~p can be defined as p → F, where → is material implication and F is absolute falsehood. Conversely, one can define F as p & ~p for any proposition p, where & is logical conjunction. The idea here is that any contradiction is false. While these ideas work in both classical and intuitionistic logic, they don't work in Brazilian logic, where contradictions are not necessarily false. But in classical logic, we get a further identity: p → q can be defined as ~p ∨ q, where ∨ is logical disjunction.
Algebraically, logical negation corresponds to the complement in a Boolean algebra (for classical logic) or a Heyting algebra (for intuitionistic logic).
Logical conjunction
Logical conjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both of its operands are true.
The truth table of p AND q (also written as p ∧ q, p & q, or p\(\cdot\)q) is as follows:
| p | q | p ∧ q |
|---|---|---|
| F | F | F |
| F | T | F |
| T | F | F |
| T | T | T |
Logical disjunction
Logical disjunction is an operation on two logical values, typically the values of two propositions, that produces a value of false if and only if both of its operands are false.
The truth table of p OR q (also written as p ∨ q) is as follows:
| p | q | p ∨ q |
|---|---|---|
| F | F | F |
| F | T | T |
| T | F | T |
| T | T | T |
Logical equality
Logical equality is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both operands are false or both operands are true.
The truth table of p EQ q (also written as p = q, p ↔ q, or p ≡ q) is as follows:
| p | q | p = q |
|---|---|---|
| F | F | T |
| F | T | F |
| T | F | F |
| T | T | T |
Exclusive disjunction
Exclusive disjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true just in case exactly one of its operands is true.
The truth table of p XOR q (also written as p + q, p ⊕ q, or p ≠ q) is as follows:
| p | q | p XOR q |
|---|---|---|
| F | F | F |
| F | T | T |
| T | F | T |
| T | T | F |
The following equivalents can then be deduced:
\[\begin{matrix} p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\ \\ & = & (p \lor q) & \land & (\lnot p \lor \lnot q) \\ \\ & = & (p \lor q) & \land & \lnot (p \land q) \end{matrix}\]
Generalized or n-ary XOR is true when the number of 1-bits is odd.
A + B = (A ∧ !B) ∨ (!A ∧ B)
= {(A ∧ !B) ∨ !A} ∧ {(A ∧ !B) ∨ B}
= {(A ∨ !A) ∧ (!B ∨ !A)} ∧ {(A ∨ B) ∧ (!B ∨ B)}
= (!A ∨ !B) ∧ (A ∨ B)
= !(A ∧ B) ∧ (A ∨ B)
p + q = (p ∧ !q) ∨ (!p ∧ B)
= {(p ∧ !q) ∨ !p} ∧ {(p ∧ !q) ∨ q}
= {(p ∨ !q) ∧ (!q ∨ !p)} ∧ {(p ∨ q) ∧ (!q ∨ q)}
= (!p ∨ !q) ∧ (p ∨ q)
= !(p ∧ q) ∧ (p ∨ q)
p + q = (p ∧ ~q) ∨ (~p ∧ q)
= ((p ∧ ~q) ∨ ~p) ∧ ((p ∧ ~q) ∨ q)
= ((p ∨ ~q) ∧ (~q ∨ ~p)) ∧ ((p ∨ q) ∧ (~q ∨ q))
= (~p ∨ ~q) ∧ (p ∨ q)
= ~(p ∧ q) ∧ (p ∨ q)
\[\begin{matrix} p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\ & = & ((p \land \lnot q) \lor \lnot p) & \and & ((p \land \lnot q) \lor q) \\ & = & ((p \lor \lnot q) \land (\lnot q \lor \lnot p)) & \land & ((p \lor q) \land (\lnot q \lor q)) \\ & = & (\lnot p \lor \lnot q) & \land & (p \lor q) \\ & = & \lnot (p \land q) & \land & (p \lor q) \end{matrix}\]
Logical implication
The material conditional and logical implication are both associated with an operation on two logical values, typically the values of two propositions, that produces a value of false if and only if the first operand is true and the second operand is false.
The truth table associated with the material conditional if p then q (symbolized as p → q) and the logical implication p implies q (symbolized as p ⇒ q) is as follows:
| p | q | p ⇒ q |
|---|---|---|
| F | F | T |
| F | T | T |
| T | F | F |
| T | T | T |
Logical NAND
The NAND operation is a logical operation on two logical values, typically the values of two propositions, that produces a value of false if and only if both of its operands are true. In other words, it produces a value of true if and only if at least one of its operands is false.
The truth table of p NAND q (also written as p | q or p ↑ q) is as follows:
| p | q | p ↑ q |
|---|---|---|
| F | F | T |
| F | T | T |
| T | F | T |
| T | T | F |
Logical NNOR
The NNOR operation is a logical operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both of its operands are false. In other words, it produces a value of false if and only if at least one of its operands is true.
The truth table of p NNOR q (also written as p ⊥ q or p ↓ q) is as follows:
| p | q | p ↓ q |
|---|---|---|
| F | F | T |
| F | T | F |
| T | F | F |
| T | T | F |
Relational Tables
Sign Relations
| O | = | Object Domain | |
| S | = | Sign Domain | |
| I | = | Interpretant Domain |
| O | = | {Ann, Bob} | = | {A, B} | |
| S | = | {"Ann", "Bob", "I", "You"} | = | {"A", "B", "i", "u"} | |
| I | = | {"Ann", "Bob", "I", "You"} | = | {"A", "B", "i", "u"} |
| 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" |
| 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
| X | Y | Z |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
| X | Y | Z |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
Semiotic Examples
| 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" |
| 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
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Method 2 : Subtitles as Top Rows
projOS(LA)
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projOS(LB)
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projSI(LA)
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projSI(LB)
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projOI(LA)
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projOI(LB)
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Relation Reduction
Method 1 : Subtitles as Captions
| X | Y | Z |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
| X | Y | Z |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
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| projXY(L0) = projXY(L1) | projXZ(L0) = projXZ(L1) | projYZ(L0) = projYZ(L1) |
| 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" |
| 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) ≠ projXY(LB) | projXZ(LA) ≠ projXZ(LB) | projYZ(LA) ≠ projYZ(LB) |
Method 2 : Subtitles as Top Rows
| X | Y | Z |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
| X | Y | Z |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
projXY(L0)
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projXZ(L0)
|
projYZ(L0)
|
projXY(L1)
|
projXZ(L1)
|
projYZ(L1)
|
| projXY(L0) = projXY(L1) | projXZ(L0) = projXZ(L1) | projYZ(L0) = projYZ(L1) |
| 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" |
| 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)
|
projXZ(LA)
|
projYZ(LA)
|
projXY(LB)
|
projXZ(LB)
|
projYZ(LB)
|
| 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
| 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
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Draft 2
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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
| \(x\): | 1 0 | \(f\) | \(m_0\) | \(m_1\) | \(m_2\) | \(m_3\) | \(m_4\) | \(m_5\) | \(m_6\) | \(m_7\) | \(m_8\) | \(m_9\) | \(m_{10}\) | \(m_{11}\) | \(m_{12}\) | \(m_{13}\) | \(m_{14}\) | \(m_{15}\) |
| \(f_0\) | 0 0 | \(0\!\) | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| \(f_1\) | 0 1 | \((x)\!\) | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| \(f_2\) | 1 0 | \(x\!\) | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| \(f_3\) | 1 1 | \(1\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| \(x:\) | 1 0 | \(f\!\) | \(m_0\) | \(m_1\) | \(m_2\) | \(m_3\) | \(m_4\) | \(m_5\) | \(m_6\) | \(m_7\) | \(m_8\) | \(m_9\) | \(m_{10}\) | \(m_{11}\) | \(m_{12}\) | \(m_{13}\) | \(m_{14}\) | \(m_{15}\) |
| \(f_0\) | 0 0 | \(0\!\) | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| \(f_1\) | 0 1 | \((x)\!\) | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| \(f_2\) | 1 0 | \(x\!\) | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| \(f_3\) | 1 1 | \(1\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Table 11
| Measure | Happening | Exactness | Existence | Linearity | Uniformity | Information |
| \(m_0\!\) | Nothing happens | |||||
| \(m_1\!\) | Just false | Nothing exists | ||||
| \(m_2\!\) | Just not \(x\!\) | |||||
| \(m_3\!\) | Nothing is \(x\!\) | |||||
| \(m_4\!\) | Just \(x\!\) | |||||
| \(m_5\!\) | Everything is \(x\!\) | \(f\!\) is linear | ||||
| \(m_6\!\) | \(f\!\) is not uniform | \(f\!\) is informed | ||||
| \(m_7\!\) | Not just true | |||||
| \(m_8\!\) | Just true | |||||
| \(m_9\!\) | \(f\!\) is uniform | \(f\!\) is not informed | ||||
| \(m_{10}\!\) | Something is not \(x\!\) | \(f\!\) is not linear | ||||
| \(m_{11}\!\) | Not just \(x\!\) | |||||
| \(m_{12}\!\) | Something is \(x\!\) | |||||
| \(m_{13}\!\) | Not just not \(x\!\) | |||||
| \(m_{14}\!\) | Not just false | Something exists | ||||
| \(m_{15}\!\) | Anything happens |
Table 12
| \(x:\) \(y:\) |
1100 1010 |
\(f\!\) | \(m_0\) | \(m_1\) | \(m_2\) | \(m_3\) | \(m_4\) | \(m_5\) | \(m_6\) | \(m_7\) | \(m_8\) | \(m_9\) | \(m_{10}\) | \(m_{11}\) | \(m_{12}\) | \(m_{13}\) | \(m_{14}\) | \(m_{15}\) | \(m_{16}\) | \(m_{17}\) | \(m_{18}\) | \(m_{19}\) | \(m_{20}\) | \(m_{21}\) | \(m_{22}\) | \(m_{23}\) |
| \(f_0\) | 0000 | \((~)\) | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| \(f_1\) | 0001 | \((x)(y)\!\) | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | ||
| \(f_2\) | 0010 | \((x) y\!\) | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | ||||
| \(f_3\) | 0011 | \((x)\!\) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||||||||
| \(f_4\) | 0100 | \(x (y)\!\) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||||||||||
| \(f_5\) | 0101 | \((y)\!\) | ||||||||||||||||||||||||
| \(f_6\) | 0110 | \((x, y)\!\) | ||||||||||||||||||||||||
| \(f_7\) | 0111 | \((x y)\!\) | ||||||||||||||||||||||||
| \(f_8\) | 1000 | \(x y\!\) | ||||||||||||||||||||||||
| \(f_9\) | 1001 | \(((x, y))\!\) | ||||||||||||||||||||||||
| \(f_{10}\) | 1010 | \(y\!\) | ||||||||||||||||||||||||
| \(f_{11}\) | 1011 | \((x (y))\!\) | ||||||||||||||||||||||||
| \(f_{12}\) | 1100 | \(x\!\) | ||||||||||||||||||||||||
| \(f_{13}\) | 1101 | \(((x) y)\!\) | ||||||||||||||||||||||||
| \(f_{14}\) | 1110 | \(((x)(y))\!\) | ||||||||||||||||||||||||
| \(f_{15}\) | 1111 | \(((~))\!\) |
| \(u:\) \(v:\) |
1100 1010 |
\(f\!\) | \(m_0\) | \(m_1\) | \(m_2\) | \(m_3\) | \(m_4\) | \(m_5\) | \(m_6\) | \(m_7\) | \(m_8\) | \(m_9\) | \(m_{10}\) | \(m_{11}\) | \(m_{12}\) | \(m_{13}\) | \(m_{14}\) | \(m_{15}\) | \(m_{16}\) | \(m_{17}\) | \(m_{18}\) | \(m_{19}\) | \(m_{20}\) | \(m_{21}\) | \(m_{22}\) | \(m_{23}\) |
| \(f_0\) | 0000 | \((~)\) | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| \(f_1\) | 0001 | \((u)(v)\!\) | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| \(f_2\) | 0010 | \((u) v\!\) | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| \(f_3\) | 0011 | \((u)\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(f_4\) | 0100 | \(u (v)\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| \(f_5\) | 0101 | \((v)\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(f_6\) | 0110 | \((u, v)\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(f_7\) | 0111 | \((u v)\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(f_8\) | 1000 | \(u v\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(f_9\) | 1001 | \(((u, v))\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(f_{10}\) | 1010 | \(v\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(f_{11}\) | 1011 | \((u (v))\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(f_{12}\) | 1100 | \(u\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(f_{13}\) | 1101 | \(((u) v)\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(f_{14}\) | 1110 | \(((u)(v))\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(f_{15}\) | 1111 | \(((~))\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Table 13
| \(u:\) \(v:\) |
1100 1010 |
\(f\!\) | \(\alpha_0\) | \(\alpha_1\) | \(\alpha_2\) | \(\alpha_3\) | \(\alpha_4\) | \(\alpha_5\) | \(\alpha_6\) | \(\alpha_7\) | \(\alpha_8\) | \(\alpha_9\) | \(\alpha_{10}\) | \(\alpha_{11}\) | \(\alpha_{12}\) | \(\alpha_{13}\) | \(\alpha_{14}\) | \(\alpha_{15}\) |
| \(f_0\) | 0000 | \((~)\) | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(f_1\) | 0001 | \((u)(v)\!\) | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(f_2\) | 0010 | \((u) v\!\) | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(f_3\) | 0011 | \((u)\!\) | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(f_4\) | 0100 | \(u (v)\!\) | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(f_5\) | 0101 | \((v)\!\) | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(f_6\) | 0110 | \((u, v)\!\) | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(f_7\) | 0111 | \((u v)\!\) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(f_8\) | 1000 | \(u v\!\) | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(f_9\) | 1001 | \(((u, v))\!\) | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(f_{10}\) | 1010 | \(v\!\) | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| \(f_{11}\) | 1011 | \((u (v))\!\) | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
| \(f_{12}\) | 1100 | \(u\!\) | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| \(f_{13}\) | 1101 | \(((u) v)\!\) | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 |
| \(f_{14}\) | 1110 | \(((u)(v))\!\) | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |
| \(f_{15}\) | 1111 | \(((~))\) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Table 14
| \(u:\) \(v:\) |
1100 1010 |
\(f\!\) | \(\beta_0\) | \(\beta_1\) | \(\beta_2\) | \(\beta_3\) | \(\beta_4\) | \(\beta_5\) | \(\beta_6\) | \(\beta_7\) | \(\beta_8\) | \(\beta_9\) | \(\beta_{10}\) | \(\beta_{11}\) | \(\beta_{12}\) | \(\beta_{13}\) | \(\beta_{14}\) | \(\beta_{15}\) |
| \(f_0\) | 0000 | \((~)\) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| \(f_1\) | 0001 | \((u)(v)\!\) | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| \(f_2\) | 0010 | \((u) v\!\) | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| \(f_3\) | 0011 | \((u)\!\) | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
| \(f_4\) | 0100 | \(u (v)\!\) | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| \(f_5\) | 0101 | \((v)\!\) | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
| \(f_6\) | 0110 | \((u, v)\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| \(f_7\) | 0111 | \((u v)\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| \(f_8\) | 1000 | \(u v\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| \(f_9\) | 1001 | \(((u, v))\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| \(f_{10}\) | 1010 | \(v\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| \(f_{11}\) | 1011 | \((u (v))\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
| \(f_{12}\) | 1100 | \(u\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| \(f_{13}\) | 1101 | \(((u) v)\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
| \(f_{14}\) | 1110 | \(((u)(v))\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| \(f_{15}\) | 1111 | \(((~))\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
Figure 15
Table 16
|
\(\begin{array}{clcl} \mathrm{A} & \mathrm{Universal~Affirmative} & \mathrm{All}\ u\ \mathrm{is}\ v & \mathrm{Indicator~of}\ u (v) = 0 \\ \mathrm{E} & \mathrm{Universal~Negative} & \mathrm{All}\ u\ \mathrm{is}\ (v) & \mathrm{Indicator~of}\ u \cdot v = 0 \\ \mathrm{I} & \mathrm{Particular~Affirmative} & \mathrm{Some}\ u\ \mathrm{is}\ v & \mathrm{Indicator~of}\ u \cdot v = 1 \\ \mathrm{O} & \mathrm{Particular~Negative} & \mathrm{Some}\ u\ \mathrm{is}\ (v) & \mathrm{Indicator~of}\ u (v) = 1 \\ \end{array}\) |
Table 17
| \(u:\) \(v:\) |
1100 1010 |
\(f\!\) | \((\ell_{11})\) \(\text{No } u \) \(\text{is } v \) |
\((\ell_{10})\) \(\text{No } u \) \(\text{is }(v)\) |
\((\ell_{01})\) \(\text{No }(u)\) \(\text{is } v \) |
\((\ell_{00})\) \(\text{No }(u)\) \(\text{is }(v)\) |
\( \ell_{00} \) \(\text{Some }(u)\) \(\text{is }(v)\) |
\( \ell_{01} \) \(\text{Some }(u)\) \(\text{is } v \) |
\( \ell_{10} \) \(\text{Some } u \) \(\text{is }(v)\) |
\( \ell_{11} \) \(\text{Some } u \) \(\text{is } v \) |
| \(f_0\) | 0000 | \((~)\) | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
| \(f_1\) | 0001 | \((u)(v)\!\) | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 |
| \(f_2\) | 0010 | \((u) v\!\) | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 |
| \(f_3\) | 0011 | \((u)\!\) | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 |
| \(f_4\) | 0100 | \(u (v)\!\) | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 |
| \(f_5\) | 0101 | \((v)\!\) | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |
| \(f_6\) | 0110 | \((u, v)\!\) | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 |
| \(f_7\) | 0111 | \((u v)\!\) | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 |
| \(f_8\) | 1000 | \(u v\!\) | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 |
| \(f_9\) | 1001 | \(((u, v))\!\) | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 |
| \(f_{10}\) | 1010 | \(v\!\) | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| \(f_{11}\) | 1011 | \((u (v))\!\) | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 |
| \(f_{12}\) | 1100 | \(u\!\) | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| \(f_{13}\) | 1101 | \(((u) v)\!\) | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 |
| \(f_{14}\) | 1110 | \(((u)(v))\!\) | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 1 |
| \(f_{15}\) | 1111 | \(((~))\) | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
Table 18
| \(u:\) \(v:\) |
1100 1010 |
\(f\!\) | \((\ell_{11})\) \(\text{No } u \) \(\text{is } v \) |
\((\ell_{10})\) \(\text{No } u \) \(\text{is }(v)\) |
\((\ell_{01})\) \(\text{No }(u)\) \(\text{is } v \) |
\((\ell_{00})\) \(\text{No }(u)\) \(\text{is }(v)\) |
\( \ell_{00} \) \(\text{Some }(u)\) \(\text{is }(v)\) |
\( \ell_{01} \) \(\text{Some }(u)\) \(\text{is } v \) |
\( \ell_{10} \) \(\text{Some } u \) \(\text{is }(v)\) |
\( \ell_{11} \) \(\text{Some } u \) \(\text{is } v \) |
| \(f_0\) | 0000 | \((~)\) | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
| \(f_1\) | 0001 | \((u)(v)\!\) | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 |
| \(f_2\) | 0010 | \((u) v\!\) | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 |
| \(f_4\) | 0100 | \(u (v)\!\) | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 |
| \(f_8\) | 1000 | \(u v\!\) | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 |
| \(f_3\) | 0011 | \((u)\!\) | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 |
| \(f_{12}\) | 1100 | \(u\!\) | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| \(f_6\) | 0110 | \((u, v)\!\) | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 |
| \(f_9\) | 1001 | \(((u, v))\!\) | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 |
| \(f_5\) | 0101 | \((v)\!\) | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |
| \(f_{10}\) | 1010 | \(v\!\) | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| \(f_7\) | 0111 | \((u v)\!\) | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 |
| \(f_{11}\) | 1011 | \((u (v))\!\) | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 |
| \(f_{13}\) | 1101 | \(((u) v)\!\) | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 |
| \(f_{14}\) | 1110 | \(((u)(v))\!\) | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 1 |
| \(f_{15}\) | 1111 | \(((~))\) | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
Table 19
| \(\text{Mnemonic}\) | \(\text{Category}\) | \(\text{Classical Form}\) | \(\text{Alternate Form}\) | \(\text{Symmetric Form}\) | \(\text{Operator}\) |
| \(\text{E}\!\) \(\text{Exclusive}\) |
\(\text{Universal}\) \(\text{Negative}\) |
\(\text{All}\ u\ \text{is}\ (v)\) | \(\text{No}\ u\ \text{is}\ v \) | \((\ell_{11})\) | |
| \(\text{A}\!\) \(\text{Absolute}\) |
\(\text{Universal}\) \(\text{Affirmative}\) |
\(\text{All}\ u\ \text{is}\ v \) | \(\text{No}\ u\ \text{is}\ (v)\) | \((\ell_{10})\) | |
| \(\text{All}\ v\ \text{is}\ u \) | \(\text{No}\ v\ \text{is}\ (u)\) | \(\text{No}\ (u)\ \text{is}\ v \) | \((\ell_{01})\) | ||
| \(\text{All}\ (v)\ \text{is}\ u \) | \(\text{No}\ (v)\ \text{is}\ (u)\) | \(\text{No}\ (u)\ \text{is}\ (v)\) | \((\ell_{00})\) | ||
| \(\text{Some}\ (u)\ \text{is}\ (v)\) | \(\text{Some}\ (u)\ \text{is}\ (v)\) | \(\ell_{00}\!\) | |||
| \(\text{Some}\ (u)\ \text{is}\ v\) | \(\text{Some}\ (u)\ \text{is}\ v\) | \(\ell_{01}\!\) | |||
| \(\text{O}\!\) \(\text{Obtrusive}\) |
\(\text{Particular}\) \(\text{Negative}\) |
\(\text{Some}\ u\ \text{is}\ (v)\) | \(\text{Some}\ u\ \text{is}\ (v)\) | \(\ell_{10}\!\) | |
| \(\text{I}\!\) \(\text{Indefinite}\) |
\(\text{Particular}\) \(\text{Affirmative}\) |
\(\text{Some}\ u\ \text{is}\ v\) | \(\text{Some}\ u\ \text{is}\ v\) | \(\ell_{11}\!\) |
