Line 1: |
Line 1: |
− | ==Differential Logic== | + | ==Cactus Language== |
| | | |
| ===Ascii Tables=== | | ===Ascii Tables=== |
| | | |
| + | {| align="center" cellpadding="6" style="text-align:center; width:90%" |
| + | | |
| <pre> | | <pre> |
− | Table 1. Propositional Forms On Two Variables
| + | o-------------------o |
− | 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------------------o----------o | + | | | |
− | | | x : 1 1 0 0 | | | | | + | | o | |
− | | | y : 1 0 1 0 | | | | | + | | | | |
− | o---------o---------o---------o----------o------------------o----------o | + | | @ | |
− | | | | | | | | | + | | | |
− | | f_0 | f_0000 | 0 0 0 0 | () | false | 0 | | + | o-------------------o |
− | | | | | | | | | + | | | |
− | | f_1 | f_0001 | 0 0 0 1 | (x)(y) | neither x nor y | ~x & ~y | | + | | a | |
− | | | | | | | | | + | | @ | |
− | | f_2 | f_0010 | 0 0 1 0 | (x) y | y and not x | ~x & y | | + | | | |
− | | | | | | | | | + | o-------------------o |
− | | f_3 | f_0011 | 0 0 1 1 | (x) | not x | ~x | | + | | | |
− | | | | | | | | | + | | a | |
− | | f_4 | f_0100 | 0 1 0 0 | x (y) | x and not y | x & ~y | | + | | o | |
− | | | | | | | | | + | | | | |
− | | 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 | | + | o-------------------o |
− | | | | | | | | | + | | | |
− | | f_7 | f_0111 | 0 1 1 1 | (x y) | not both x and y | ~x v ~y | | + | | a b c | |
− | | | | | | | | | + | | @ | |
− | | f_8 | f_1000 | 1 0 0 0 | x y | x and y | x & y | | + | | | |
− | | | | | | | | | + | o-------------------o |
− | | f_9 | f_1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y |
| + | | | |
− | | | | | | | |
| + | | a b c | |
− | | f_10 | f_1010 | 1 0 1 0 | y | y | y | | + | | o o o | |
− | | | | | | | | | + | | \|/ | |
− | | f_11 | f_1011 | 1 0 1 1 | (x (y)) | not x without y | x => y |
| + | | o | |
− | | | | | | | | | + | | | | |
− | | 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 | | + | o-------------------o |
− | | | | | | | | | + | | | |
− | | f_14 | f_1110 | 1 1 1 0 | ((x)(y)) | x or y | x v y |
| + | | a b | |
− | | | | | | | | | + | | o---o | |
− | | f_15 | f_1111 | 1 1 1 1 | (()) | true | 1 | | + | | | | |
− | | | | | | | | | + | | @ | |
− | o---------o---------o---------o----------o------------------o----------o | + | | | |
− | </pre>
| + | o-------------------o |
− | <pre>
| + | | | |
− | Table 2. Ef Expanded Over Ordinary Features {x, y}
| + | | a b | |
− | o------o------------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 | + | o-------------------o |
− | | | | | | | | | + | | | |
− | | f_0 | () | () | () | () | () | | + | | a b | |
− | | | | | | | | | + | | o---o | |
− | o------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 | | + | | | |
− | | | | | | | |
| + | o-------------------o |
− | | f_4 | x (y) | (dx) dy | (dx)(dy) | dx dy | dx (dy) |
| + | | | |
− | | | | | | | |
| + | | a b c | |
− | | f_8 | x y | (dx)(dy) | (dx) dy | dx (dy) | dx dy |
| + | | o--o--o | |
− | | | | | | | |
| + | | \ / | |
− | o------o------------o------------o------------o------------o------------o
| + | | \ / | |
− | | | | | | | | | + | | @ | |
− | | f_3 | (x) | dx | dx | (dx) | (dx) | | + | | | |
− | | | | | | | | | + | o-------------------o |
− | | f_12 | x | (dx) | (dx) | dx | dx | | + | | | |
− | | | | | | | | | + | | a b c | |
− | o------o------------o------------o------------o------------o------------o | + | | o o o | |
− | | | | | | | | | + | | | | | | |
− | | f_6 | (x, y) | (dx, dy) | ((dx, dy)) | ((dx, dy)) | (dx, dy) | | + | | o--o--o | |
− | | | | | | | | | + | | \ / | |
− | | f_9 | ((x, y)) | ((dx, dy)) | (dx, dy) | (dx, dy) | ((dx, dy)) | | + | | \ / | |
− | | | | | | | | | + | | @ | |
− | o------o------------o------------o------------o------------o------------o | + | | | |
− | | | | | | | | | + | o-------------------o |
− | | f_5 | (y) | dy | (dy) | dy | (dy) | | + | | | |
− | | | | | | | | | + | | b c | |
− | | f_10 | y | (dy) | dy | (dy) | dy | | + | | o o | |
− | | | | | | | | | + | | a | | | |
− | o------o------------o------------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)) |
| + | | | |
− | | | | | | | | | + | o-------------------o |
− | | f_13 | ((x) y) | (dx (dy)) | (dx dy) | ((dx)(dy)) | ((dx) dy) | | + | </pre> |
− | | | | | | | | | + | |} |
− | | f_14 | ((x)(y)) | (dx dy) | (dx (dy)) | ((dx) dy) | ((dx)(dy)) | | + | |
− | | | | | | | | | + | {| align="center" cellpadding="6" style="text-align:center; width:90%" |
− | o------o------------o------------o------------o------------o------------o | + | | |
− | | | | | | | | | + | <pre> |
− | | f_15 | (()) | (()) | (()) | (()) | (()) | | + | Table 13. The Existential Interpretation |
− | | | | | | | | | + | o----o-------------------o-------------------o-------------------o |
− | o------o------------o------------o------------o------------o------------o | + | | Ex | Cactus Graph | Cactus Expression | Existential | |
− | </pre>
| + | | | | | Interpretation | |
− | <pre> | + | o----o-------------------o-------------------o-------------------o |
− | Table 3. Df Expanded Over Ordinary Features {x, y} | + | | | | | | |
− | o------o------------o------------o------------o------------o------------o | + | | 1 | @ | " " | true. | |
− | | | | | | | | | + | | | | | | |
− | | | f | Df | xy | Df | x(y) | Df | (x)y | Df | (x)(y)| | + | o----o-------------------o-------------------o-------------------o |
− | | | | | | | |
| + | | | | | | |
− | o------o------------o------------o------------o------------o------------o
| + | | | o | | | |
− | | | | | | | | | + | | | | | | | |
− | | f_0 | () | () | () | () | () | | + | | 2 | @ | ( ) | untrue. | |
− | | | | | | | | | + | | | | | | |
− | o------o------------o------------o------------o------------o------------o | + | o----o-------------------o-------------------o-------------------o |
− | | | | | | | | | + | | | | | | |
− | | f_1 | (x)(y) | dx dy | dx (dy) | (dx) dy | ((dx)(dy)) | | + | | | a | | | |
− | | | | | | | | | + | | 3 | @ | a | a. | |
− | | f_2 | (x) y | dx (dy) | dx dy | ((dx)(dy)) | (dx) dy |
| + | | | | | | |
− | | | | | | | | | + | o----o-------------------o-------------------o-------------------o |
− | | f_4 | x (y) | (dx) dy | ((dx)(dy)) | dx dy | dx (dy) | | + | | | | | | |
− | | | | | | | | | + | | | a | | | |
− | | f_8 | x y | ((dx)(dy)) | (dx) dy | dx (dy) | dx dy |
| + | | | o | | | |
− | | | | | | | |
| + | | | | | | | |
− | o------o------------o------------o------------o------------o------------o | + | | 4 | @ | (a) | not a. | |
− | | | | | | | | | + | | | | | | |
− | | f_3 | (x) | dx | dx | dx | dx | | + | o----o-------------------o-------------------o-------------------o |
− | | | | | | | | | + | | | | | | |
− | | f_12 | x | dx | dx | dx | dx | | + | | | a b c | | | |
− | | | | | | | | | + | | 5 | @ | a b c | a and b and c. | |
− | o------o------------o------------o------------o------------o------------o | + | | | | | | |
− | | | | | | | | | + | o----o-------------------o-------------------o-------------------o |
− | | f_6 | (x, y) | (dx, dy) | (dx, dy) | (dx, dy) | (dx, dy) | | + | | | | | | |
− | | | | | | | | | + | | | a b c | | | |
− | | f_9 | ((x, y)) | (dx, dy) | (dx, dy) | (dx, dy) | (dx, dy) |
| + | | | o o o | | | |
− | | | | | | | |
| + | | | \|/ | | | |
− | o------o------------o------------o------------o------------o------------o
| + | | | o | | | |
− | | | | | | | | | + | | | | | | | |
− | | f_5 | (y) | dy | dy | dy | dy | | + | | 6 | @ | ((a)(b)(c)) | a or b or c. | |
− | | | | | | | | | + | | | | | | |
− | | f_10 | y | dy | dy | dy | dy | | + | o----o-------------------o-------------------o-------------------o |
− | | | | | | | | | + | | | | | | |
− | o------o------------o------------o------------o------------o------------o | + | | | | | a implies b. | |
− | | | | | | | | | + | | | a b | | | |
− | | f_7 | (x y) | ((dx)(dy)) | (dx) dy | dx (dy) | dx dy | | + | | | o---o | | if a then b. | |
− | | | | | | | | | + | | | | | | | |
− | | f_11 | (x (y)) | (dx) dy | ((dx)(dy)) | dx dy | dx (dy) | | + | | 7 | @ | ( a (b)) | no a sans b. | |
− | | | | | | | |
| + | | | | | | |
− | | f_13 | ((x) y) | dx (dy) | dx dy | ((dx)(dy)) | (dx) dy | | + | o----o-------------------o-------------------o-------------------o |
− | | | | | | | |
| + | | | | | | |
− | | f_14 | ((x)(y)) | dx dy | dx (dy) | (dx) dy | ((dx)(dy)) |
| + | | | a b | | | |
− | | | | | | | |
| + | | | o---o | | a exclusive-or b. | |
− | o------o------------o------------o------------o------------o------------o
| + | | | \ / | | | |
− | | | | | | | | | + | | 8 | @ | ( a , b ) | a not equal to b. | |
− | | f_15 | (()) | () | () | () | () | | + | | | | | | |
− | | | | | | | | | + | o----o-------------------o-------------------o-------------------o |
− | o------o------------o------------o------------o------------o------------o | + | | | | | | |
− | </pre>
| + | | | a b | | | |
| + | | | o---o | | | |
| + | | | \ / | | | |
| + | | | o | | a if & only if b. | |
| + | | | | | | | |
| + | | 9 | @ | (( a , b )) | a equates with b. | |
| + | | | | | | |
| + | o----o-------------------o-------------------o-------------------o |
| + | | | | | | |
| + | | | a b c | | | |
| + | | | o--o--o | | | |
| + | | | \ / | | | |
| + | | | \ / | | just one false | |
| + | | 10 | @ | ( a , b , c ) | out of a, b, c. | |
| + | | | | | | |
| + | o----o-------------------o-------------------o-------------------o |
| + | | | | | | |
| + | | | a b c | | | |
| + | | | o o o | | | |
| + | | | | | | | | | |
| + | | | o--o--o | | | |
| + | | | \ / | | | |
| + | | | \ / | | just one true | |
| + | | 11 | @ | ((a),(b),(c)) | among a, b, c. | |
| + | | | | | | |
| + | o----o-------------------o-------------------o-------------------o |
| + | | | | | | |
| + | | | | | genus a over | |
| + | | | b c | | species b, c. | |
| + | | | o o | | | |
| + | | | a | | | | partition a | |
| + | | | o--o--o | | among b & c. | |
| + | | | \ / | | | |
| + | | | \ / | | whole pie a: | |
| + | | 12 | @ | ( a ,(b),(c)) | slices b, c. | |
| + | | | | | | |
| + | o----o-------------------o-------------------o-------------------o |
| + | </pre> |
| + | |} |
| + | |
| + | {| align="center" cellpadding="6" style="text-align:center; width:90%" |
| + | | |
| + | <pre> |
| + | Table 14. The Entitative Interpretation |
| + | o----o-------------------o-------------------o-------------------o |
| + | | En | Cactus Graph | Cactus Expression | Entitative | |
| + | | | | | Interpretation | |
| + | o----o-------------------o-------------------o-------------------o |
| + | | | | | | |
| + | | 1 | @ | " " | untrue. | |
| + | | | | | | |
| + | o----o-------------------o-------------------o-------------------o |
| + | | | | | | |
| + | | | o | | | |
| + | | | | | | | |
| + | | 2 | @ | ( ) | true. | |
| + | | | | | | |
| + | o----o-------------------o-------------------o-------------------o |
| + | | | | | | |
| + | | | a | | | |
| + | | 3 | @ | a | a. | |
| + | | | | | | |
| + | o----o-------------------o-------------------o-------------------o |
| + | | | | | | |
| + | | | a | | | |
| + | | | o | | | |
| + | | | | | | | |
| + | | 4 | @ | (a) | not a. | |
| + | | | | | | |
| + | o----o-------------------o-------------------o-------------------o |
| + | | | | | | |
| + | | | a b c | | | |
| + | | 5 | @ | a b c | a or b or c. | |
| + | | | | | | |
| + | o----o-------------------o-------------------o-------------------o |
| + | | | | | | |
| + | | | a b c | | | |
| + | | | o o o | | | |
| + | | | \|/ | | | |
| + | | | o | | | |
| + | | | | | | | |
| + | | 6 | @ | ((a)(b)(c)) | a and b and c. | |
| + | | | | | | |
| + | o----o-------------------o-------------------o-------------------o |
| + | | | | | | |
| + | | | | | a implies b. | |
| + | | | | | | |
| + | | | o a | | if a then b. | |
| + | | | | | | | |
| + | | 7 | @ b | (a) b | not a, or b. | |
| + | | | | | | |
| + | o----o-------------------o-------------------o-------------------o |
| + | | | | | | |
| + | | | a b | | | |
| + | | | o---o | | a if & only if b. | |
| + | | | \ / | | | |
| + | | 8 | @ | ( a , b ) | a equates with b. | |
| + | | | | | | |
| + | o----o-------------------o-------------------o-------------------o |
| + | | | | | | |
| + | | | a b | | | |
| + | | | o---o | | | |
| + | | | \ / | | | |
| + | | | o | | a exclusive-or b. | |
| + | | | | | | | |
| + | | 9 | @ | (( a , b )) | a not equal to b. | |
| + | | | | | | |
| + | o----o-------------------o-------------------o-------------------o |
| + | | | | | | |
| + | | | a b c | | | |
| + | | | o--o--o | | | |
| + | | | \ / | | | |
| + | | | \ / | | not just one true | |
| + | | 10 | @ | ( a , b , c ) | out of a, b, c. | |
| + | | | | | | |
| + | o----o-------------------o-------------------o-------------------o |
| + | | | | | | |
| + | | | a b c | | | |
| + | | | o--o--o | | | |
| + | | | \ / | | | |
| + | | | \ / | | | |
| + | | | o | | | |
| + | | | | | | just one true | |
| + | | 11 | @ | (( a , b , c )) | among a, b, c. | |
| + | | | | | | |
| + | o----o-------------------o-------------------o-------------------o |
| + | | | | | | |
| + | | | a | | | |
| + | | | o | | genus a over | |
| + | | | | b c | | species b, c. | |
| + | | | o--o--o | | | |
| + | | | \ / | | partition a | |
| + | | | \ / | | among b & c. | |
| + | | | o | | | |
| + | | | | | | whole pie a: | |
| + | | 12 | @ | (((a), b , c )) | slices b, c. | |
| + | | | | | | |
| + | o----o-------------------o-------------------o-------------------o |
| + | </pre> |
| + | |} |
| + | |
| + | {| align="center" cellpadding="6" style="text-align:center; width:90%" |
| + | | |
| + | <pre> |
| + | Table 15. Existential & Entitative Interpretations of Cactus Structures |
| + | o-----------------o-----------------o-----------------o-----------------o |
| + | | Cactus Graph | Cactus String | Existential | Entitative | |
| + | | | | Interpretation | Interpretation | |
| + | o-----------------o-----------------o-----------------o-----------------o |
| + | | | | | | |
| + | | @ | " " | true | false | |
| + | | | | | | |
| + | o-----------------o-----------------o-----------------o-----------------o |
| + | | | | | | |
| + | | o | | | | |
| + | | | | | | | |
| + | | @ | ( ) | false | true | |
| + | | | | | | |
| + | o-----------------o-----------------o-----------------o-----------------o |
| + | | | | | | |
| + | | C_1 ... C_k | | | | |
| + | | @ | C_1 ... C_k | C_1 & ... & C_k | C_1 v ... v C_k | |
| + | | | | | | |
| + | o-----------------o-----------------o-----------------o-----------------o |
| + | | | | | | |
| + | | C_1 C_2 C_k | | Just one | Not just one | |
| + | | o---o-...-o | | | | |
| + | | \ / | | of the C_j, | of the C_j, | |
| + | | \ / | | | | |
| + | | \ / | | j = 1 to k, | j = 1 to k, | |
| + | | \ / | | | | |
| + | | @ | (C_1, ..., C_k) | is not true. | is true. | |
| + | | | | | | |
| + | o-----------------o-----------------o-----------------o-----------------o |
| + | </pre> |
| + | |} |
| + | |
| + | ===Wiki TeX Tables=== |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%" |
| + | |+ <math>\text{Table A.}~~\text{Existential Interpretation}</math> |
| + | |- style="background:#f0f0ff" |
| + | | <math>\text{Cactus Graph}\!</math> |
| + | | <math>\text{Cactus Expression}\!</math> |
| + | | <math>\text{Interpretation}\!</math> |
| + | |- |
| + | | height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]] |
| + | | <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math> |
| + | | <math>\operatorname{true}.</math> |
| + | |- |
| + | | height="100px" | [[Image:Cactus Spike Big Fat.jpg|20px]] |
| + | | <math>\texttt{(~)}</math> |
| + | | <math>\operatorname{false}.</math> |
| + | |- |
| + | | height="100px" | [[Image:Cactus A Big.jpg|20px]] |
| + | | <math>a\!</math> |
| + | | <math>a.\!</math> |
| + | |- |
| + | | height="120px" | [[Image:Cactus (A) Big.jpg|20px]] |
| + | | <math>\texttt{(} a \texttt{)}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \tilde{a} |
| + | \\[2pt] |
| + | a^\prime |
| + | \\[2pt] |
| + | \lnot a |
| + | \\[2pt] |
| + | \operatorname{not}~ a. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="100px" | [[Image:Cactus ABC Big.jpg|50px]] |
| + | | <math>a~b~c</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | a \land b \land c |
| + | \\[6pt] |
| + | a ~\operatorname{and}~ b ~\operatorname{and}~ c. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]] |
| + | | <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | a \lor b \lor c |
| + | \\[6pt] |
| + | a ~\operatorname{or}~ b ~\operatorname{or}~ c. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="120px" | [[Image:Cactus (A(B)) Big.jpg|60px]] |
| + | | <math>\texttt{(} a \texttt{(} b \texttt{))}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | a \Rightarrow b |
| + | \\[2pt] |
| + | a ~\operatorname{implies}~ b. |
| + | \\[2pt] |
| + | \operatorname{if}~ a ~\operatorname{then}~ b. |
| + | \\[2pt] |
| + | \operatorname{not}~ a ~\operatorname{without}~ b. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]] |
| + | | <math>\texttt{(} a, b \texttt{)}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | a + b |
| + | \\[2pt] |
| + | a \neq b |
| + | \\[2pt] |
| + | a ~\operatorname{exclusive-or}~ b. |
| + | \\[2pt] |
| + | a ~\operatorname{not~equal~to}~ b. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]] |
| + | | <math>\texttt{((} a, b \texttt{))}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | a = b |
| + | \\[2pt] |
| + | a \iff b |
| + | \\[2pt] |
| + | a ~\operatorname{equals}~ b. |
| + | \\[2pt] |
| + | a ~\operatorname{if~and~only~if}~ b. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="120px" | [[Image:Cactus (A,B,C) Big.jpg|65px]] |
| + | | <math>\texttt{(} a, b, c \texttt{)}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{just~one~of} |
| + | \\ |
| + | a, b, c |
| + | \\ |
| + | \operatorname{is~false}. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="160px" | [[Image:Cactus ((A),(B),(C)) Big.jpg|65px]] |
| + | | <math>\texttt{((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{just~one~of} |
| + | \\ |
| + | a, b, c |
| + | \\ |
| + | \operatorname{is~true}. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="160px" | [[Image:Cactus (A,(B),(C)) Big.jpg|65px]] |
| + | | <math>\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{genus}~ a ~\operatorname{of~species}~ b, c. |
| + | \\[6pt] |
| + | \operatorname{partition}~ a ~\operatorname{into}~ b, c. |
| + | \\[6pt] |
| + | \operatorname{pie}~ a ~\operatorname{of~slices}~ b, c. |
| + | \end{matrix}</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%" |
| + | |+ <math>\text{Table B.}~~\text{Entitative Interpretation}</math> |
| + | |- style="background:#f0f0ff" |
| + | | <math>\text{Cactus Graph}\!</math> |
| + | | <math>\text{Cactus Expression}\!</math> |
| + | | <math>\text{Interpretation}\!</math> |
| + | |- |
| + | | height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]] |
| + | | <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math> |
| + | | <math>\operatorname{false}.</math> |
| + | |- |
| + | | height="100px" | [[Image:Cactus Spike Big Fat.jpg|20px]] |
| + | | <math>\texttt{(~)}</math> |
| + | | <math>\operatorname{true}.</math> |
| + | |- |
| + | | height="100px" | [[Image:Cactus A Big.jpg|20px]] |
| + | | <math>a\!</math> |
| + | | <math>a.\!</math> |
| + | |- |
| + | | height="120px" | [[Image:Cactus (A) Big.jpg|20px]] |
| + | | <math>\texttt{(} a \texttt{)}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \tilde{a} |
| + | \\[2pt] |
| + | a^\prime |
| + | \\[2pt] |
| + | \lnot a |
| + | \\[2pt] |
| + | \operatorname{not}~ a. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="100px" | [[Image:Cactus ABC Big.jpg|50px]] |
| + | | <math>a~b~c</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | a \lor b \lor c |
| + | \\[6pt] |
| + | a ~\operatorname{or}~ b ~\operatorname{or}~ c. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]] |
| + | | <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | a \land b \land c |
| + | \\[6pt] |
| + | a ~\operatorname{and}~ b ~\operatorname{and}~ c. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="120px" | [[Image:Cactus (A)B Big.jpg|35px]] |
| + | | <math>\texttt{(} a \texttt{)} b</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | a \Rightarrow b |
| + | \\[2pt] |
| + | a ~\operatorname{implies}~ b. |
| + | \\[2pt] |
| + | \operatorname{if}~ a ~\operatorname{then}~ b. |
| + | \\[2pt] |
| + | \operatorname{not}~ a, ~\operatorname{or}~ b. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]] |
| + | | <math>\texttt{(} a, b \texttt{)}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | a = b |
| + | \\[2pt] |
| + | a \iff b |
| + | \\[2pt] |
| + | a ~\operatorname{equals}~ b. |
| + | \\[2pt] |
| + | a ~\operatorname{if~and~only~if}~ b. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]] |
| + | | <math>\texttt{((} a, b \texttt{))}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | a + b |
| + | \\[2pt] |
| + | a \neq b |
| + | \\[2pt] |
| + | a ~\operatorname{exclusive-or}~ b. |
| + | \\[2pt] |
| + | a ~\operatorname{not~equal~to}~ b. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="120px" | [[Image:Cactus (A,B,C) Big.jpg|65px]] |
| + | | <math>\texttt{(} a, b, c \texttt{)}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{not~just~one~of} |
| + | \\ |
| + | a, b, c |
| + | \\ |
| + | \operatorname{is~true}. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="160px" | [[Image:Cactus ((A,B,C)) Big.jpg|65px]] |
| + | | <math>\texttt{((} a, b, c \texttt{))}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{just~one~of} |
| + | \\ |
| + | a, b, c |
| + | \\ |
| + | \operatorname{is~true}. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="200px" | [[Image:Cactus (((A),B,C)) Big.jpg|65px]] |
| + | | <math>\texttt{(((} a \texttt{)}, b, c \texttt{))}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{genus}~ a ~\operatorname{of~species}~ b, c. |
| + | \\[6pt] |
| + | \operatorname{partition}~ a ~\operatorname{into}~ b, c. |
| + | \\[6pt] |
| + | \operatorname{pie}~ a ~\operatorname{of~slices}~ b, c. |
| + | \end{matrix}</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%" |
| + | |+ <math>\text{Table C.}~~\text{Dualing Interpretations}</math> |
| + | |- style="background:#f0f0ff" |
| + | | <math>\text{Graph}\!</math> |
| + | | <math>\text{String}\!</math> |
| + | | <math>\text{Existential}\!</math> |
| + | | <math>\text{Entitative}\!</math> |
| + | |- |
| + | | height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]] |
| + | | <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math> |
| + | | <math>\operatorname{true}.</math> |
| + | | <math>\operatorname{false}.</math> |
| + | |- |
| + | | height="100px" | [[Image:Cactus Spike Big Fat.jpg|20px]] |
| + | | <math>\texttt{(~)}</math> |
| + | | <math>\operatorname{false}.</math> |
| + | | <math>\operatorname{true}.</math> |
| + | |- |
| + | | height="100px" | [[Image:Cactus A Big.jpg|20px]] |
| + | | <math>a\!</math> |
| + | | <math>a.\!</math> |
| + | | <math>a.\!</math> |
| + | |- |
| + | | height="120px" | [[Image:Cactus (A) Big.jpg|20px]] |
| + | | <math>\texttt{(} a \texttt{)}</math> |
| + | | <math>\lnot a</math> |
| + | | <math>\lnot a</math> |
| + | |- |
| + | | height="100px" | [[Image:Cactus ABC Big.jpg|50px]] |
| + | | <math>a~b~c</math> |
| + | | <math>a \land b \land c</math> |
| + | | <math>a \lor b \lor c</math> |
| + | |- |
| + | | height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]] |
| + | | <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math> |
| + | | <math>a \lor b \lor c</math> |
| + | | <math>a \land b \land c</math> |
| + | |- |
| + | | height="120px" | [[Image:Cactus (A(B)) Big.jpg|60px]] |
| + | | <math>\texttt{(} a \texttt{(} b \texttt{))}</math> |
| + | | <math>a \Rightarrow b</math> |
| + | | |
| + | |- |
| + | | height="120px" | [[Image:Cactus (A)B Big.jpg|35px]] |
| + | | <math>\texttt{(} a \texttt{)} b</math> |
| + | | |
| + | | <math>a \Rightarrow b</math> |
| + | |- |
| + | | height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]] |
| + | | <math>\texttt{(} a, b \texttt{)}</math> |
| + | | <math>a \neq b</math> |
| + | | <math>a = b\!</math> |
| + | |- |
| + | | height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]] |
| + | | <math>\texttt{((} a, b \texttt{))}</math> |
| + | | <math>a = b\!</math> |
| + | | <math>a \neq b\!</math> |
| + | |- |
| + | | height="120px" | [[Image:Cactus (A,B,C) Big.jpg|65px]] |
| + | | <math>\texttt{(} a, b, c \texttt{)}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{just~one} |
| + | \\ |
| + | \operatorname{of}~ a, b, c |
| + | \\ |
| + | \operatorname{is~false}. |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{not~just~one} |
| + | \\ |
| + | \operatorname{of}~ a, b, c |
| + | \\ |
| + | \operatorname{is~true}. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="160px" | [[Image:Cactus ((A),(B),(C)) Big.jpg|65px]] |
| + | | <math>\texttt{((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{just~one} |
| + | \\ |
| + | \operatorname{of}~ a, b, c |
| + | \\ |
| + | \operatorname{is~true}. |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{not~just~one} |
| + | \\ |
| + | \operatorname{of}~ a, b, c |
| + | \\ |
| + | \operatorname{is~false}. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="160px" | [[Image:Cactus ((A,B,C)) Big.jpg|65px]] |
| + | | <math>\texttt{((} a, b, c \texttt{))}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{not~just~one} |
| + | \\ |
| + | \operatorname{of}~ a, b, c |
| + | \\ |
| + | \operatorname{is~false}. |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{just~one} |
| + | \\ |
| + | \operatorname{of}~ a, b, c |
| + | \\ |
| + | \operatorname{is~true}. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="200px" | [[Image:Cactus (((A),(B),(C))) Big.jpg|65px]] |
| + | | <math>\texttt{(((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{)))}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{not~just~one} |
| + | \\ |
| + | \operatorname{of}~ a, b, c |
| + | \\ |
| + | \operatorname{is~true}. |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{just~one} |
| + | \\ |
| + | \operatorname{of}~ a, b, c |
| + | \\ |
| + | \operatorname{is~false}. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="160px" | [[Image:Cactus (A,(B),(C)) Big.jpg|65px]] |
| + | | <math>\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{partition}~ a |
| + | \\ |
| + | \operatorname{into}~ b, c. |
| + | \end{matrix}</math> |
| + | | |
| + | |- |
| + | | height="200px" | [[Image:Cactus (((A),B,C)) Big.jpg|65px]] |
| + | | <math>\texttt{(((} a \texttt{)}, b, c \texttt{))}</math> |
| + | | |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{partition}~ a |
| + | \\ |
| + | \operatorname{into}~ b, c. |
| + | \end{matrix}</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | ==Differential Logic== |
| + | |
| + | ===Ascii Tables=== |
| + | |
| <pre> | | <pre> |
− | Table 4. Ef Expanded Over Differential Features {dx, dy} | + | Table A1. Propositional Forms On Two Variables |
− | o------o------------o------------o------------o------------o------------o | + | o---------o---------o---------o----------o------------------o----------o |
− | | | | | | | | | + | | L_1 | L_2 | L_3 | L_4 | L_5 | L_6 | |
− | | | f | T_11 f | T_10 f | T_01 f | T_00 f | | + | | | | | | | | |
− | | | | | | | | | + | | Decimal | Binary | Vector | Cactus | English | Ordinary | |
− | | | | Ef| dx dy | Ef| dx(dy) | Ef| (dx)dy | Ef|(dx)(dy)|
| + | o---------o---------o---------o----------o------------------o----------o |
− | | | | | | | |
| + | | | x : 1 1 0 0 | | | | |
− | o------o------------o------------o------------o------------o------------o
| + | | | y : 1 0 1 0 | | | | |
− | | | | | | | |
| + | o---------o---------o---------o----------o------------------o----------o |
− | | f_0 | () | () | () | () | () |
| + | | | | | | | | |
− | | | | | | | |
| + | | 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_1 | (x)(y) | x y | x (y) | (x) y | (x)(y) | | + | | | | | | | | |
− | | | | | | | | | + | | f_2 | f_0010 | 0 0 1 0 | (x) y | y and not x | ~x & y | |
− | | f_2 | (x) y | x (y) | x y | (x)(y) | (x) y | | + | | | | | | | | |
− | | | | | | | | | + | | f_3 | f_0011 | 0 0 1 1 | (x) | not x | ~x | |
− | | f_4 | x (y) | (x) y | (x)(y) | x y | x (y) | | + | | | | | | | | |
− | | | | | | | | | + | | f_4 | f_0100 | 0 1 0 0 | x (y) | x and not y | x & ~y | |
− | | f_8 | x y | (x)(y) | (x) y | x (y) | x y | | + | | | | | | | | |
− | | | | | | | | | + | | f_5 | f_0101 | 0 1 0 1 | (y) | not y | ~y | |
− | 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_3 | (x) | x | x | (x) | (x) | | + | | | | | | | | |
− | | | | | | | | | + | | f_7 | f_0111 | 0 1 1 1 | (x y) | not both x and y | ~x v ~y | |
− | | f_12 | x | (x) | (x) | x | x | | + | | | | | | | | |
− | | | | | | | | | + | | f_8 | f_1000 | 1 0 0 0 | x y | x and y | x & y | |
− | o------o------------o------------o------------o------------o------------o
| + | | | | | | | | |
− | | | | | | | | | + | | f_9 | f_1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y | |
− | | f_6 | (x, y) | (x, y) | ((x, y)) | ((x, y)) | (x, y) | | + | | | | | | | | |
− | | | | | | | | | + | | f_10 | f_1010 | 1 0 1 0 | y | y | y | |
− | | f_9 | ((x, y)) | ((x, y)) | (x, y) | (x, y) | ((x, y)) | | + | | | | | | | | |
− | | | | | | | | | + | | f_11 | f_1011 | 1 0 1 1 | (x (y)) | not x without y | x => y | |
− | o------o------------o------------o------------o------------o------------o | + | | | | | | | | |
− | | | | | | | |
| + | | f_12 | f_1100 | 1 1 0 0 | x | x | x | |
− | | f_5 | (y) | y | (y) | y | (y) |
| + | | | | | | | | |
− | | | | | | | |
| + | | f_13 | f_1101 | 1 1 0 1 | ((x) y) | not y without x | x <= y | |
− | | f_10 | y | (y) | y | (y) | 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 | |
− | | f_7 | (x y) | ((x)(y)) | ((x) y) | (x (y)) | (x y) | | + | | | | | | | | |
− | | | | | | | |
| + | o---------o---------o---------o----------o------------------o----------o |
− | | f_11 | (x (y)) | ((x) y) | ((x)(y)) | (x y) | (x (y)) |
| + | </pre> |
− | | | | | | | | | + | |
− | | f_13 | ((x) y) | (x (y)) | (x y) | ((x)(y)) | ((x) y) |
| + | <pre> |
− | | | | | | | | | + | Table A2. Propositional Forms On Two Variables |
− | | f_14 | ((x)(y)) | (x y) | (x (y)) | ((x) y) | ((x)(y)) | | + | o---------o---------o---------o----------o------------------o----------o |
− | | | | | | | | | + | | L_1 | L_2 | L_3 | L_4 | L_5 | L_6 | |
− | o------o------------o------------o------------o------------o------------o
| + | | | | | | | | |
− | | | | | | | |
| + | | Decimal | Binary | Vector | Cactus | English | Ordinary | |
− | | f_15 | (()) | (()) | (()) | (()) | (()) | | + | o---------o---------o---------o----------o------------------o----------o |
− | | | | | | | | | + | | | x : 1 1 0 0 | | | | |
− | o------o------------o------------o------------o------------o------------o
| + | | | y : 1 0 1 0 | | | | |
− | | | | | | | | + | o---------o---------o---------o----------o------------------o----------o |
− | | Fixed Point Total | 4 | 4 | 4 | 16 | | + | | | | | | | | |
− | | | | | | | | + | | f_0 | f_0000 | 0 0 0 0 | () | false | 0 | |
− | o-------------------o------------o------------o------------o------------o
| + | | | | | | | | |
− | </pre>
| + | o---------o---------o---------o----------o------------------o----------o |
− | <pre>
| + | | | | | | | | |
− | Table 5. Df Expanded Over Differential Features {dx, dy}
| + | | f_1 | f_0001 | 0 0 0 1 | (x)(y) | neither x nor y | ~x & ~y | |
− | o------o------------o------------o------------o------------o------------o
| + | | | | | | | | |
− | | | | | | | | | + | | f_2 | f_0010 | 0 0 1 0 | (x) y | y and not x | ~x & y | |
− | | | f | Df| dx dy | Df| dx(dy) | Df| (dx)dy | Df|(dx)(dy)| | + | | | | | | | | |
− | | | | | | | | | + | | f_4 | f_0100 | 0 1 0 0 | x (y) | x and not y | x & ~y | |
− | o------o------------o------------o------------o------------o------------o
| + | | | | | | | | |
− | | | | | | | | | + | | f_8 | f_1000 | 1 0 0 0 | x y | x and y | x & y | |
− | | f_0 | () | () | () | () | () | | + | | | | | | | | |
− | | | | | | | | | + | 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 |
| + | </pre> |
| + | |
| + | <pre> |
| + | Table A3. Ef Expanded Over Differential Features {dx, dy} |
| o------o------------o------------o------------o------------o------------o | | o------o------------o------------o------------o------------o------------o |
| | | | | | | | | | | | | | | | | |
− | | f_1 | (x)(y) | ((x, y)) | (y) | (x) | () | | + | | | f | T_11 f | T_10 f | T_01 f | T_00 f | |
| | | | | | | | | | | | | | | | | |
− | | f_2 | (x) y | (x, y) | y | (x) | () | | + | | | | Ef| dx dy | Ef| dx(dy) | Ef| (dx)dy | Ef|(dx)(dy)| |
| | | | | | | | | | | | | | | | | |
− | | f_4 | x (y) | (x, y) | (y) | x | () |
| + | o------o------------o------------o------------o------------o------------o |
| | | | | | | | | | | | | | | | | |
− | | f_8 | x y | ((x, y)) | y | x | () | | + | | f_0 | () | () | () | () | () | |
| | | | | | | | | | | | | | | | | |
| o------o------------o------------o------------o------------o------------o | | o------o------------o------------o------------o------------o------------o |
| | | | | | | | | | | | | | | | | |
− | | f_3 | (x) | (()) | (()) | () | () | | + | | 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 | (()) | (()) | () | () | | + | | f_12 | x | (x) | (x) | x | x | |
| | | | | | | | | | | | | | | | | |
| o------o------------o------------o------------o------------o------------o | | o------o------------o------------o------------o------------o------------o |
| | | | | | | | | | | | | | | | | |
− | | f_6 | (x, y) | () | (()) | (()) | () | | + | | f_6 | (x, y) | (x, y) | ((x, y)) | ((x, y)) | (x, y) | |
| | | | | | | | | | | | | | | | | |
− | | f_9 | ((x, y)) | () | (()) | (()) | () | | + | | f_9 | ((x, y)) | ((x, y)) | (x, y) | (x, y) | ((x, y)) | |
| | | | | | | | | | | | | | | | | |
| o------o------------o------------o------------o------------o------------o | | o------o------------o------------o------------o------------o------------o |
| | | | | | | | | | | | | | | | | |
− | | f_5 | (y) | (()) | () | (()) | () | | + | | f_5 | (y) | y | (y) | y | (y) | |
| | | | | | | | | | | | | | | | | |
− | | f_10 | y | (()) | () | (()) | () | | + | | f_10 | y | (y) | y | (y) | y | |
| | | | | | | | | | | | | | | | | |
| o------o------------o------------o------------o------------o------------o | | o------o------------o------------o------------o------------o------------o |
| | | | | | | | | | | | | | | | | |
− | | f_7 | (x y) | ((x, y)) | y | x | () | | + | | f_7 | (x y) | ((x)(y)) | ((x) y) | (x (y)) | (x y) | |
| | | | | | | | | | | | | | | | | |
− | | f_11 | (x (y)) | (x, y) | (y) | x | () | | + | | f_11 | (x (y)) | ((x) y) | ((x)(y)) | (x y) | (x (y)) | |
| | | | | | | | | | | | | | | | | |
− | | f_13 | ((x) y) | (x, y) | y | (x) | () | | + | | f_13 | ((x) y) | (x (y)) | (x y) | ((x)(y)) | ((x) y) | |
| | | | | | | | | | | | | | | | | |
− | | f_14 | ((x)(y)) | ((x, y)) | (y) | (x) | () | | + | | f_14 | ((x)(y)) | (x y) | (x (y)) | ((x) y) | ((x)(y)) | |
| | | | | | | | | | | | | | | | | |
| o------o------------o------------o------------o------------o------------o | | o------o------------o------------o------------o------------o------------o |
| | | | | | | | | | | | | | | | | |
− | | f_15 | (()) | () | () | () | () | | + | | f_15 | (()) | (()) | (()) | (()) | (()) | |
| | | | | | | | | | | | | | | | | |
| o------o------------o------------o------------o------------o------------o | | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | |
| + | | Fixed Point Total | 4 | 4 | 4 | 16 | |
| + | | | | | | | |
| + | o-------------------o------------o------------o------------o------------o |
| </pre> | | </pre> |
| | | |
− | ===Wiki Tables===
| + | <pre> |
− | | + | Table A4. Df Expanded Over Differential Features {dx, dy} |
− | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
| + | o------o------------o------------o------------o------------o------------o |
− | |+ '''Table 1. Propositional Forms on Two Variables''' | + | | | | | | | | |
− | |- style="background:paleturquoise" | + | | | f | Df| dx dy | Df| dx(dy) | Df| (dx)dy | Df|(dx)(dy)| |
− | ! style="width:15%" | L<sub>1</sub>
| + | | | | | | | | |
− | ! style="width:15%" | L<sub>2</sub>
| + | o------o------------o------------o------------o------------o------------o |
− | ! style="width:15%" | L<sub>3</sub>
| + | | | | | | | | |
− | ! style="width:15%" | L<sub>4</sub>
| + | | f_0 | () | () | () | () | () | |
− | ! style="width:15%" | L<sub>5</sub>
| + | | | | | | | | |
− | ! style="width:15%" | L<sub>6</sub>
| + | o------o------------o------------o------------o------------o------------o |
− | |- style="background:paleturquoise" | + | | | | | | | | |
− | | | + | | f_1 | (x)(y) | ((x, y)) | (y) | (x) | () | |
− | | align="right" | x : | + | | | | | | | | |
− | | 1 1 0 0 | + | | f_2 | (x) y | (x, y) | y | (x) | () | |
− | | | + | | | | | | | | |
− | | | + | | f_4 | x (y) | (x, y) | (y) | x | () | |
− | | | + | | | | | | | | |
− | |- style="background:paleturquoise" | + | | f_8 | x y | ((x, y)) | y | x | () | |
− | | | + | | | | | | | | |
− | | align="right" | y : | + | o------o------------o------------o------------o------------o------------o |
− | | 1 0 1 0 | + | | | | | | | | |
− | | | + | | f_3 | (x) | (()) | (()) | () | () | |
− | | | + | | | | | | | | |
− | | | + | | f_12 | x | (()) | (()) | () | () | |
− | |-
| + | | | | | | | | |
− | | f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || ( ) || false || 0 | + | o------o------------o------------o------------o------------o------------o |
− | |- | + | | | | | | | | |
− | | f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || ¬x ∧ ¬y | + | | f_6 | (x, y) | () | (()) | (()) | () | |
− | |- | + | | | | | | | | |
− | | f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || ¬x ∧ y | + | | f_9 | ((x, y)) | () | (()) | (()) | () | |
− | |- | + | | | | | | | | |
− | | f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || ¬x | + | o------o------------o------------o------------o------------o------------o |
− | |- | + | | | | | | | | |
− | | f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x ∧ ¬y | + | | f_5 | (y) | (()) | () | (()) | () | |
− | |- | + | | | | | | | | |
− | | f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || ¬y | + | | f_10 | y | (()) | () | (()) | () | |
− | |- | + | | | | | | | | |
− | | f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x ≠ y | + | o------o------------o------------o------------o------------o------------o |
− | |- | + | | | | | | | | |
− | | f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x y) || not both x and y || ¬x ∨ ¬y | + | | f_7 | (x y) | ((x, y)) | y | x | () | |
− | |- | + | | | | | | | | |
− | | f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x y || x and y || x ∧ y | + | | f_11 | (x (y)) | (x, y) | (y) | x | () | |
− | |- | + | | | | | | | | |
− | | f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y | + | | f_13 | ((x) y) | (x, y) | y | (x) | () | |
− | |- | + | | | | | | | | |
− | | f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y | + | | f_14 | ((x)(y)) | ((x, y)) | (y) | (x) | () | |
− | |- | + | | | | | | | | |
− | | f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x → y | + | o------o------------o------------o------------o------------o------------o |
− | |- | + | | | | | | | | |
− | | f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x | + | | f_15 | (()) | () | () | () | () | |
− | |- | + | | | | | | | | |
− | | f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x ← y | + | o------o------------o------------o------------o------------o------------o |
− | |- | + | </pre> |
− | | f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y || x ∨ y | |
− | |- | |
− | | f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || (( )) || true || 1 | |
− | |} | |
− | <br> | |
| | | |
− | ==Inquiry Driven Systems==
| + | <pre> |
− | | + | Table A5. Ef Expanded Over Ordinary Features {x, y} |
− | ===Table 1. Sign Relation of Interpreter ''A''===
| + | o------o------------o------------o------------o------------o------------o |
− | | + | | | | | | | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
| + | | | f | Ef | xy | Ef | x(y) | Ef | (x)y | Ef | (x)(y)| |
− | |+ Table 1. Sign Relation of Interpreter ''A'' | + | | | | | | | | |
− | |- style="background:paleturquoise" | + | o------o------------o------------o------------o------------o------------o |
− | ! style="width:20%" | Object
| + | | | | | | | | |
− | ! style="width:20%" | Sign
| + | | f_0 | () | () | () | () | () | |
− | ! style="width:20%" | Interpretant
| + | | | | | | | | |
− | |- | + | o------o------------o------------o------------o------------o------------o |
− | | ''A'' || "A" || "A" | + | | | | | | | | |
− | |- | + | | f_1 | (x)(y) | dx dy | dx (dy) | (dx) dy | (dx)(dy) | |
− | | ''A'' || "A" || "i" | + | | | | | | | | |
− | |- | + | | f_2 | (x) y | dx (dy) | dx dy | (dx)(dy) | (dx) dy | |
− | | ''A'' || "i" || "A" | + | | | | | | | | |
− | |- | + | | f_4 | x (y) | (dx) dy | (dx)(dy) | dx dy | dx (dy) | |
− | | ''A'' || "i" || "i" | + | | | | | | | | |
− | |- | + | | f_8 | x y | (dx)(dy) | (dx) dy | dx (dy) | dx dy | |
− | | ''B'' || "B" || "B" | + | | | | | | | | |
− | |- | + | o------o------------o------------o------------o------------o------------o |
− | | ''B'' || "B" || "u" | + | | | | | | | | |
− | |- | + | | f_3 | (x) | dx | dx | (dx) | (dx) | |
− | | ''B'' || "u" || "B" | + | | | | | | | | |
− | |- | + | | f_12 | x | (dx) | (dx) | dx | dx | |
− | | ''B'' || "u" || "u" | + | | | | | | | | |
− | |} | + | o------o------------o------------o------------o------------o------------o |
− | <br> | + | | | | | | | | |
− | | + | | f_6 | (x, y) | (dx, dy) | ((dx, dy)) | ((dx, dy)) | (dx, dy) | |
− | ===Table 2. Sign Relation of Interpreter ''B''===
| + | | | | | | | | |
| + | | 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 |
| + | </pre> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" | + | <pre> |
− | |+ Table 2. Sign Relation of Interpreter ''B'' | + | Table A6. Df Expanded Over Ordinary Features {x, y} |
− | |- style="background:paleturquoise" | + | o------o------------o------------o------------o------------o------------o |
− | ! style="width:20%" | Object
| + | | | | | | | | |
− | ! style="width:20%" | Sign
| + | | | f | Df | xy | Df | x(y) | Df | (x)y | Df | (x)(y)| |
− | ! style="width:20%" | Interpretant
| + | | | | | | | | |
− | |- | + | o------o------------o------------o------------o------------o------------o |
− | | ''A'' || "A" || "A" | + | | | | | | | | |
− | |- | + | | f_0 | () | () | () | () | () | |
− | | ''A'' || "A" || "u" | + | | | | | | | | |
− | |- | + | o------o------------o------------o------------o------------o------------o |
− | | ''A'' || "u" || "A" | + | | | | | | | | |
− | |- | + | | f_1 | (x)(y) | dx dy | dx (dy) | (dx) dy | ((dx)(dy)) | |
− | | ''A'' || "u" || "u" | + | | | | | | | | |
− | |- | + | | f_2 | (x) y | dx (dy) | dx dy | ((dx)(dy)) | (dx) dy | |
− | | ''B'' || "B" || "B" | + | | | | | | | | |
− | |- | + | | f_4 | x (y) | (dx) dy | ((dx)(dy)) | dx dy | dx (dy) | |
− | | ''B'' || "B" || "i" | + | | | | | | | | |
− | |- | + | | f_8 | x y | ((dx)(dy)) | (dx) dy | dx (dy) | dx dy | |
− | | ''B'' || "i" || "B" | + | | | | | | | | |
− | |- | + | o------o------------o------------o------------o------------o------------o |
− | | ''B'' || "i" || "i" | + | | | | | | | | |
− | |} | + | | f_3 | (x) | dx | dx | dx | dx | |
− | <br> | + | | | | | | | | |
− | | + | | f_12 | x | dx | dx | dx | dx | |
− | ===Table 3. Semiotic Partition of Interpreter ''A''=== | + | | | | | | | | |
− | | + | o------o------------o------------o------------o------------o------------o |
− | <pre> | + | | | | | | | | |
− | Table 3. Semiotic Partition of Interpreter ''A''
| + | | f_6 | (x, y) | (dx, dy) | (dx, dy) | (dx, dy) | (dx, dy) | |
− | "A"
| + | | | | | | | | |
− | "i"
| + | | f_9 | ((x, y)) | (dx, dy) | (dx, dy) | (dx, dy) | (dx, dy) | |
− | "u"
| + | | | | | | | | |
− | "B"
| + | 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 |
| + | </pre> |
| + | |
| + | <pre> |
| + | 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 |
| + | </pre> |
| + | |
| + | <pre> |
| + | 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 |
| + | </pre> |
| + | |
| + | <pre> |
| + | 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 |
| </pre> | | </pre> |
| | | |
− | ===Table 4. Semiotic Partition of Interpreter ''B''=== | + | <pre> |
| + | Matrix Representations of Permutations in Sym(3) |
| + | o---------o---------o---------o---------o---------o---------o |
| + | | | | | | | | |
| + | | e | f | g | h | i | j | |
| + | | | | | | | | |
| + | o=========o=========o=========o=========o=========o=========o |
| + | | | | | | | | |
| + | | 1 0 0 | 0 0 1 | 0 1 0 | 1 0 0 | 0 0 1 | 0 1 0 | |
| + | | 0 1 0 | 1 0 0 | 0 0 1 | 0 0 1 | 0 1 0 | 1 0 0 | |
| + | | 0 0 1 | 0 1 0 | 1 0 0 | 0 1 0 | 1 0 0 | 0 0 1 | |
| + | | | | | | | | |
| + | o---------o---------o---------o---------o---------o---------o |
| + | </pre> |
| | | |
| <pre> | | <pre> |
− | Table 4. Semiotic Partition of Interpreter ''B''
| + | Symmetric Group S_3 |
− | "A"
| + | o-------------------------------------------------o |
− | "i"
| + | | | |
− | "u"
| + | | ^ | |
− | "B"
| + | | 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 |
| </pre> | | </pre> |
| | | |
− | ==Logical Tables== | + | ===Wiki Tables : New Versions=== |
| | | |
− | ===Higher Order Propositions=== | + | ====Propositional Forms on Two Variables==== |
| | | |
− | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" | + | <br> |
− | |+ '''Table 7. Higher Order Propositions (n = 1)''' | + | |
− | |- style="background:paleturquoise" | + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" |
− | | \ ''x'' || 1 0 || ''F'' | + | |+ '''Table A1. Propositional Forms on Two Variables''' |
− | |''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m'' | + | |- style="background:#f0f0ff" |
− | |''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m'' | + | ! width="15%" | L<sub>1</sub> |
− | |- style="background:paleturquoise" | + | ! width="15%" | L<sub>2</sub> |
− | | ''F'' \ || || | + | ! width="15%" | L<sub>3</sub> |
− | |00||01||02||03||04||05||06||07||08||09||10||11||12||13||14||15 | + | ! width="15%" | L<sub>4</sub> |
| + | ! width="25%" | L<sub>5</sub> |
| + | ! width="15%" | L<sub>6</sub> |
| + | |- style="background:#f0f0ff" |
| + | | |
| + | | align="right" | x : |
| + | | 1 1 0 0 |
| + | | |
| + | | |
| + | | |
| + | |- style="background:#f0f0ff" |
| + | | |
| + | | align="right" | y : |
| + | | 1 0 1 0 |
| + | | |
| + | | |
| + | | |
| |- | | |- |
− | | ''F<sub>0</sub> || 0 0 || 0 ||0||1||0||1||0||1||0||1||0||1||0||1||0||1||0||1 | + | | f<sub>0</sub> |
| + | | f<sub>0000</sub> |
| + | | 0 0 0 0 |
| + | | ( ) |
| + | | false |
| + | | 0 |
| |- | | |- |
− | | ''F<sub>1</sub> || 0 1 || (x) ||0||0||1||1||0||0||1||1||0||0||1||1||0||0||1||1 | + | | f<sub>1</sub> |
| + | | f<sub>0001</sub> |
| + | | 0 0 0 1 |
| + | | (x)(y) |
| + | | neither x nor y |
| + | | ¬x ∧ ¬y |
| |- | | |- |
− | | ''F<sub>2</sub> || 1 0 || x ||0||0||0||0||1||1||1||1||0||0||0||0||1||1||1||1 | + | | f<sub>2</sub> |
| + | | f<sub>0010</sub> |
| + | | 0 0 1 0 |
| + | | (x) y |
| + | | y and not x |
| + | | ¬x ∧ y |
| |- | | |- |
− | | ''F<sub>3</sub> || 1 1 || 1 ||0||0||0||0||0||0||0||0||1||1||1||1||1||1||1||1 | + | | f<sub>3</sub> |
− | |} | + | | f<sub>0011</sub> |
− | <br> | + | | 0 0 1 1 |
− | | + | | (x) |
− | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
| + | | not x |
− | |+ '''Table 8. Interpretive Categories for Higher Order Propositions (n = 1)''' | + | | ¬x |
− | |- style="background:paleturquoise" | |
− | |Measure||Happening||Exactness||Existence||Linearity||Uniformity||Information | |
| |- | | |- |
− | |''m''<sub>0</sub>||nothing happens|| || || || || | + | | f<sub>4</sub> |
| + | | f<sub>0100</sub> |
| + | | 0 1 0 0 |
| + | | x (y) |
| + | | x and not y |
| + | | x ∧ ¬y |
| |- | | |- |
− | |''m''<sub>1</sub>|| ||just false||nothing exists|| || || | + | | f<sub>5</sub> |
| + | | f<sub>0101</sub> |
| + | | 0 1 0 1 |
| + | | (y) |
| + | | not y |
| + | | ¬y |
| |- | | |- |
− | |''m''<sub>2</sub>|| ||just not x|| || || || | + | | f<sub>6</sub> |
| + | | f<sub>0110</sub> |
| + | | 0 1 1 0 |
| + | | (x, y) |
| + | | x not equal to y |
| + | | x ≠ y |
| |- | | |- |
− | |''m''<sub>3</sub>|| || ||nothing is x|| || || | + | | f<sub>7</sub> |
| + | | f<sub>0111</sub> |
| + | | 0 1 1 1 |
| + | | (x y) |
| + | | not both x and y |
| + | | ¬x ∨ ¬y |
| |- | | |- |
− | |''m''<sub>4</sub>|| ||just x|| || || || | + | | f<sub>8</sub> |
| + | | f<sub>1000</sub> |
| + | | 1 0 0 0 |
| + | | x y |
| + | | x and y |
| + | | x ∧ y |
| |- | | |- |
− | |''m''<sub>5</sub>|| || ||everything is x||F is linear|| || | + | | f<sub>9</sub> |
| + | | f<sub>1001</sub> |
| + | | 1 0 0 1 |
| + | | ((x, y)) |
| + | | x equal to y |
| + | | x = y |
| |- | | |- |
− | |''m''<sub>6</sub>|| || || || ||F is not uniform||F is informed | + | | f<sub>10</sub> |
| + | | f<sub>1010</sub> |
| + | | 1 0 1 0 |
| + | | y |
| + | | y |
| + | | y |
| |- | | |- |
− | |''m''<sub>7</sub>|| ||not just true|| || || || | + | | f<sub>11</sub> |
| + | | f<sub>1011</sub> |
| + | | 1 0 1 1 |
| + | | (x (y)) |
| + | | not x without y |
| + | | x ⇒ y |
| |- | | |- |
− | |''m''<sub>8</sub>|| ||just true|| || || || | + | | f<sub>12</sub> |
| + | | f<sub>1100</sub> |
| + | | 1 1 0 0 |
| + | | x |
| + | | x |
| + | | x |
| |- | | |- |
− | |''m''<sub>9</sub>|| || || || ||F is uniform||F is not informed | + | | f<sub>13</sub> |
| + | | f<sub>1101</sub> |
| + | | 1 1 0 1 |
| + | | ((x) y) |
| + | | not y without x |
| + | | x ⇐ y |
| |- | | |- |
− | |''m''<sub>10</sub>|| || ||something is not x||F is not linear|| || | + | | f<sub>14</sub> |
| + | | f<sub>1110</sub> |
| + | | 1 1 1 0 |
| + | | ((x)(y)) |
| + | | x or y |
| + | | x ∨ y |
| |- | | |- |
− | |''m''<sub>11</sub>|| ||not just x|| || || || | + | | f<sub>15</sub> |
− | |-
| + | | f<sub>1111</sub> |
− | |''m''<sub>12</sub>|| || ||something is x|| || || | + | | 1 1 1 1 |
− | |-
| + | | (( )) |
− | |''m''<sub>13</sub>|| ||not just not x|| || || ||
| + | | true || 1 |
− | |- | |
− | |''m''<sub>14</sub>|| ||not just false||something exists|| || || | |
− | |-
| |
− | |''m''<sub>15</sub>||anything happens|| || || || || | |
| |} | | |} |
| + | |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; 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 9. Higher Order Propositions (n = 2)''' | + | |+ '''Table A2. Propositional Forms on Two Variables''' |
− | |- style="background:paleturquoise" | + | |- style="background:#f0f0ff" |
− | | align=right | ''x'' : || 1100 || ''f'' | + | ! width="15%" | L<sub>1</sub> |
− | |''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m'' | + | ! width="15%" | L<sub>2</sub> |
− | |''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m'' | + | ! width="15%" | L<sub>3</sub> |
− | |''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m'' | + | ! width="15%" | L<sub>4</sub> |
− | |- style="background:paleturquoise" | + | ! width="25%" | L<sub>5</sub> |
− | | align=right | ''y'' : || 1010 || | + | ! width="15%" | L<sub>6</sub> |
− | |0||1||2||3||4||5||6||7||8||9||10||11||12 | + | |- style="background:#f0f0ff" |
− | |13||14||15||16||17||18||19||20||21||22||23 | + | | |
| + | | align="right" | x : |
| + | | 1 1 0 0 |
| + | | |
| + | | |
| + | | |
| + | |- style="background:#f0f0ff" |
| + | | |
| + | | align="right" | y : |
| + | | 1 0 1 0 |
| + | | |
| + | | |
| + | | |
| |- | | |- |
− | | ''f<sub>0</sub> || 0000 || ( ) | + | | f<sub>0</sub> |
− | | 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 | + | | f<sub>0000</sub> |
− | | 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 | + | | 0 0 0 0 |
− | | 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 | + | | ( ) |
| + | | false |
| + | | 0 |
| |- | | |- |
− | | ''f<sub>1</sub> || 0001 || (x)(y) | + | | |
− | | || || 1 || 1 || 0 || 0 || 1 || 1 | + | {| align="center" |
− | | 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1
| + | | |
− | | 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 | + | <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" |
| + | | |
| + | <p>0 0 0 1</p> |
| + | <p>0 0 1 0</p> |
| + | <p>0 1 0 0</p> |
| + | <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> |
| + | |} |
| |- | | |- |
− | | ''f<sub>2</sub> || 0010 || (x) y | + | | |
− | | || || || || 1 || 1 || 1 || 1 | + | {| align="center" |
− | | 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1
| + | | |
− | | 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 | + | <p>f<sub>3</sub></p> |
| + | <p>f<sub>12</sub></p> |
| + | |} |
| + | | |
| + | {| align="center" |
| + | | |
| + | <p>f<sub>0011</sub></p> |
| + | <p>f<sub>1100</sub></p> |
| + | |} |
| + | | |
| + | {| align="center" |
| + | | |
| + | <p>0 0 1 1</p> |
| + | <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> |
| + | |} |
| |- | | |- |
− | | ''f<sub>3</sub> || 0011 || (x) | + | | |
− | | || || || || || || || | + | {| align="center" |
− | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
| + | | |
− | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 | + | <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" |
| + | | |
| + | <p>0 1 1 0</p> |
| + | <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> |
| + | |} |
| |- | | |- |
− | | ''f<sub>4</sub> || 0100 || x (y) | + | | |
− | | || || || || || || || | + | {| align="center" |
− | | || || || || || || || | + | | |
− | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
| + | <p>f<sub>5</sub></p> |
− | |- | + | <p>f<sub>10</sub></p> |
− | | ''f<sub>5</sub> || 0101 || (y)
| + | |} |
− | | || || || || || || || | + | | |
− | | || || || || || || || | + | {| align="center" |
− | | || || || || || || || | + | | |
− | |- | + | <p>f<sub>0101</sub></p> |
− | | ''f<sub>6</sub> || 0110 || (x, y)
| + | <p>f<sub>1010</sub></p> |
− | | || || || || || || || | + | |} |
− | | || || || || || || ||
| + | | |
− | | || || || || || || || | + | {| align="center" |
| + | | |
| + | <p>0 1 0 1</p> |
| + | <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> |
| + | |} |
| |- | | |- |
− | | ''f<sub>7</sub> || 0111 || (x y) | + | | |
− | | || || || || || || || | + | {| 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" |
| + | | |
| + | <p>0 1 1 1</p> |
| + | <p>1 0 1 1</p> |
| + | <p>1 1 0 1</p> |
| + | <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>8</sub> || 1000 || x y | + | | f<sub>15</sub> |
− | | || || || || || || || | + | | f<sub>1111</sub> |
− | | || || || || || || || | + | | 1 1 1 1 |
− | | || || || || || || ||
| + | | (( )) |
| + | | true |
| + | | 1 |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | ====Differential Propositions==== |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="6" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" |
| + | |+ '''Table 14. Differential Propositions''' |
| + | |- style="background:#f0f0ff" |
| + | | |
| + | | align="right" | A : |
| + | | 1 1 0 0 |
| + | | |
| + | | |
| + | | |
| + | |- style="background:#f0f0ff" |
| + | | |
| + | | align="right" | dA : |
| + | | 1 0 1 0 |
| + | | |
| + | | |
| + | | |
| |- | | |- |
− | | ''f<sub>9</sub> || 1001 || ((x, y)) | + | | f<sub>0</sub> |
− | | || || || || || || ||
| + | | g<sub>0</sub> |
− | | || || || || || || || | + | | 0 0 0 0 |
− | | || || || || || || || | + | | ( ) |
| + | | False |
| + | | 0 |
| |- | | |- |
− | | ''f<sub>10</sub> || 1010 || y | + | | |
− | | || || || || || || ||
| + | {| |
− | | || || || || || || || | + | | |
− | | || || || || || || ||
| + | <br> |
| + | <br> |
| + | <br> |
| + | |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | g<sub>1</sub><br> |
| + | g<sub>2</sub><br> |
| + | g<sub>4</sub><br> |
| + | g<sub>8</sub> |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | 0 0 0 1<br> |
| + | 0 0 1 0<br> |
| + | 0 1 0 0<br> |
| + | 1 0 0 0 |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | (A)(dA)<br> |
| + | (A) dA <br> |
| + | A (dA)<br> |
| + | A dA |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | Neither A nor dA<br> |
| + | Not A but dA<br> |
| + | A but not dA<br> |
| + | A and dA |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | ¬A ∧ ¬dA<br> |
| + | ¬A ∧ dA<br> |
| + | A ∧ ¬dA<br> |
| + | A ∧ dA |
| + | |} |
| |- | | |- |
− | | ''f<sub>11</sub> || 1011 || (x (y)) | + | | |
− | | || || || || || || || | + | {| |
− | | || || || || || || || | + | | |
− | | || || || || || || || | + | f<sub>1</sub><br> |
| + | f<sub>2</sub> |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | g<sub>3</sub><br> |
| + | g<sub>12</sub> |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | 0 0 1 1<br> |
| + | 1 1 0 0 |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | (A)<br> |
| + | A |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | Not A<br> |
| + | A |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | ¬A<br> |
| + | A |
| + | |} |
| |- | | |- |
− | | ''f<sub>12</sub> || 1100 || x | + | | |
− | | || || || || || || ||
| + | {| |
− | | || || || || || || || | + | | |
− | | || || || || || || || | + | <br> |
− | |-
| + | |
− | | ''f<sub>13</sub> || 1101 || ((x) y)
| + | |} |
− | | || || || || || || || | + | | |
− | | || || || || || || || | + | {| |
− | | || || || || || || || | + | | |
− | |- | + | g<sub>6</sub><br> |
− | | ''f<sub>14</sub> || 1110 || ((x)(y))
| + | g<sub>9</sub> |
− | | || || || || || || || | + | |} |
− | | || || || || || || || | + | | |
− | | || || || || || || || | + | {| |
− | |- | + | | |
− | | ''f<sub>15</sub> || 1111 || (( ))
| + | 0 1 1 0<br> |
− | | || || || || || || || | + | 1 0 0 1 |
− | | || || || || || || ||
| + | |} |
− | | || || || || || || ||
| + | | |
| + | {| |
| + | | |
| + | (A, dA)<br> |
| + | ((A, dA)) |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | A not equal to dA<br> |
| + | A equal to dA |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | A ≠ dA<br> |
| + | A = dA |
| |} | | |} |
− | <br>
| |
− |
| |
− | {| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
| |
− | |+ '''Table 10. Qualifiers of Implication Ordering: α<sub>''i'' </sub>''f'' = Υ(''f''<sub>''i''</sub> ⇒ ''f'')'''
| |
− | |- style="background:paleturquoise"
| |
− | | align=right | ''x'' : || 1100 || ''f''
| |
− | |α||α||α||α||α||α||α||α
| |
− | |α||α||α||α||α||α||α||α
| |
− | |- style="background:paleturquoise"
| |
− | | align=right | ''y'' : || 1010 ||
| |
− | |15||14||13||12||11||10||9||8||7||6||5||4||3||2||1||0
| |
| |- | | |- |
− | | ''f<sub>0</sub> || 0000 || ( ) | + | | |
− | | || || || || || || ||
| + | {| |
− | | || || || || || || || 1 | + | | |
− | |-
| + | <br> |
− | | ''f<sub>1</sub> || 0001 || (x)(y)
| + | |
− | | || || || || || || || | + | |} |
− | | || || || || || || 1 || 1 | + | | |
| + | {| |
| + | | |
| + | g<sub>5</sub><br> |
| + | g<sub>10</sub> |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | 0 1 0 1<br> |
| + | 1 0 1 0 |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | (dA)<br> |
| + | dA |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | Not dA<br> |
| + | dA |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | ¬dA<br> |
| + | dA |
| + | |} |
| |- | | |- |
− | | ''f<sub>2</sub> || 0010 || (x) y | + | | |
− | | || || || || || || || | + | {| |
− | | || || || || || 1 || || 1 | + | | |
| + | <br> |
| + | <br> |
| + | <br> |
| + | |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | g<sub>7</sub><br> |
| + | g<sub>11</sub><br> |
| + | g<sub>13</sub><br> |
| + | g<sub>14</sub> |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | 0 1 1 1<br> |
| + | 1 0 1 1<br> |
| + | 1 1 0 1<br> |
| + | 1 1 1 0 |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | (A dA)<br> |
| + | (A (dA))<br> |
| + | ((A) dA)<br> |
| + | ((A)(dA)) |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | Not both A and dA<br> |
| + | Not A without dA<br> |
| + | Not dA without A<br> |
| + | A or dA |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | ¬A ∨ ¬dA<br> |
| + | A ⇒ dA<br> |
| + | A ⇐ dA<br> |
| + | A ∨ dA |
| + | |} |
| |- | | |- |
− | | ''f<sub>3</sub> || 0011 || (x) | + | | f<sub>3</sub> |
− | | || || || || || || ||
| + | | g<sub>15</sub> |
− | | || || || || 1 || 1 || 1 || 1 | + | | 1 1 1 1 |
− | |- | + | | (( )) |
− | | ''f<sub>4</sub> || 0100 || x (y) | + | | True |
− | | || || || || || || || | + | | 1 |
− | | || || || 1 || || || || 1 | + | |} |
| + | |
| + | <br> |
| + | |
| + | ===Wiki Tables : Old Versions=== |
| + | |
| + | ====Propositional Forms on Two Variables==== |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |+ '''Table 1. Propositional Forms on Two Variables''' |
| + | |- style="background:paleturquoise" |
| + | ! 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:paleturquoise" |
| + | | |
| + | | align="right" | x : |
| + | | 1 1 0 0 |
| + | | |
| + | | |
| + | | |
| + | |- style="background:paleturquoise" |
| + | | |
| + | | align="right" | y : |
| + | | 1 0 1 0 |
| + | | |
| + | | |
| + | | |
| |- | | |- |
− | | ''f<sub>5</sub> || 0101 || (y) | + | | f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || ( ) || false || 0 |
− | | || || || || || || || | |
− | | || || 1 || 1 || || || 1 || 1
| |
| |- | | |- |
− | | ''f<sub>6</sub> || 0110 || (x, y) | + | | f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || ¬x ∧ ¬y |
− | | || || || || || || || | |
− | | || 1 || || 1 || || 1 || || 1
| |
| |- | | |- |
− | | ''f<sub>7</sub> || 0111 || (x y) | + | | f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || ¬x ∧ y |
− | | || || || || || || || | |
− | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
| |
| |- | | |- |
− | | ''f<sub>8</sub> || 1000 || x y | + | | f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || ¬x |
− | | || || || || || || || 1
| |
− | | || || || || || || || 1
| |
| |- | | |- |
− | | ''f<sub>9</sub> || 1001 || ((x, y)) | + | | f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x ∧ ¬y |
− | | || || || || || || 1 || 1 | + | |- |
− | | || || || || || || 1 || 1 | + | | f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || ¬y |
| + | |- |
| + | | f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x ≠ y |
| + | |- |
| + | | f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x y) || not both x and y || ¬x ∨ ¬y |
| + | |- |
| + | | f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x y || x and y || x ∧ y |
| + | |- |
| + | | f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y |
| |- | | |- |
− | | ''f<sub>10</sub> || 1010 || y | + | | f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y |
− | | || || || || || 1 || || 1
| |
− | | || || || || || 1 || || 1 | |
| |- | | |- |
− | | ''f<sub>11</sub> || 1011 || (x (y)) | + | | f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x → y |
− | | || || || || 1 || 1 || 1 || 1 | |
− | | || || || || 1 || 1 || 1 || 1
| |
| |- | | |- |
− | | ''f<sub>12</sub> || 1100 || x | + | | f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x |
− | | || || || 1 || || || || 1
| |
− | | || || || 1 || || || || 1
| |
| |- | | |- |
− | | ''f<sub>13</sub> || 1101 || ((x) y) | + | | f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x ← y |
− | | || || 1 || 1 || || || 1 || 1 | |
− | | || || 1 || 1 || || || 1 || 1
| |
| |- | | |- |
− | | ''f<sub>14</sub> || 1110 || ((x)(y)) | + | | f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y || x ∨ y |
− | | || 1 || || 1 || || 1 || || 1 | |
− | | || 1 || || 1 || || 1 || || 1
| |
| |- | | |- |
− | | ''f<sub>15</sub> || 1111 || (( )) | + | | f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || (( )) || true || 1 |
− | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 | |
− | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
| |
| |} | | |} |
| + | |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" | + | ====Differential Propositions==== |
− | |+ '''Table 11. Qualifiers of Implication Ordering: β<sub>''i'' </sub>''f'' = Υ(''f'' ⇒ ''f''<sub>''i''</sub>)''' | + | |
− | |- style="background:paleturquoise" | + | <br> |
− | | align=right | ''x'' : || 1100 || ''f'' | + | |
− | |β||β||β||β||β||β||β||β | + | {| align="center" border="1" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:90%" |
− | |β||β||β||β||β||β||β||β | + | |+ '''Table 14. Differential Propositions''' |
− | |- style="background:paleturquoise" | + | |- style="background:ghostwhite" |
− | | align=right | ''y'' : || 1010 || | + | | |
− | |0||1||2||3||4||5||6||7||8||9||10||11||12||13||14||15 | + | | align="right" | A : |
| + | | 1 1 0 0 |
| + | | |
| + | | |
| + | | |
| + | |- style="background:ghostwhite" |
| + | | |
| + | | align="right" | dA : |
| + | | 1 0 1 0 |
| + | | |
| + | | |
| + | | |
| |- | | |- |
− | | ''f<sub>0</sub> || 0000 || ( ) | + | | f<sub>0</sub> |
− | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 | + | | g<sub>0</sub> |
− | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 | + | | 0 0 0 0 |
| + | | ( ) |
| + | | False |
| + | | 0 |
| |- | | |- |
− | | ''f<sub>1</sub> || 0001 || (x)(y) | + | | |
− | | || 1 || || 1 || || 1 || || 1
| + | {| |
− | | || 1 || || 1 || || 1 || || 1 | + | | |
− | |-
| + | <br> |
− | | ''f<sub>2</sub> || 0010 || (x) y
| + | <br> |
− | | || || 1 || 1 || || || 1 || 1
| + | <br> |
− | | || || 1 || 1 || || || 1 || 1 | + | |
− | |- | + | |} |
− | | ''f<sub>3</sub> || 0011 || (x)
| + | | |
− | | || || || 1 || || || || 1 | + | {| |
− | | || || || 1 || || || || 1
| + | | |
| + | g<sub>1</sub><br> |
| + | g<sub>2</sub><br> |
| + | g<sub>4</sub><br> |
| + | g<sub>8</sub> |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | 0 0 0 1<br> |
| + | 0 0 1 0<br> |
| + | 0 1 0 0<br> |
| + | 1 0 0 0 |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | (A)(dA)<br> |
| + | (A) dA <br> |
| + | A (dA)<br> |
| + | A dA |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | Neither A nor dA<br> |
| + | Not A but dA<br> |
| + | A but not dA<br> |
| + | A and dA |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | ¬A ∧ ¬dA<br> |
| + | ¬A ∧ dA<br> |
| + | A ∧ ¬dA<br> |
| + | A ∧ dA |
| + | |} |
| |- | | |- |
− | | ''f<sub>4</sub> || 0100 || x (y) | + | | |
− | | || || || || 1 || 1 || 1 || 1 | + | {| |
− | | || || || || 1 || 1 || 1 || 1 | + | | |
| + | f<sub>1</sub><br> |
| + | f<sub>2</sub> |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | g<sub>3</sub><br> |
| + | g<sub>12</sub> |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | 0 0 1 1<br> |
| + | 1 1 0 0 |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | (A)<br> |
| + | A |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | Not A<br> |
| + | A |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | ¬A<br> |
| + | A |
| + | |} |
| |- | | |- |
− | | ''f<sub>5</sub> || 0101 || (y) | + | | |
− | | || || || || || 1 || || 1
| + | {| |
− | | || || || || || 1 || || 1 | + | | |
− | |- | + | <br> |
− | | ''f<sub>6</sub> || 0110 || (x, y)
| + | |
− | | || || || || || || 1 || 1
| + | |} |
− | | || || || || || || 1 || 1 | + | | |
| + | {| |
| + | | |
| + | g<sub>6</sub><br> |
| + | g<sub>9</sub> |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | 0 1 1 0<br> |
| + | 1 0 0 1 |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | (A, dA)<br> |
| + | ((A, dA)) |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | A not equal to dA<br> |
| + | A equal to dA |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | A ≠ dA<br> |
| + | A = dA |
| + | |} |
| |- | | |- |
− | | ''f<sub>7</sub> || 0111 || (x y) | + | | |
− | | || || || || || || || 1
| + | {| |
− | | || || || || || || || 1 | + | | |
| + | <br> |
| + | |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | g<sub>5</sub><br> |
| + | g<sub>10</sub> |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | 0 1 0 1<br> |
| + | 1 0 1 0 |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | (dA)<br> |
| + | dA |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | Not dA<br> |
| + | dA |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | ¬dA<br> |
| + | dA |
| + | |} |
| |- | | |- |
− | | ''f<sub>8</sub> || 1000 || x y | + | | |
− | | || || || || || || ||
| + | {| |
− | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
| + | | |
| + | <br> |
| + | <br> |
| + | <br> |
| + | |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | g<sub>7</sub><br> |
| + | g<sub>11</sub><br> |
| + | g<sub>13</sub><br> |
| + | g<sub>14</sub> |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | 0 1 1 1<br> |
| + | 1 0 1 1<br> |
| + | 1 1 0 1<br> |
| + | 1 1 1 0 |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | (A dA)<br> |
| + | (A (dA))<br> |
| + | ((A) dA)<br> |
| + | ((A)(dA)) |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | Not both A and dA<br> |
| + | Not A without dA<br> |
| + | Not dA without A<br> |
| + | A or dA |
| + | |} |
| + | | |
| + | {| |
| + | | |
| + | ¬A ∨ ¬dA<br> |
| + | A → dA<br> |
| + | A ← dA<br> |
| + | A ∨ dA |
| + | |} |
| |- | | |- |
− | | ''f<sub>9</sub> || 1001 || ((x, y)) | + | | f<sub>3</sub> |
− | | || || || || || || || | + | | g<sub>15</sub> |
− | | || 1 || || 1 || || 1 || || 1 | + | | 1 1 1 1 |
| + | | (( )) |
| + | | True |
| + | | 1 |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | ===Wiki TeX Tables : PQ=== |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
| + | |+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math> |
| + | |- style="background:#f0f0ff" |
| + | | width="15%" | |
| + | <p><math>\mathcal{L}_1</math></p> |
| + | <p><math>\text{Decimal}</math></p> |
| + | | width="15%" | |
| + | <p><math>\mathcal{L}_2</math></p> |
| + | <p><math>\text{Binary}</math></p> |
| + | | width="15%" | |
| + | <p><math>\mathcal{L}_3</math></p> |
| + | <p><math>\text{Vector}</math></p> |
| + | | width="15%" | |
| + | <p><math>\mathcal{L}_4</math></p> |
| + | <p><math>\text{Cactus}</math></p> |
| + | | width="25%" | |
| + | <p><math>\mathcal{L}_5</math></p> |
| + | <p><math>\text{English}</math></p> |
| + | | width="15%" | |
| + | <p><math>\mathcal{L}_6</math></p> |
| + | <p><math>\text{Ordinary}</math></p> |
| + | |- style="background:#f0f0ff" |
| + | | |
| + | | align="right" | <math>p\colon\!</math> |
| + | | <math>1~1~0~0\!</math> |
| + | | |
| + | | |
| + | | |
| + | |- style="background:#f0f0ff" |
| + | | |
| + | | align="right" | <math>q\colon\!</math> |
| + | | <math>1~0~1~0\!</math> |
| + | | |
| + | | |
| + | | |
| |- | | |- |
− | | ''f<sub>10</sub> || 1010 || y | + | | |
− | | || || || || || || ||
| + | <math>\begin{matrix} |
− | | || || 1 || 1 || || || 1 || 1
| + | f_0 |
− | |-
| + | \\[4pt] |
− | | ''f<sub>11</sub> || 1011 || (x (y))
| + | f_1 |
− | | || || || || || || || | + | \\[4pt] |
− | | || || || 1 || || || || 1
| + | f_2 |
| + | \\[4pt] |
| + | f_3 |
| + | \\[4pt] |
| + | f_4 |
| + | \\[4pt] |
| + | f_5 |
| + | \\[4pt] |
| + | f_6 |
| + | \\[4pt] |
| + | f_7 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | f_{0000} |
| + | \\[4pt] |
| + | f_{0001} |
| + | \\[4pt] |
| + | f_{0010} |
| + | \\[4pt] |
| + | f_{0011} |
| + | \\[4pt] |
| + | f_{0100} |
| + | \\[4pt] |
| + | f_{0101} |
| + | \\[4pt] |
| + | f_{0110} |
| + | \\[4pt] |
| + | f_{0111} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | 0~0~0~0 |
| + | \\[4pt] |
| + | 0~0~0~1 |
| + | \\[4pt] |
| + | 0~0~1~0 |
| + | \\[4pt] |
| + | 0~0~1~1 |
| + | \\[4pt] |
| + | 0~1~0~0 |
| + | \\[4pt] |
| + | 0~1~0~1 |
| + | \\[4pt] |
| + | 0~1~1~0 |
| + | \\[4pt] |
| + | 0~1~1~1 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~) |
| + | \\[4pt] |
| + | (p)(q) |
| + | \\[4pt] |
| + | (p)~q~ |
| + | \\[4pt] |
| + | (p)~~~ |
| + | \\[4pt] |
| + | ~p~(q) |
| + | \\[4pt] |
| + | ~~~(q) |
| + | \\[4pt] |
| + | (p,~q) |
| + | \\[4pt] |
| + | (p~~q) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \text{false} |
| + | \\[4pt] |
| + | \text{neither}~ p ~\text{nor}~ q |
| + | \\[4pt] |
| + | q ~\text{without}~ p |
| + | \\[4pt] |
| + | \text{not}~ p |
| + | \\[4pt] |
| + | p ~\text{without}~ q |
| + | \\[4pt] |
| + | \text{not}~ q |
| + | \\[4pt] |
| + | p ~\text{not equal to}~ q |
| + | \\[4pt] |
| + | \text{not both}~ p ~\text{and}~ q |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | 0 |
| + | \\[4pt] |
| + | \lnot p \land \lnot q |
| + | \\[4pt] |
| + | \lnot p \land q |
| + | \\[4pt] |
| + | \lnot p |
| + | \\[4pt] |
| + | p \land \lnot q |
| + | \\[4pt] |
| + | \lnot q |
| + | \\[4pt] |
| + | p \ne q |
| + | \\[4pt] |
| + | \lnot p \lor \lnot q |
| + | \end{matrix}</math> |
| |- | | |- |
− | | ''f<sub>12</sub> || 1100 || x | + | | |
− | | || || || || || || ||
| + | <math>\begin{matrix} |
− | | || || || || 1 || 1 || 1 || 1
| + | f_8 |
− | |-
| + | \\[4pt] |
− | | ''f<sub>13</sub> || 1101 || ((x) y)
| + | f_9 |
− | | || || || || || || ||
| + | \\[4pt] |
− | | || || || || || 1 || || 1
| + | f_{10} |
− | |-
| + | \\[4pt] |
− | | ''f<sub>14</sub> || 1110 || ((x)(y))
| + | f_{11} |
− | | || || || || || || ||
| + | \\[4pt] |
− | | || || || || || || 1 || 1
| + | f_{12} |
− | |- | + | \\[4pt] |
− | | ''f<sub>15</sub> || 1111 || (( ))
| + | f_{13} |
− | | || || || || || || ||
| + | \\[4pt] |
− | | || || || || || || || 1
| + | f_{14} |
| + | \\[4pt] |
| + | f_{15} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | f_{1000} |
| + | \\[4pt] |
| + | f_{1001} |
| + | \\[4pt] |
| + | f_{1010} |
| + | \\[4pt] |
| + | f_{1011} |
| + | \\[4pt] |
| + | f_{1100} |
| + | \\[4pt] |
| + | f_{1101} |
| + | \\[4pt] |
| + | f_{1110} |
| + | \\[4pt] |
| + | f_{1111} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | 1~0~0~0 |
| + | \\[4pt] |
| + | 1~0~0~1 |
| + | \\[4pt] |
| + | 1~0~1~0 |
| + | \\[4pt] |
| + | 1~0~1~1 |
| + | \\[4pt] |
| + | 1~1~0~0 |
| + | \\[4pt] |
| + | 1~1~0~1 |
| + | \\[4pt] |
| + | 1~1~1~0 |
| + | \\[4pt] |
| + | 1~1~1~1 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~~p~~q~~ |
| + | \\[4pt] |
| + | ((p,~q)) |
| + | \\[4pt] |
| + | ~~~~~q~~ |
| + | \\[4pt] |
| + | ~(p~(q)) |
| + | \\[4pt] |
| + | ~~p~~~~~ |
| + | \\[4pt] |
| + | ((p)~q)~ |
| + | \\[4pt] |
| + | ((p)(q)) |
| + | \\[4pt] |
| + | ((~)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | p ~\text{and}~ q |
| + | \\[4pt] |
| + | p ~\text{equal to}~ q |
| + | \\[4pt] |
| + | q |
| + | \\[4pt] |
| + | \text{not}~ p ~\text{without}~ q |
| + | \\[4pt] |
| + | p |
| + | \\[4pt] |
| + | \text{not}~ q ~\text{without}~ p |
| + | \\[4pt] |
| + | p ~\text{or}~ q |
| + | \\[4pt] |
| + | \text{true} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | p \land q |
| + | \\[4pt] |
| + | p = q |
| + | \\[4pt] |
| + | q |
| + | \\[4pt] |
| + | p \Rightarrow q |
| + | \\[4pt] |
| + | p |
| + | \\[4pt] |
| + | p \Leftarrow q |
| + | \\[4pt] |
| + | p \lor q |
| + | \\[4pt] |
| + | 1 |
| + | \end{matrix}</math> |
| |} | | |} |
| + | |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" | + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
− | |+ '''Table 13. Syllogistic Premisses as Higher Order Indicator Functions''' | + | |+ <math>\text{Table A2.}~~\text{Propositional Forms on Two Variables}</math> |
− | | A | + | |- style="background:#f0f0ff" |
− | | align=left | Universal Affirmative | + | | width="15%" | |
− | | align=left | All | + | <p><math>\mathcal{L}_1</math></p> |
− | | x || is || y | + | <p><math>\text{Decimal}</math></p> |
− | | align=left | Indicator of " x (y)" = 0 | + | | width="15%" | |
| + | <p><math>\mathcal{L}_2</math></p> |
| + | <p><math>\text{Binary}</math></p> |
| + | | width="15%" | |
| + | <p><math>\mathcal{L}_3</math></p> |
| + | <p><math>\text{Vector}</math></p> |
| + | | width="15%" | |
| + | <p><math>\mathcal{L}_4</math></p> |
| + | <p><math>\text{Cactus}</math></p> |
| + | | width="25%" | |
| + | <p><math>\mathcal{L}_5</math></p> |
| + | <p><math>\text{English}</math></p> |
| + | | width="15%" | |
| + | <p><math>\mathcal{L}_6</math></p> |
| + | <p><math>\text{Ordinary}</math></p> |
| + | |- style="background:#f0f0ff" |
| + | | |
| + | | align="right" | <math>p\colon\!</math> |
| + | | <math>1~1~0~0\!</math> |
| + | | |
| + | | |
| + | | |
| + | |- style="background:#f0f0ff" |
| + | | |
| + | | align="right" | <math>q\colon\!</math> |
| + | | <math>1~0~1~0\!</math> |
| + | | |
| + | | |
| + | | |
| |- | | |- |
− | | E | + | | <math>f_0\!</math> |
− | | align=left | Universal Negative | + | | <math>f_{0000}\!</math> |
− | | align=left | All | + | | <math>0~0~0~0</math> |
− | | x || is || (y) | + | | <math>(~)</math> |
− | | align=left | Indicator of " x y " = 0 | + | | <math>\text{false}\!</math> |
| + | | <math>0\!</math> |
| |- | | |- |
− | | I | + | | |
− | | align=left | Particular Affirmative | + | <math>\begin{matrix} |
− | | align=left | Some | + | f_1 |
− | | x || is || y | + | \\[4pt] |
− | | align=left | Indicator of " x y " = 1
| + | f_2 |
| + | \\[4pt] |
| + | f_4 |
| + | \\[4pt] |
| + | f_8 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | f_{0001} |
| + | \\[4pt] |
| + | f_{0010} |
| + | \\[4pt] |
| + | f_{0100} |
| + | \\[4pt] |
| + | f_{1000} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | 0~0~0~1 |
| + | \\[4pt] |
| + | 0~0~1~0 |
| + | \\[4pt] |
| + | 0~1~0~0 |
| + | \\[4pt] |
| + | 1~0~0~0 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (p)(q) |
| + | \\[4pt] |
| + | (p)~q~ |
| + | \\[4pt] |
| + | ~p~(q) |
| + | \\[4pt] |
| + | ~p~~q~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \text{neither}~ p ~\text{nor}~ q |
| + | \\[4pt] |
| + | q ~\text{without}~ p |
| + | \\[4pt] |
| + | p ~\text{without}~ q |
| + | \\[4pt] |
| + | p ~\text{and}~ q |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \lnot p \land \lnot q |
| + | \\[4pt] |
| + | \lnot p \land q |
| + | \\[4pt] |
| + | p \land \lnot q |
| + | \\[4pt] |
| + | p \land q |
| + | \end{matrix}</math> |
| |- | | |- |
− | | O | + | | |
− | | align=left | Particular Negative | + | <math>\begin{matrix} |
− | | align=left | Some | + | f_3 |
− | | x || is || (y)
| + | \\[4pt] |
− | | align=left | Indicator of " x (y)" = 1 | + | f_{12} |
− | |}
| + | \end{matrix}</math> |
− | <br> | + | | |
− | | + | <math>\begin{matrix} |
− | {| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" | + | f_{0011} |
− | |+ '''Table 14. Relation of Quantifiers to Higher Order Propositions'''
| + | \\[4pt] |
− | |- style="background:paleturquoise"
| + | f_{1100} |
− | |Mnemonic||Category||Classical Form||Alternate Form||Symmetric Form||Operator
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | 0~0~1~1 |
| + | \\[4pt] |
| + | 1~1~0~0 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (p) |
| + | \\[4pt] |
| + | ~p~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \text{not}~ p |
| + | \\[4pt] |
| + | p |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \lnot p |
| + | \\[4pt] |
| + | p |
| + | \end{matrix}</math> |
| |- | | |- |
− | | E<br>Exclusive | + | | |
− | | Universal<br>Negative | + | <math>\begin{matrix} |
− | | align=left | All x is (y) | + | f_6 |
− | | align=left | | + | \\[4pt] |
− | | align=left | No x is y | + | f_9 |
− | | (''L''<sub>11</sub>)
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | f_{0110} |
| + | \\[4pt] |
| + | f_{1001} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | 0~1~1~0 |
| + | \\[4pt] |
| + | 1~0~0~1 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~(p,~q)~ |
| + | \\[4pt] |
| + | ((p,~q)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | p ~\text{not equal to}~ q |
| + | \\[4pt] |
| + | p ~\text{equal to}~ q |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | p \ne q |
| + | \\[4pt] |
| + | p = q |
| + | \end{matrix}</math> |
| |- | | |- |
− | | A<br>Absolute | + | | |
− | | Universal<br>Affirmative | + | <math>\begin{matrix} |
− | | align=left | All x is y | + | f_5 |
− | | align=left |
| + | \\[4pt] |
− | | align=left | No x is (y)
| + | f_{10} |
− | | (''L''<sub>10</sub>) | + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | f_{0101} |
| + | \\[4pt] |
| + | f_{1010} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | 0~1~0~1 |
| + | \\[4pt] |
| + | 1~0~1~0 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (q) |
| + | \\[4pt] |
| + | ~q~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \text{not}~ q |
| + | \\[4pt] |
| + | q |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \lnot q |
| + | \\[4pt] |
| + | q |
| + | \end{matrix}</math> |
| |- | | |- |
− | | | + | | |
− | | | + | <math>\begin{matrix} |
− | | align=left | All y is x | + | f_7 |
− | | align=left | No y is (x)
| + | \\[4pt] |
− | | align=left | No (x) is y
| + | f_{11} |
− | | (''L''<sub>01</sub>) | + | \\[4pt] |
| + | f_{13} |
| + | \\[4pt] |
| + | f_{14} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | f_{0111} |
| + | \\[4pt] |
| + | f_{1011} |
| + | \\[4pt] |
| + | f_{1101} |
| + | \\[4pt] |
| + | f_{1110} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | 0~1~1~1 |
| + | \\[4pt] |
| + | 1~0~1~1 |
| + | \\[4pt] |
| + | 1~1~0~1 |
| + | \\[4pt] |
| + | 1~1~1~0 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~(p~~q)~ |
| + | \\[4pt] |
| + | ~(p~(q)) |
| + | \\[4pt] |
| + | ((p)~q)~ |
| + | \\[4pt] |
| + | ((p)(q)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \text{not both}~ p ~\text{and}~ q |
| + | \\[4pt] |
| + | \text{not}~ p ~\text{without}~ q |
| + | \\[4pt] |
| + | \text{not}~ q ~\text{without}~ p |
| + | \\[4pt] |
| + | p ~\text{or}~ q |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \lnot p \lor \lnot q |
| + | \\[4pt] |
| + | p \Rightarrow q |
| + | \\[4pt] |
| + | p \Leftarrow q |
| + | \\[4pt] |
| + | p \lor q |
| + | \end{matrix}</math> |
| |- | | |- |
− | | | + | | <math>f_{15}\!</math> |
− | | | + | | <math>f_{1111}\!</math> |
− | | align=left | All (y) is x | + | | <math>1~1~1~1</math> |
− | | align=left | No (y) is (x) | + | | <math>((~))</math> |
− | | align=left | No (x) is (y) | + | | <math>\text{true}\!</math> |
− | | (''L''<sub>00</sub>)
| + | | <math>1\!</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
| + | |+ <math>\text{Table A3.}~~\operatorname{E}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}p, \operatorname{d}q \}</math> |
| + | |- style="background:#f0f0ff" |
| + | | width="10%" | |
| + | | width="18%" | <math>f\!</math> |
| + | | width="18%" | |
| + | <p><math>\operatorname{T}_{11} f</math></p> |
| + | <p><math>\operatorname{E}f|_{\operatorname{d}p~\operatorname{d}q}</math></p> |
| + | | width="18%" | |
| + | <p><math>\operatorname{T}_{10} f</math></p> |
| + | <p><math>\operatorname{E}f|_{\operatorname{d}p(\operatorname{d}q)}</math></p> |
| + | | width="18%" | |
| + | <p><math>\operatorname{T}_{01} f</math></p> |
| + | <p><math>\operatorname{E}f|_{(\operatorname{d}p)\operatorname{d}q}</math></p> |
| + | | width="18%" | |
| + | <p><math>\operatorname{T}_{00} f</math></p> |
| + | <p><math>\operatorname{E}f|_{(\operatorname{d}p)(\operatorname{d}q)}</math></p> |
| |- | | |- |
− | | | + | | <math>f_0\!</math> |
− | | | + | | <math>(~)</math> |
− | | align=left | Some (x) is (y) | + | | <math>(~)</math> |
− | | align=left | | + | | <math>(~)</math> |
− | | align=left | Some (x) is (y)
| + | | <math>(~)</math> |
− | | ''L''<sub>00</sub> | + | | <math>(~)</math> |
| |- | | |- |
− | | | + | | |
− | | | + | <math>\begin{matrix} |
− | | align=left | Some (x) is y | + | f_1 |
− | | align=left | | + | \\[4pt] |
− | | align=left | Some (x) is y
| + | f_2 |
− | | ''L''<sub>01</sub> | + | \\[4pt] |
| + | f_4 |
| + | \\[4pt] |
| + | f_8 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (p)(q) |
| + | \\[4pt] |
| + | (p)~q~ |
| + | \\[4pt] |
| + | ~p~(q) |
| + | \\[4pt] |
| + | ~p~~q~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~p~~q~ |
| + | \\[4pt] |
| + | ~p~(q) |
| + | \\[4pt] |
| + | (p)~q~ |
| + | \\[4pt] |
| + | (p)(q) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~p~(q) |
| + | \\[4pt] |
| + | ~p~~q~ |
| + | \\[4pt] |
| + | (p)(q) |
| + | \\[4pt] |
| + | (p)~q~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (p)~q~ |
| + | \\[4pt] |
| + | (p)(q) |
| + | \\[4pt] |
| + | ~p~~q~ |
| + | \\[4pt] |
| + | ~p~(q) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (p)(q) |
| + | \\[4pt] |
| + | (p)~q~ |
| + | \\[4pt] |
| + | ~p~(q) |
| + | \\[4pt] |
| + | ~p~~q~ |
| + | \end{matrix}</math> |
| |- | | |- |
− | | O<br>Obtrusive | + | | |
− | | Particular<br>Negative
| + | <math>\begin{matrix} |
− | | align=left | Some x is (y)
| + | f_3 |
− | | align=left |
| + | \\[4pt] |
− | | align=left | Some x is (y)
| + | f_{12} |
− | | ''L''<sub>10</sub>
| + | \end{matrix}</math> |
− | |- | + | | |
− | | I<br>Indefinite
| + | <math>\begin{matrix} |
− | | Particular<br>Affirmative
| + | (p) |
− | | align=left | Some x is y
| + | \\[4pt] |
− | | align=left |
| + | ~p~ |
− | | align=left | Some x is y
| + | \end{matrix}</math> |
− | | ''L''<sub>11</sub>
| + | | |
− | |} | + | <math>\begin{matrix} |
− | <br> | + | ~p~ |
− | | + | \\[4pt] |
− | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
| + | (p) |
− | |+ '''Table 15. Simple Qualifiers of Propositions (n = 2)'''
| + | \end{matrix}</math> |
− | |- style="background:paleturquoise"
| + | | |
− | | align=right | ''x'' : || 1100 || ''f''
| + | <math>\begin{matrix} |
− | | (''L''<sub>11</sub>)
| + | ~p~ |
− | | (''L''<sub>10</sub>) | + | \\[4pt] |
− | | (''L''<sub>01</sub>)
| + | (p) |
− | | (''L''<sub>00</sub>)
| + | \end{matrix}</math> |
− | | ''L''<sub>00</sub>
| + | | |
− | | ''L''<sub>01</sub> | + | <math>\begin{matrix} |
− | | ''L''<sub>10</sub>
| + | (p) |
− | | ''L''<sub>11</sub>
| + | \\[4pt] |
− | |- style="background:paleturquoise"
| + | ~p~ |
− | | align=right | ''y'' : || 1010 ||
| + | \end{matrix}</math> |
− | | align=left | no x <br> is y
| + | | |
− | | align=left | no x <br> is (y) | + | <math>\begin{matrix} |
− | | align=left | no (x) <br> is y
| + | (p) |
− | | align=left | no (x) <br> is (y)
| + | \\[4pt] |
− | | align=left | some (x) <br> is (y)
| + | ~p~ |
− | | align=left | some (x) <br> is y
| + | \end{matrix}</math> |
− | | align=left | some x <br> is (y)
| |
− | | align=left | some x <br> is y
| |
| |- | | |- |
− | | ''f<sub>0</sub> || 0000 || ( ) | + | | |
− | | 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0
| + | <math>\begin{matrix} |
− | |- | + | f_6 |
− | | ''f<sub>1</sub> || 0001 || (x)(y)
| + | \\[4pt] |
− | | 1 || 1 || 1 || 0 || 1 || 0 || 0 || 0
| + | f_9 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~(p,~q)~ |
| + | \\[4pt] |
| + | ((p,~q)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~(p,~q)~ |
| + | \\[4pt] |
| + | ((p,~q)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((p,~q)) |
| + | \\[4pt] |
| + | ~(p,~q)~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((p,~q)) |
| + | \\[4pt] |
| + | ~(p,~q)~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~(p,~q)~ |
| + | \\[4pt] |
| + | ((p,~q)) |
| + | \end{matrix}</math> |
| |- | | |- |
− | | ''f<sub>2</sub> || 0010 || (x) y | + | | |
− | | 1 || 1 || 0 || 1 || 0 || 1 || 0 || 0 | + | <math>\begin{matrix} |
| + | f_5 |
| + | \\[4pt] |
| + | f_{10} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (q) |
| + | \\[4pt] |
| + | ~q~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~q~ |
| + | \\[4pt] |
| + | (q) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (q) |
| + | \\[4pt] |
| + | ~q~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~q~ |
| + | \\[4pt] |
| + | (q) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (q) |
| + | \\[4pt] |
| + | ~q~ |
| + | \end{matrix}</math> |
| |- | | |- |
− | | ''f<sub>3</sub> || 0011 || (x) | + | | |
− | | 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0 | + | <math>\begin{matrix} |
| + | f_7 |
| + | \\[4pt] |
| + | f_{11} |
| + | \\[4pt] |
| + | f_{13} |
| + | \\[4pt] |
| + | f_{14} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~p~~q~) |
| + | \\[4pt] |
| + | (~p~(q)) |
| + | \\[4pt] |
| + | ((p)~q~) |
| + | \\[4pt] |
| + | ((p)(q)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((p)(q)) |
| + | \\[4pt] |
| + | ((p)~q~) |
| + | \\[4pt] |
| + | (~p~(q)) |
| + | \\[4pt] |
| + | (~p~~q~) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((p)~q~) |
| + | \\[4pt] |
| + | ((p)(q)) |
| + | \\[4pt] |
| + | (~p~~q~) |
| + | \\[4pt] |
| + | (~p~(q)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~p~(q)) |
| + | \\[4pt] |
| + | (~p~~q~) |
| + | \\[4pt] |
| + | ((p)(q)) |
| + | \\[4pt] |
| + | ((p)~q~) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~p~~q~) |
| + | \\[4pt] |
| + | (~p~(q)) |
| + | \\[4pt] |
| + | ((p)~q~) |
| + | \\[4pt] |
| + | ((p)(q)) |
| + | \end{matrix}</math> |
| |- | | |- |
− | | ''f<sub>4</sub> || 0100 || x (y) | + | | <math>f_{15}\!</math> |
− | | 1 || 0 || 1 || 1 || 0 || 0 || 1 || 0 | + | | <math>((~))</math> |
| + | | <math>((~))</math> |
| + | | <math>((~))</math> |
| + | | <math>((~))</math> |
| + | | <math>((~))</math> |
| + | |- style="background:#f0f0ff" |
| + | | colspan="2" | <math>\text{Fixed Point Total}\!</math> |
| + | | <math>4\!</math> |
| + | | <math>4\!</math> |
| + | | <math>4\!</math> |
| + | | <math>16\!</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
| + | |+ <math>\text{Table A4.}~~\operatorname{D}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}p, \operatorname{d}q \}</math> |
| + | |- style="background:#f0f0ff" |
| + | | width="10%" | |
| + | | width="18%" | <math>f\!</math> |
| + | | width="18%" | |
| + | <math>\operatorname{D}f|_{\operatorname{d}p~\operatorname{d}q}</math> |
| + | | width="18%" | |
| + | <math>\operatorname{D}f|_{\operatorname{d}p(\operatorname{d}q)}</math> |
| + | | width="18%" | |
| + | <math>\operatorname{D}f|_{(\operatorname{d}p)\operatorname{d}q}</math> |
| + | | width="18%" | |
| + | <math>\operatorname{D}f|_{(\operatorname{d}p)(\operatorname{d}q)}</math> |
| |- | | |- |
− | | ''f<sub>5</sub> || 0101 || (y) | + | | <math>f_0\!</math> |
− | | 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 | + | | <math>(~)</math> |
| + | | <math>(~)</math> |
| + | | <math>(~)</math> |
| + | | <math>(~)</math> |
| + | | <math>(~)</math> |
| |- | | |- |
− | | ''f<sub>6</sub> || 0110 || (x, y) | + | | |
− | | 1 || 0 || 0 || 1 || 0 || 1 || 1 || 0 | + | <math>\begin{matrix} |
| + | f_1 |
| + | \\[4pt] |
| + | f_2 |
| + | \\[4pt] |
| + | f_4 |
| + | \\[4pt] |
| + | f_8 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (p)(q) |
| + | \\[4pt] |
| + | (p)~q~ |
| + | \\[4pt] |
| + | ~p~(q) |
| + | \\[4pt] |
| + | ~p~~q~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((p,~q)) |
| + | \\[4pt] |
| + | ~(p,~q)~ |
| + | \\[4pt] |
| + | ~(p,~q)~ |
| + | \\[4pt] |
| + | ((p,~q)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (q) |
| + | \\[4pt] |
| + | ~q~ |
| + | \\[4pt] |
| + | (q) |
| + | \\[4pt] |
| + | ~q~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (p) |
| + | \\[4pt] |
| + | (p) |
| + | \\[4pt] |
| + | ~p~ |
| + | \\[4pt] |
| + | ~p~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~) |
| + | \\[4pt] |
| + | (~) |
| + | \\[4pt] |
| + | (~) |
| + | \\[4pt] |
| + | (~) |
| + | \end{matrix}</math> |
| |- | | |- |
− | | ''f<sub>7</sub> || 0111 || (x y) | + | | |
− | | 1 || 0 || 0 || 0 || 1 || 1 || 1 || 0 | + | <math>\begin{matrix} |
| + | f_3 |
| + | \\[4pt] |
| + | f_{12} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (p) |
| + | \\[4pt] |
| + | ~p~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((~)) |
| + | \\[4pt] |
| + | ((~)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((~)) |
| + | \\[4pt] |
| + | ((~)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~) |
| + | \\[4pt] |
| + | (~) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~) |
| + | \\[4pt] |
| + | (~) |
| + | \end{matrix}</math> |
| |- | | |- |
− | | ''f<sub>8</sub> || 1000 || x y | + | | |
− | | 0 || 1 || 1 || 1 || 0 || 0 || 0 || 1
| + | <math>\begin{matrix} |
− | |- | + | f_6 |
− | | ''f<sub>9</sub> || 1001 || ((x, y))
| + | \\[4pt] |
− | | 0 || 1 || 1 || 0 || 1 || 0 || 0 || 1 | + | f_9 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~(p,~q)~ |
| + | \\[4pt] |
| + | ((p,~q)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~) |
| + | \\[4pt] |
| + | (~) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((~)) |
| + | \\[4pt] |
| + | ((~)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((~)) |
| + | \\[4pt] |
| + | ((~)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~) |
| + | \\[4pt] |
| + | (~) |
| + | \end{matrix}</math> |
| |- | | |- |
− | | ''f<sub>10</sub> || 1010 || y | + | | |
− | | 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 | + | <math>\begin{matrix} |
| + | f_5 |
| + | \\[4pt] |
| + | f_{10} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (q) |
| + | \\[4pt] |
| + | ~q~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((~)) |
| + | \\[4pt] |
| + | ((~)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~) |
| + | \\[4pt] |
| + | (~) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((~)) |
| + | \\[4pt] |
| + | ((~)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~) |
| + | \\[4pt] |
| + | (~) |
| + | \end{matrix}</math> |
| |- | | |- |
− | | ''f<sub>11</sub> || 1011 || (x (y)) | + | | |
− | | 0 || 1 || 0 || 0 || 1 || 1 || 0 || 1 | + | <math>\begin{matrix} |
| + | f_7 |
| + | \\[4pt] |
| + | f_{11} |
| + | \\[4pt] |
| + | f_{13} |
| + | \\[4pt] |
| + | f_{14} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~(p~~q)~ |
| + | \\[4pt] |
| + | ~(p~(q)) |
| + | \\[4pt] |
| + | ((p)~q)~ |
| + | \\[4pt] |
| + | ((p)(q)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((p,~q)) |
| + | \\[4pt] |
| + | ~(p,~q)~ |
| + | \\[4pt] |
| + | ~(p,~q)~ |
| + | \\[4pt] |
| + | ((p,~q)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~q~ |
| + | \\[4pt] |
| + | (q) |
| + | \\[4pt] |
| + | ~q~ |
| + | \\[4pt] |
| + | (q) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~p~ |
| + | \\[4pt] |
| + | ~p~ |
| + | \\[4pt] |
| + | (p) |
| + | \\[4pt] |
| + | (p) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~) |
| + | \\[4pt] |
| + | (~) |
| + | \\[4pt] |
| + | (~) |
| + | \\[4pt] |
| + | (~) |
| + | \end{matrix}</math> |
| |- | | |- |
− | | ''f<sub>12</sub> || 1100 || x | + | | <math>f_{15}\!</math> |
− | | 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 | + | | <math>((~))</math> |
| + | | <math>(~)</math> |
| + | | <math>(~)</math> |
| + | | <math>(~)</math> |
| + | | <math>(~)</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
| + | |+ <math>\text{Table A5.}~~\operatorname{E}f ~\text{Expanded Over Ordinary Features}~ \{ p, q \}</math> |
| + | |- style="background:#f0f0ff" |
| + | | width="10%" | |
| + | | width="18%" | <math>f\!</math> |
| + | | width="18%" | <math>\operatorname{E}f|_{xy}</math> |
| + | | width="18%" | <math>\operatorname{E}f|_{p(q)}</math> |
| + | | width="18%" | <math>\operatorname{E}f|_{(p)q}</math> |
| + | | width="18%" | <math>\operatorname{E}f|_{(p)(q)}</math> |
| |- | | |- |
− | | ''f<sub>13</sub> || 1101 || ((x) y) | + | | <math>f_0\!</math> |
− | | 0 || 0 || 1 || 0 || 1 || 0 || 1 || 1 | + | | <math>(~)</math> |
| + | | <math>(~)</math> |
| + | | <math>(~)</math> |
| + | | <math>(~)</math> |
| + | | <math>(~)</math> |
| |- | | |- |
− | | ''f<sub>14</sub> || 1110 || ((x)(y)) | + | | |
− | | 0 || 0 || 0 || 1 || 0 || 1 || 1 || 1 | + | <math>\begin{matrix} |
| + | f_1 |
| + | \\[4pt] |
| + | f_2 |
| + | \\[4pt] |
| + | f_4 |
| + | \\[4pt] |
| + | f_8 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (p)(q) |
| + | \\[4pt] |
| + | (p)~q~ |
| + | \\[4pt] |
| + | ~p~(q) |
| + | \\[4pt] |
| + | ~p~~q~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~\operatorname{d}p~~\operatorname{d}q~ |
| + | \\[4pt] |
| + | ~\operatorname{d}p~(\operatorname{d}q) |
| + | \\[4pt] |
| + | (\operatorname{d}p)~\operatorname{d}q~ |
| + | \\[4pt] |
| + | (\operatorname{d}p)(\operatorname{d}q) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~\operatorname{d}p~(\operatorname{d}q) |
| + | \\[4pt] |
| + | ~\operatorname{d}p~~\operatorname{d}q~ |
| + | \\[4pt] |
| + | (\operatorname{d}p)(\operatorname{d}q) |
| + | \\[4pt] |
| + | (\operatorname{d}p)~\operatorname{d}q~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (\operatorname{d}p)~\operatorname{d}q~ |
| + | \\[4pt] |
| + | (\operatorname{d}p)(\operatorname{d}q) |
| + | \\[4pt] |
| + | ~\operatorname{d}p~~\operatorname{d}q~ |
| + | \\[4pt] |
| + | ~\operatorname{d}p~(\operatorname{d}q) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (\operatorname{d}p)(\operatorname{d}q) |
| + | \\[4pt] |
| + | (\operatorname{d}p)~\operatorname{d}q~ |
| + | \\[4pt] |
| + | ~\operatorname{d}p~(\operatorname{d}q) |
| + | \\[4pt] |
| + | ~\operatorname{d}p~~\operatorname{d}q~ |
| + | \end{matrix}</math> |
| |- | | |- |
− | | ''f<sub>15</sub> || 1111 || (( )) | + | | |
− | | 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 | + | <math>\begin{matrix} |
− | |} | + | f_3 |
− | <br> | + | \\[4pt] |
− | | + | f_{12} |
− | Table 7. Higher Order Propositions (n = 1)
| + | \end{matrix}</math> |
− | 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 |
| + | <math>\begin{matrix} |
− | | F \ | | |00|01|02|03|04|05|06|07|08|09|10|11|12|13|14|15 |
| + | (p) |
− | o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o
| + | \\[4pt] |
− | | | | | |
| + | ~p~ |
− | | F_0 | 0 0 | 0 | 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 |
| + | \end{matrix}</math> |
− | | | | | |
| + | | |
− | | F_1 | 0 1 | (x) | 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 |
| + | <math>\begin{matrix} |
− | | | | | |
| + | ~\operatorname{d}p~ |
− | | F_2 | 1 0 | x | 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 |
| + | \\[4pt] |
− | | | | | |
| + | (\operatorname{d}p) |
− | | F_3 | 1 1 | 1 | 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 |
| + | \end{matrix}</math> |
− | | | | | |
| + | | |
− | o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o
| + | <math>\begin{matrix} |
− | <br> | + | ~\operatorname{d}p~ |
− | | + | \\[4pt] |
− | Table 8. Interpretive Categories for Higher Order Propositions (n = 1)
| + | (\operatorname{d}p) |
− | o-------o----------o------------o------------o----------o----------o-----------o
| + | \end{matrix}</math> |
− | |Measure| Happening| Exactness | Existence | Linearity|Uniformity|Information|
| + | | |
− | o-------o----------o------------o------------o----------o----------o-----------o
| + | <math>\begin{matrix} |
− | | m_0 | nothing | | | | | |
| + | (\operatorname{d}p) |
− | | | happens | | | | | |
| + | \\[4pt] |
− | o-------o----------o------------o------------o----------o----------o-----------o
| + | ~\operatorname{d}p~ |
− | | m_1 | | | nothing | | | |
| + | \end{matrix}</math> |
− | | | | just false | exists | | | |
| + | | |
− | o-------o----------o------------o------------o----------o----------o-----------o
| + | <math>\begin{matrix} |
− | | m_2 | | | | | | |
| + | (\operatorname{d}p) |
− | | | | just not x | | | | |
| + | \\[4pt] |
− | o-------o----------o------------o------------o----------o----------o-----------o
| + | ~\operatorname{d}p~ |
− | | m_3 | | | nothing | | | |
| + | \end{matrix}</math> |
− | | | | | is x | | | |
| + | |- |
− | o-------o----------o------------o------------o----------o----------o-----------o
| + | | |
− | | m_4 | | | | | | |
| + | <math>\begin{matrix} |
− | | | | just x | | | | |
| + | f_6 |
− | o-------o----------o------------o------------o----------o----------o-----------o
| + | \\[4pt] |
− | | m_5 | | | everything | F is | | |
| + | f_9 |
− | | | | | is x | linear | | |
| + | \end{matrix}</math> |
− | o-------o----------o------------o------------o----------o----------o-----------o
| + | | |
− | | m_6 | | | | | F is not | F is |
| + | <math>\begin{matrix} |
− | | | | | | | uniform | informed |
| + | ~(p,~q)~ |
− | o-------o----------o------------o------------o----------o----------o-----------o
| + | \\[4pt] |
− | | m_7 | | not | | | | |
| + | ((p,~q)) |
− | | | | just true | | | | |
| + | \end{matrix}</math> |
− | o-------o----------o------------o------------o----------o----------o-----------o
| + | | |
− | | m_8 | | | | | | |
| + | <math>\begin{matrix} |
− | | | | just true | | | | |
| + | ~(\operatorname{d}p,~\operatorname{d}q)~ |
− | o-------o----------o------------o------------o----------o----------o-----------o
| + | \\[4pt] |
− | | m_9 | | | | | F is | F is not |
| + | ((\operatorname{d}p,~\operatorname{d}q)) |
− | | | | | | | uniform | informed |
| + | \end{matrix}</math> |
− | o-------o----------o------------o------------o----------o----------o-----------o
| + | | |
− | | m_10 | | | something | F is not | | |
| + | <math>\begin{matrix} |
− | | | | | is not x | linear | | |
| + | ((\operatorname{d}p,~\operatorname{d}q)) |
− | o-------o----------o------------o------------o----------o----------o-----------o
| + | \\[4pt] |
− | | m_11 | | not | | | | |
| + | ~(\operatorname{d}p,~\operatorname{d}q)~ |
− | | | | just x | | | | |
| + | \end{matrix}</math> |
− | o-------o----------o------------o------------o----------o----------o-----------o
| + | | |
− | | m_12 | | | something | | | |
| + | <math>\begin{matrix} |
− | | | | | is x | | | |
| + | ((\operatorname{d}p,~\operatorname{d}q)) |
− | o-------o----------o------------o------------o----------o----------o-----------o
| + | \\[4pt] |
− | | m_13 | | not | | | | |
| + | ~(\operatorname{d}p,~\operatorname{d}q)~ |
− | | | | just not x | | | | |
| + | \end{matrix}</math> |
− | o-------o----------o------------o------------o----------o----------o-----------o
| + | | |
− | | m_14 | | not | something | | | |
| + | <math>\begin{matrix} |
− | | | | just false | exists | | | |
| + | ~(\operatorname{d}p,~\operatorname{d}q)~ |
− | o-------o----------o------------o------------o----------o----------o-----------o
| + | \\[4pt] |
− | | m_15 | anything | | | | | |
| + | ((\operatorname{d}p,~\operatorname{d}q)) |
− | | | happens | | | | | |
| + | \end{matrix}</math> |
− | o-------o----------o------------o------------o----------o----------o-----------o
| + | |- |
| + | | |
| + | <math>\begin{matrix} |
| + | f_5 |
| + | \\[4pt] |
| + | f_{10} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (q) |
| + | \\[4pt] |
| + | ~q~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~\operatorname{d}q~ |
| + | \\[4pt] |
| + | (\operatorname{d}q) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (\operatorname{d}q) |
| + | \\[4pt] |
| + | ~\operatorname{d}q~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~\operatorname{d}q~ |
| + | \\[4pt] |
| + | (\operatorname{d}q) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (\operatorname{d}q) |
| + | \\[4pt] |
| + | ~\operatorname{d}q~ |
| + | \end{matrix}</math> |
| + | |- |
| + | | |
| + | <math>\begin{matrix} |
| + | f_7 |
| + | \\[4pt] |
| + | f_{11} |
| + | \\[4pt] |
| + | f_{13} |
| + | \\[4pt] |
| + | f_{14} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~p~~q~) |
| + | \\[4pt] |
| + | (~p~(q)) |
| + | \\[4pt] |
| + | ((p)~q~) |
| + | \\[4pt] |
| + | ((p)(q)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((\operatorname{d}p)(\operatorname{d}q)) |
| + | \\[4pt] |
| + | ((\operatorname{d}p)~\operatorname{d}q~) |
| + | \\[4pt] |
| + | (~\operatorname{d}p~(\operatorname{d}q)) |
| + | \\[4pt] |
| + | (~\operatorname{d}p~~\operatorname{d}q~) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((\operatorname{d}p)~\operatorname{d}q~) |
| + | \\[4pt] |
| + | ((\operatorname{d}p)(\operatorname{d}q)) |
| + | \\[4pt] |
| + | (~\operatorname{d}p~~\operatorname{d}q~) |
| + | \\[4pt] |
| + | (~\operatorname{d}p~(\operatorname{d}q)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~\operatorname{d}p~(\operatorname{d}q)) |
| + | \\[4pt] |
| + | (~\operatorname{d}p~~\operatorname{d}q~) |
| + | \\[4pt] |
| + | ((\operatorname{d}p)(\operatorname{d}q)) |
| + | \\[4pt] |
| + | ((\operatorname{d}p)~\operatorname{d}q~) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~\operatorname{d}p~~\operatorname{d}q~) |
| + | \\[4pt] |
| + | (~\operatorname{d}p~(\operatorname{d}q)) |
| + | \\[4pt] |
| + | ((\operatorname{d}p)~\operatorname{d}q~) |
| + | \\[4pt] |
| + | ((\operatorname{d}p)(\operatorname{d}q)) |
| + | \end{matrix}</math> |
| + | |- |
| + | | <math>f_{15}\!</math> |
| + | | <math>((~))</math> |
| + | | <math>((~))</math> |
| + | | <math>((~))</math> |
| + | | <math>((~))</math> |
| + | | <math>((~))</math> |
| + | |} |
| + | |
| <br> | | <br> |
| | | |
− | Table 9. Higher Order Propositions (n = 2)
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
− | o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
| + | |+ <math>\text{Table A6.}~~\operatorname{D}f ~\text{Expanded Over Ordinary Features}~ \{ p, q \}</math> |
− | | | x | 1100 | f |m|m|m|m|m|m|m|m|m|m|m|m|m|m|m|m|.|
| + | |- style="background:#f0f0ff" |
− | | | y | 1010 | |0|0|0|0|0|0|0|0|0|0|1|1|1|1|1|1|.|
| + | | width="10%" | |
− | | f \ | | |0|1|2|3|4|5|6|7|8|9|0|1|2|3|4|5|.|
| + | | width="18%" | <math>f\!</math> |
− | o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
| + | | width="18%" | <math>\operatorname{D}f|_{xy}</math> |
− | | | | | |
| + | | width="18%" | <math>\operatorname{D}f|_{p(q)}</math> |
− | | f_0 | 0000 | () |0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 |
| + | | width="18%" | <math>\operatorname{D}f|_{(p)q}</math> |
− | | | | | |
| + | | width="18%" | <math>\operatorname{D}f|_{(p)(q)}</math> |
− | | f_1 | 0001 | (x)(y) | 1 1 0 0 1 1 0 0 1 1 0 0 1 1 |
| + | |- |
− | | | | | |
| + | | <math>f_0\!</math> |
− | | f_2 | 0010 | (x) y | 1 1 1 1 0 0 0 0 1 1 1 1 |
| + | | <math>(~)</math> |
− | | | | | |
| + | | <math>(~)</math> |
− | | f_3 | 0011 | (x) | 1 1 1 1 1 1 1 1 |
| + | | <math>(~)</math> |
− | | | | | |
| + | | <math>(~)</math> |
− | | f_4 | 0100 | x (y) | |
| + | | <math>(~)</math> |
− | | | | | |
| + | |- |
− | | f_5 | 0101 | (y) | |
| + | | |
− | | | | | |
| + | <math>\begin{matrix} |
− | | f_6 | 0110 | (x, y) | |
| + | f_1 |
− | | | | | |
| + | \\[4pt] |
− | | f_7 | 0111 | (x y) | |
| + | f_2 |
− | | | | | |
| + | \\[4pt] |
− | o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
| + | f_4 |
− | | | | | |
| + | \\[4pt] |
− | | f_8 | 1000 | x y | |
| + | f_8 |
− | | | | | |
| + | \end{matrix}</math> |
− | | f_9 | 1001 | ((x, y)) | |
| + | | |
− | | | | | |
| + | <math>\begin{matrix} |
− | | f_10 | 1010 | y | |
| + | (p)(q) |
− | | | | | |
| + | \\[4pt] |
− | | f_11 | 1011 | (x (y)) | |
| + | (p)~q~ |
− | | | | | |
| + | \\[4pt] |
− | | f_12 | 1100 | x | |
| + | ~p~(q) |
− | | | | | |
| + | \\[4pt] |
− | | f_13 | 1101 | ((x) y) | |
| + | ~p~~q~ |
− | | | | | |
| + | \end{matrix}</math> |
− | | f_14 | 1110 | ((x)(y)) | |
| + | | |
− | | | | | |
| + | <math>\begin{matrix} |
− | | f_15 | 1111 | (()) | |
| + | ~~\operatorname{d}p~~\operatorname{d}q~~ |
− | | | | | |
| + | \\[4pt] |
− | o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
| + | ~~\operatorname{d}p~(\operatorname{d}q)~ |
− | <br> | + | \\[4pt] |
− | | + | ~(\operatorname{d}p)~\operatorname{d}q~~ |
− | Table 10. Qualifiers of Implication Ordering: !a!_i f = !Y!(f_i => f)
| + | \\[4pt] |
− | o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
| + | ((\operatorname{d}p)(\operatorname{d}q)) |
− | | | x | 1100 | f |a |a |a |a |a |a |a |a |a |a |a |a |a |a |a |a |
| + | \end{matrix}</math> |
− | | | 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 |
| + | <math>\begin{matrix} |
− | o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
| + | ~~\operatorname{d}p~(\operatorname{d}q)~ |
− | | | | | |
| + | \\[4pt] |
− | | f_0 | 0000 | () | 1 |
| + | ~~\operatorname{d}p~~\operatorname{d}q~~ |
− | | | | | |
| + | \\[4pt] |
− | | f_1 | 0001 | (x)(y) | 1 1 |
| + | ((\operatorname{d}p)(\operatorname{d}q)) |
− | | | | | |
| + | \\[4pt] |
− | | f_2 | 0010 | (x) y | 1 1 |
| + | ~(\operatorname{d}p)~\operatorname{d}q~~ |
− | | | | | |
| + | \end{matrix}</math> |
− | | f_3 | 0011 | (x) | 1 1 1 1 |
| + | | |
− | | | | | |
| + | <math>\begin{matrix} |
− | | f_4 | 0100 | x (y) | 1 1 |
| + | ~(\operatorname{d}p)~\operatorname{d}q~~ |
− | | | | | |
| + | \\[4pt] |
− | | f_5 | 0101 | (y) | 1 1 1 1 |
| + | ((\operatorname{d}p)(\operatorname{d}q)) |
− | | | | | |
| + | \\[4pt] |
− | | f_6 | 0110 | (x, y) | 1 1 1 1 |
| + | ~~\operatorname{d}p~~\operatorname{d}q~~ |
− | | | | | |
| + | \\[4pt] |
− | | f_7 | 0111 | (x y) | 1 1 1 1 1 1 1 1 |
| + | ~~\operatorname{d}p~(\operatorname{d}q)~ |
− | | | | | |
| + | \end{matrix}</math> |
− | | f_8 | 1000 | x y | 1 1 |
| + | | |
− | | | | | |
| + | <math>\begin{matrix} |
− | | f_9 | 1001 | ((x, y)) | 1 1 1 1 |
| + | ((\operatorname{d}p)(\operatorname{d}q)) |
− | | | | | |
| + | \\[4pt] |
− | | f_10 | 1010 | y | 1 1 1 1 |
| + | ~(\operatorname{d}p)~\operatorname{d}q~~ |
− | | | | | |
| + | \\[4pt] |
− | | f_11 | 1011 | (x (y)) | 1 1 1 1 1 1 1 1 |
| + | ~~\operatorname{d}p~(\operatorname{d}q)~ |
− | | | | | |
| + | \\[4pt] |
− | | f_12 | 1100 | x | 1 1 1 1 |
| + | ~~\operatorname{d}p~~\operatorname{d}q~~ |
− | | | | | |
| + | \end{matrix}</math> |
− | | 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 |
| + | <math>\begin{matrix} |
− | | | | | |
| + | f_3 |
− | | f_15 | 1111 | (()) |1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
| + | \\[4pt] |
− | | | | | |
| + | f_{12} |
− | o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
| + | \end{matrix}</math> |
− | <br> | + | | |
| + | <math>\begin{matrix} |
| + | (p) |
| + | \\[4pt] |
| + | ~p~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{d}p |
| + | \\[4pt] |
| + | \operatorname{d}p |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{d}p |
| + | \\[4pt] |
| + | \operatorname{d}p |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{d}p |
| + | \\[4pt] |
| + | \operatorname{d}p |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{d}p |
| + | \\[4pt] |
| + | \operatorname{d}p |
| + | \end{matrix}</math> |
| + | |- |
| + | | |
| + | <math>\begin{matrix} |
| + | f_6 |
| + | \\[4pt] |
| + | f_9 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~(p,~q)~ |
| + | \\[4pt] |
| + | ((p,~q)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (\operatorname{d}p,~\operatorname{d}q) |
| + | \\[4pt] |
| + | (\operatorname{d}p,~\operatorname{d}q) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (\operatorname{d}p,~\operatorname{d}q) |
| + | \\[4pt] |
| + | (\operatorname{d}p,~\operatorname{d}q) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (\operatorname{d}p,~\operatorname{d}q) |
| + | \\[4pt] |
| + | (\operatorname{d}p,~\operatorname{d}q) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (\operatorname{d}p,~\operatorname{d}q) |
| + | \\[4pt] |
| + | (\operatorname{d}p,~\operatorname{d}q) |
| + | \end{matrix}</math> |
| + | |- |
| + | | |
| + | <math>\begin{matrix} |
| + | f_5 |
| + | \\[4pt] |
| + | f_{10} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (q) |
| + | \\[4pt] |
| + | ~q~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{d}q |
| + | \\[4pt] |
| + | \operatorname{d}q |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{d}q |
| + | \\[4pt] |
| + | \operatorname{d}q |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{d}q |
| + | \\[4pt] |
| + | \operatorname{d}q |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{d}q |
| + | \\[4pt] |
| + | \operatorname{d}q |
| + | \end{matrix}</math> |
| + | |- |
| + | | |
| + | <math>\begin{matrix} |
| + | f_7 |
| + | \\[4pt] |
| + | f_{11} |
| + | \\[4pt] |
| + | f_{13} |
| + | \\[4pt] |
| + | f_{14} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~p~~q~) |
| + | \\[4pt] |
| + | (~p~(q)) |
| + | \\[4pt] |
| + | ((p)~q~) |
| + | \\[4pt] |
| + | ((p)(q)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((\operatorname{d}p)(\operatorname{d}q)) |
| + | \\[4pt] |
| + | ~(\operatorname{d}p)~\operatorname{d}q~~ |
| + | \\[4pt] |
| + | ~~\operatorname{d}p~(\operatorname{d}q)~ |
| + | \\[4pt] |
| + | ~~\operatorname{d}p~~\operatorname{d}q~~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~(\operatorname{d}p)~\operatorname{d}q~~ |
| + | \\[4pt] |
| + | ((\operatorname{d}p)(\operatorname{d}q)) |
| + | \\[4pt] |
| + | ~~\operatorname{d}p~~\operatorname{d}q~~ |
| + | \\[4pt] |
| + | ~~\operatorname{d}p~(\operatorname{d}q)~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~~\operatorname{d}p~(\operatorname{d}q)~ |
| + | \\[4pt] |
| + | ~~\operatorname{d}p~~\operatorname{d}q~~ |
| + | \\[4pt] |
| + | ((\operatorname{d}p)(\operatorname{d}q)) |
| + | \\[4pt] |
| + | ~(\operatorname{d}p)~\operatorname{d}q~~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~~\operatorname{d}p~~\operatorname{d}q~~ |
| + | \\[4pt] |
| + | ~~\operatorname{d}p~(\operatorname{d}q)~ |
| + | \\[4pt] |
| + | ~(\operatorname{d}p)~\operatorname{d}q~~ |
| + | \\[4pt] |
| + | ((\operatorname{d}p)(\operatorname{d}q)) |
| + | \end{matrix}</math> |
| + | |- |
| + | | <math>f_{15}\!</math> |
| + | | <math>((~))</math> |
| + | | <math>((~))</math> |
| + | | <math>((~))</math> |
| + | | <math>((~))</math> |
| + | | <math>((~))</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | ===Wiki TeX Tables : XY=== |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
| + | |+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math> |
| + | |- style="background:#f0f0ff" |
| + | | width="15%" | |
| + | <p><math>\mathcal{L}_1</math></p> |
| + | <p><math>\text{Decimal}</math></p> |
| + | | width="15%" | |
| + | <p><math>\mathcal{L}_2</math></p> |
| + | <p><math>\text{Binary}</math></p> |
| + | | width="15%" | |
| + | <p><math>\mathcal{L}_3</math></p> |
| + | <p><math>\text{Vector}</math></p> |
| + | | width="15%" | |
| + | <p><math>\mathcal{L}_4</math></p> |
| + | <p><math>\text{Cactus}</math></p> |
| + | | width="25%" | |
| + | <p><math>\mathcal{L}_5</math></p> |
| + | <p><math>\text{English}</math></p> |
| + | | width="15%" | |
| + | <p><math>\mathcal{L}_6</math></p> |
| + | <p><math>\text{Ordinary}</math></p> |
| + | |- style="background:#f0f0ff" |
| + | | |
| + | | align="right" | <math>x\colon\!</math> |
| + | | <math>1~1~0~0\!</math> |
| + | | |
| + | | |
| + | | |
| + | |- style="background:#f0f0ff" |
| + | | |
| + | | align="right" | <math>y\colon\!</math> |
| + | | <math>1~0~1~0\!</math> |
| + | | |
| + | | |
| + | | |
| + | |- |
| + | | <math>f_{0}\!</math> |
| + | | <math>f_{0000}\!</math> |
| + | | <math>0~0~0~0\!</math> |
| + | | <math>(~)\!</math> |
| + | | <math>\text{false}\!</math> |
| + | | <math>0\!</math> |
| + | |- |
| + | | <math>f_{1}\!</math> |
| + | | <math>f_{0001}\!</math> |
| + | | <math>0~0~0~1\!</math> |
| + | | <math>(x)(y)\!</math> |
| + | | <math>\text{neither}~ x ~\text{nor}~ y\!</math> |
| + | | <math>\lnot x \land \lnot y\!</math> |
| + | |- |
| + | | <math>f_{2}\!</math> |
| + | | <math>f_{0010}\!</math> |
| + | | <math>0~0~1~0\!</math> |
| + | | <math>(x)~y\!</math> |
| + | | <math>y ~\text{without}~ x\!</math> |
| + | | <math>\lnot x \land y\!</math> |
| + | |- |
| + | | <math>f_{3}\!</math> |
| + | | <math>f_{0011}\!</math> |
| + | | <math>0~0~1~1\!</math> |
| + | | <math>(x)\!</math> |
| + | | <math>\text{not}~ x\!</math> |
| + | | <math>\lnot x\!</math> |
| + | |- |
| + | | <math>f_{4}\!</math> |
| + | | <math>f_{0100}\!</math> |
| + | | <math>0~1~0~0\!</math> |
| + | | <math>x~(y)\!</math> |
| + | | <math>x ~\text{without}~ y\!</math> |
| + | | <math>x \land \lnot y\!</math> |
| + | |- |
| + | | <math>f_{5}\!</math> |
| + | | <math>f_{0101}\!</math> |
| + | | <math>0~1~0~1\!</math> |
| + | | <math>(y)\!</math> |
| + | | <math>\text{not}~ y\!</math> |
| + | | <math>\lnot y\!</math> |
| + | |- |
| + | | <math>f_{6}\!</math> |
| + | | <math>f_{0110}\!</math> |
| + | | <math>0~1~1~0\!</math> |
| + | | <math>(x,~y)\!</math> |
| + | | <math>x ~\text{not equal to}~ y\!</math> |
| + | | <math>x \ne y\!</math> |
| + | |- |
| + | | <math>f_{7}\!</math> |
| + | | <math>f_{0111}\!</math> |
| + | | <math>0~1~1~1\!</math> |
| + | | <math>(x~y)\!</math> |
| + | | <math>\text{not both}~ x ~\text{and}~ y\!</math> |
| + | | <math>\lnot x \lor \lnot y\!</math> |
| + | |- |
| + | | <math>f_{8}\!</math> |
| + | | <math>f_{1000}\!</math> |
| + | | <math>1~0~0~0\!</math> |
| + | | <math>x~y\!</math> |
| + | | <math>x ~\text{and}~ y\!</math> |
| + | | <math>x \land y\!</math> |
| + | |- |
| + | | <math>f_{9}\!</math> |
| + | | <math>f_{1001}\!</math> |
| + | | <math>1~0~0~1\!</math> |
| + | | <math>((x,~y))\!</math> |
| + | | <math>x ~\text{equal to}~ y\!</math> |
| + | | <math>x = y\!</math> |
| + | |- |
| + | | <math>f_{10}\!</math> |
| + | | <math>f_{1010}\!</math> |
| + | | <math>1~0~1~0\!</math> |
| + | | <math>y\!</math> |
| + | | <math>y\!</math> |
| + | | <math>y\!</math> |
| + | |- |
| + | | <math>f_{11}\!</math> |
| + | | <math>f_{1011}\!</math> |
| + | | <math>1~0~1~1\!</math> |
| + | | <math>(x~(y))\!</math> |
| + | | <math>\text{not}~ x ~\text{without}~ y\!</math> |
| + | | <math>x \Rightarrow y\!</math> |
| + | |- |
| + | | <math>f_{12}\!</math> |
| + | | <math>f_{1100}\!</math> |
| + | | <math>1~1~0~0\!</math> |
| + | | <math>x\!</math> |
| + | | <math>x\!</math> |
| + | | <math>x\!</math> |
| + | |- |
| + | | <math>f_{13}\!</math> |
| + | | <math>f_{1101}\!</math> |
| + | | <math>1~1~0~1\!</math> |
| + | | <math>((x)~y)\!</math> |
| + | | <math>\text{not}~ y ~\text{without}~ x\!</math> |
| + | | <math>x \Leftarrow y\!</math> |
| + | |- |
| + | | <math>f_{14}\!</math> |
| + | | <math>f_{1110}\!</math> |
| + | | <math>1~1~1~0\!</math> |
| + | | <math>((x)(y))\!</math> |
| + | | <math>x ~\text{or}~ y\!</math> |
| + | | <math>x \lor y\!</math> |
| + | |- |
| + | | <math>f_{15}\!</math> |
| + | | <math>f_{1111}\!</math> |
| + | | <math>1~1~1~1\!</math> |
| + | | <math>((~))\!</math> |
| + | | <math>\text{true}\!</math> |
| + | | <math>1\!</math> |
| + | |} |
| | | |
− | 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
| |
| <br> | | <br> |
| | | |
− | Table 13. Syllogistic Premisses as Higher Order Indicator Functions
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
− | o---o------------------------o-----------------o---------------------------o
| + | |+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math> |
− | | | | | |
| + | |- style="background:#f0f0ff" |
− | | A | Universal Affirmative | All x is y | Indicator of " x (y)" = 0 |
| + | | width="15%" | |
− | | | | | |
| + | <p><math>\mathcal{L}_1</math></p> |
− | | E | Universal Negative | All x is (y) | Indicator of " x y " = 0 |
| + | <p><math>\text{Decimal}</math></p> |
− | | | | | |
| + | | width="15%" | |
− | | I | Particular Affirmative | Some x is y | Indicator of " x y " = 1 |
| + | <p><math>\mathcal{L}_2</math></p> |
− | | | | | |
| + | <p><math>\text{Binary}</math></p> |
− | | O | Particular Negative | Some x is (y) | Indicator of " x (y)" = 1 |
| + | | width="15%" | |
− | | | | | |
| + | <p><math>\mathcal{L}_3</math></p> |
− | o---o------------------------o-----------------o---------------------------o
| + | <p><math>\text{Vector}</math></p> |
− | <br> | + | | width="15%" | |
− | | + | <p><math>\mathcal{L}_4</math></p> |
− | Table 14. Relation of Quantifiers to Higher Order Propositions
| + | <p><math>\text{Cactus}</math></p> |
− | o------------o------------o-----------o-----------o-----------o-----------o
| + | | width="25%" | |
− | | Mnemonic | Category | Classical | Alternate | Symmetric | Operator |
| + | <p><math>\mathcal{L}_5</math></p> |
− | | | | Form | Form | Form | |
| + | <p><math>\text{English}</math></p> |
− | o============o============o===========o===========o===========o===========o
| + | | width="15%" | |
− | | E | Universal | All x | | No x | (L_11) |
| + | <p><math>\mathcal{L}_6</math></p> |
− | | Exclusive | Negative | is (y) | | is y | |
| + | <p><math>\text{Ordinary}</math></p> |
− | o------------o------------o-----------o-----------o-----------o-----------o
| + | |- style="background:#f0f0ff" |
− | | A | Universal | All x | | No x | (L_10) |
| + | | |
− | | Absolute | Affrmtve | is y | | is (y) | |
| + | | align="right" | <math>x\colon\!</math> |
− | o------------o------------o-----------o-----------o-----------o-----------o
| + | | <math>1~1~0~0\!</math> |
− | | | | 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) |
| + | |- style="background:#f0f0ff" |
− | | | | is x | is (x) | is (y) | |
| + | | |
− | o------------o------------o-----------o-----------o-----------o-----------o
| + | | align="right" | <math>y\colon\!</math> |
− | | | | Some (x) | | Some (x) | L_00 |
| + | | <math>1~0~1~0\!</math> |
− | | | | 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 |
| + | <math>\begin{matrix} |
− | | Obtrusive | Negative | is (y) | | is (y) | |
| + | f_0 |
− | o------------o------------o-----------o-----------o-----------o-----------o
| + | \\[4pt] |
− | | I | Particular | Some x | | Some x | L_11 |
| + | f_1 |
− | | Indefinite | Affrmtve | is y | | is y | |
| + | \\[4pt] |
− | o------------o------------o-----------o-----------o-----------o-----------o
| + | f_2 |
| + | \\[4pt] |
| + | f_3 |
| + | \\[4pt] |
| + | f_4 |
| + | \\[4pt] |
| + | f_5 |
| + | \\[4pt] |
| + | f_6 |
| + | \\[4pt] |
| + | f_7 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | f_{0000} |
| + | \\[4pt] |
| + | f_{0001} |
| + | \\[4pt] |
| + | f_{0010} |
| + | \\[4pt] |
| + | f_{0011} |
| + | \\[4pt] |
| + | f_{0100} |
| + | \\[4pt] |
| + | f_{0101} |
| + | \\[4pt] |
| + | f_{0110} |
| + | \\[4pt] |
| + | f_{0111} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | 0~0~0~0 |
| + | \\[4pt] |
| + | 0~0~0~1 |
| + | \\[4pt] |
| + | 0~0~1~0 |
| + | \\[4pt] |
| + | 0~0~1~1 |
| + | \\[4pt] |
| + | 0~1~0~0 |
| + | \\[4pt] |
| + | 0~1~0~1 |
| + | \\[4pt] |
| + | 0~1~1~0 |
| + | \\[4pt] |
| + | 0~1~1~1 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~) |
| + | \\[4pt] |
| + | (x)(y) |
| + | \\[4pt] |
| + | (x)~y~ |
| + | \\[4pt] |
| + | (x)~~~ |
| + | \\[4pt] |
| + | ~x~(y) |
| + | \\[4pt] |
| + | ~~~(y) |
| + | \\[4pt] |
| + | (x,~y) |
| + | \\[4pt] |
| + | (x~~y) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \text{false} |
| + | \\[4pt] |
| + | \text{neither}~ x ~\text{nor}~ y |
| + | \\[4pt] |
| + | y ~\text{without}~ x |
| + | \\[4pt] |
| + | \text{not}~ x |
| + | \\[4pt] |
| + | x ~\text{without}~ y |
| + | \\[4pt] |
| + | \text{not}~ y |
| + | \\[4pt] |
| + | x ~\text{not equal to}~ y |
| + | \\[4pt] |
| + | \text{not both}~ x ~\text{and}~ y |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | 0 |
| + | \\[4pt] |
| + | \lnot x \land \lnot y |
| + | \\[4pt] |
| + | \lnot x \land y |
| + | \\[4pt] |
| + | \lnot x |
| + | \\[4pt] |
| + | x \land \lnot y |
| + | \\[4pt] |
| + | \lnot y |
| + | \\[4pt] |
| + | x \ne y |
| + | \\[4pt] |
| + | \lnot x \lor \lnot y |
| + | \end{matrix}</math> |
| + | |- |
| + | | |
| + | <math>\begin{matrix} |
| + | f_8 |
| + | \\[4pt] |
| + | f_9 |
| + | \\[4pt] |
| + | f_{10} |
| + | \\[4pt] |
| + | f_{11} |
| + | \\[4pt] |
| + | f_{12} |
| + | \\[4pt] |
| + | f_{13} |
| + | \\[4pt] |
| + | f_{14} |
| + | \\[4pt] |
| + | f_{15} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | f_{1000} |
| + | \\[4pt] |
| + | f_{1001} |
| + | \\[4pt] |
| + | f_{1010} |
| + | \\[4pt] |
| + | f_{1011} |
| + | \\[4pt] |
| + | f_{1100} |
| + | \\[4pt] |
| + | f_{1101} |
| + | \\[4pt] |
| + | f_{1110} |
| + | \\[4pt] |
| + | f_{1111} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | 1~0~0~0 |
| + | \\[4pt] |
| + | 1~0~0~1 |
| + | \\[4pt] |
| + | 1~0~1~0 |
| + | \\[4pt] |
| + | 1~0~1~1 |
| + | \\[4pt] |
| + | 1~1~0~0 |
| + | \\[4pt] |
| + | 1~1~0~1 |
| + | \\[4pt] |
| + | 1~1~1~0 |
| + | \\[4pt] |
| + | 1~1~1~1 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~~x~~y~~ |
| + | \\[4pt] |
| + | ((x,~y)) |
| + | \\[4pt] |
| + | ~~~~~y~~ |
| + | \\[4pt] |
| + | ~(x~(y)) |
| + | \\[4pt] |
| + | ~~x~~~~~ |
| + | \\[4pt] |
| + | ((x)~y)~ |
| + | \\[4pt] |
| + | ((x)(y)) |
| + | \\[4pt] |
| + | ((~)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | x ~\text{and}~ y |
| + | \\[4pt] |
| + | x ~\text{equal to}~ y |
| + | \\[4pt] |
| + | y |
| + | \\[4pt] |
| + | \text{not}~ x ~\text{without}~ y |
| + | \\[4pt] |
| + | x |
| + | \\[4pt] |
| + | \text{not}~ y ~\text{without}~ x |
| + | \\[4pt] |
| + | x ~\text{or}~ y |
| + | \\[4pt] |
| + | \text{true} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | x \land y |
| + | \\[4pt] |
| + | x = y |
| + | \\[4pt] |
| + | y |
| + | \\[4pt] |
| + | x \Rightarrow y |
| + | \\[4pt] |
| + | x |
| + | \\[4pt] |
| + | x \Leftarrow y |
| + | \\[4pt] |
| + | x \lor y |
| + | \\[4pt] |
| + | 1 |
| + | \end{matrix}</math> |
| + | |} |
| + | |
| <br> | | <br> |
| | | |
− | Table 15. Simple Qualifiers of Propositions (n = 2)
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
− | o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o
| + | |+ <math>\text{Table A2.}~~\text{Propositional Forms on Two Variables}</math> |
− | | | x | 1100 | f |(L11)|(L10)|(L01)|(L00)| L00 | L01 | L10 | L11 |
| + | |- style="background:#f0f0ff" |
− | | | y | 1010 | |no x|no x|no ~x|no ~x|sm ~x|sm ~x|sm x|sm x|
| + | | width="15%" | |
− | | f \ | | |is y|is ~y|is y|is ~y|is ~y|is y|is ~y|is y|
| + | <p><math>\mathcal{L}_1</math></p> |
− | o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o
| + | <p><math>\text{Decimal}</math></p> |
− | | | | | |
| + | | width="15%" | |
− | | f_0 | 0000 | () | 1 1 1 1 0 0 0 0 |
| + | <p><math>\mathcal{L}_2</math></p> |
− | | | | | |
| + | <p><math>\text{Binary}</math></p> |
− | | f_1 | 0001 | (x)(y) | 1 1 1 0 1 0 0 0 |
| + | | width="15%" | |
− | | | | | |
| + | <p><math>\mathcal{L}_3</math></p> |
− | | f_2 | 0010 | (x) y | 1 1 0 1 0 1 0 0 |
| + | <p><math>\text{Vector}</math></p> |
− | | | | | |
| + | | width="15%" | |
− | | f_3 | 0011 | (x) | 1 1 0 0 1 1 0 0 |
| + | <p><math>\mathcal{L}_4</math></p> |
− | | | | | |
| + | <p><math>\text{Cactus}</math></p> |
− | | f_4 | 0100 | x (y) | 1 0 1 1 0 0 1 0 |
| + | | width="25%" | |
− | | | | | |
| + | <p><math>\mathcal{L}_5</math></p> |
− | | f_5 | 0101 | (y) | 1 0 1 0 1 0 1 0 |
| + | <p><math>\text{English}</math></p> |
− | | | | | |
| + | | width="15%" | |
− | | f_6 | 0110 | (x, y) | 1 0 0 1 0 1 1 0 |
| + | <p><math>\mathcal{L}_6</math></p> |
− | | | | | |
| + | <p><math>\text{Ordinary}</math></p> |
− | | f_7 | 0111 | (x y) | 1 0 0 0 1 1 1 0 |
| + | |- style="background:#f0f0ff" |
− | | | | | |
| + | | |
− | | f_8 | 1000 | x y | 0 1 1 1 0 0 0 1 |
| + | | align="right" | <math>x\colon\!</math> |
− | | | | | |
| + | | <math>1~1~0~0\!</math> |
− | | 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 |
| + | | |
− | | | | | |
| + | |- style="background:#f0f0ff" |
− | | f_11 | 1011 | (x (y)) | 0 1 0 0 1 1 0 1 |
| + | | |
− | | | | | |
| + | | align="right" | <math>y\colon\!</math> |
− | | f_12 | 1100 | x | 0 0 1 1 0 0 1 1 |
| + | | <math>1~0~1~0\!</math> |
− | | | | | |
| + | | |
− | | 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 |
| + | |- |
− | | | | | |
| + | | <math>f_0\!</math> |
− | | f_15 | 1111 | (()) | 0 0 0 0 1 1 1 1 |
| + | | <math>f_{0000}\!</math> |
− | | | | | |
| + | | <math>0~0~0~0</math> |
− | o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o
| + | | <math>(~)</math> |
− | <br> | + | | <math>\text{false}\!</math> |
| + | | <math>0\!</math> |
| + | |- |
| + | | |
| + | <math>\begin{matrix} |
| + | f_1 |
| + | \\[4pt] |
| + | f_2 |
| + | \\[4pt] |
| + | f_4 |
| + | \\[4pt] |
| + | f_8 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | f_{0001} |
| + | \\[4pt] |
| + | f_{0010} |
| + | \\[4pt] |
| + | f_{0100} |
| + | \\[4pt] |
| + | f_{1000} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | 0~0~0~1 |
| + | \\[4pt] |
| + | 0~0~1~0 |
| + | \\[4pt] |
| + | 0~1~0~0 |
| + | \\[4pt] |
| + | 1~0~0~0 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (x)(y) |
| + | \\[4pt] |
| + | (x)~y~ |
| + | \\[4pt] |
| + | ~x~(y) |
| + | \\[4pt] |
| + | ~x~~y~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \text{neither}~ x ~\text{nor}~ y |
| + | \\[4pt] |
| + | y ~\text{without}~ x |
| + | \\[4pt] |
| + | x ~\text{without}~ y |
| + | \\[4pt] |
| + | x ~\text{and}~ y |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \lnot x \land \lnot y |
| + | \\[4pt] |
| + | \lnot x \land y |
| + | \\[4pt] |
| + | x \land \lnot y |
| + | \\[4pt] |
| + | x \land y |
| + | \end{matrix}</math> |
| + | |- |
| + | | |
| + | <math>\begin{matrix} |
| + | f_3 |
| + | \\[4pt] |
| + | f_{12} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | f_{0011} |
| + | \\[4pt] |
| + | f_{1100} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | 0~0~1~1 |
| + | \\[4pt] |
| + | 1~1~0~0 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (x) |
| + | \\[4pt] |
| + | ~x~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \text{not}~ x |
| + | \\[4pt] |
| + | x |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \lnot x |
| + | \\[4pt] |
| + | x |
| + | \end{matrix}</math> |
| + | |- |
| + | | |
| + | <math>\begin{matrix} |
| + | f_6 |
| + | \\[4pt] |
| + | f_9 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | f_{0110} |
| + | \\[4pt] |
| + | f_{1001} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | 0~1~1~0 |
| + | \\[4pt] |
| + | 1~0~0~1 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~(x,~y)~ |
| + | \\[4pt] |
| + | ((x,~y)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | x ~\text{not equal to}~ y |
| + | \\[4pt] |
| + | x ~\text{equal to}~ y |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | x \ne y |
| + | \\[4pt] |
| + | x = y |
| + | \end{matrix}</math> |
| + | |- |
| + | | |
| + | <math>\begin{matrix} |
| + | f_5 |
| + | \\[4pt] |
| + | f_{10} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | f_{0101} |
| + | \\[4pt] |
| + | f_{1010} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | 0~1~0~1 |
| + | \\[4pt] |
| + | 1~0~1~0 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (y) |
| + | \\[4pt] |
| + | ~y~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \text{not}~ y |
| + | \\[4pt] |
| + | y |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \lnot y |
| + | \\[4pt] |
| + | y |
| + | \end{matrix}</math> |
| + | |- |
| + | | |
| + | <math>\begin{matrix} |
| + | f_7 |
| + | \\[4pt] |
| + | f_{11} |
| + | \\[4pt] |
| + | f_{13} |
| + | \\[4pt] |
| + | f_{14} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | f_{0111} |
| + | \\[4pt] |
| + | f_{1011} |
| + | \\[4pt] |
| + | f_{1101} |
| + | \\[4pt] |
| + | f_{1110} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | 0~1~1~1 |
| + | \\[4pt] |
| + | 1~0~1~1 |
| + | \\[4pt] |
| + | 1~1~0~1 |
| + | \\[4pt] |
| + | 1~1~1~0 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~(x~~y)~ |
| + | \\[4pt] |
| + | ~(x~(y)) |
| + | \\[4pt] |
| + | ((x)~y)~ |
| + | \\[4pt] |
| + | ((x)(y)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \text{not both}~ x ~\text{and}~ y |
| + | \\[4pt] |
| + | \text{not}~ x ~\text{without}~ y |
| + | \\[4pt] |
| + | \text{not}~ y ~\text{without}~ x |
| + | \\[4pt] |
| + | x ~\text{or}~ y |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \lnot x \lor \lnot y |
| + | \\[4pt] |
| + | x \Rightarrow y |
| + | \\[4pt] |
| + | x \Leftarrow y |
| + | \\[4pt] |
| + | x \lor y |
| + | \end{matrix}</math> |
| + | |- |
| + | | <math>f_{15}\!</math> |
| + | | <math>f_{1111}\!</math> |
| + | | <math>1~1~1~1</math> |
| + | | <math>((~))</math> |
| + | | <math>\text{true}\!</math> |
| + | | <math>1\!</math> |
| + | |} |
| | | |
− | ===[[Zeroth Order Logic]]===
| + | <br> |
| | | |
− | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" | + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
− | |+ '''Table 1. Propositional Forms on Two Variables''' | + | |+ <math>\text{Table A3.}~~\operatorname{E}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}x, \operatorname{d}y \}</math> |
− | |- style="background:paleturquoise" | + | |- style="background:#f0f0ff" |
− | ! style="width:15%" | L<sub>1</sub>
| + | | width="10%" | |
− | ! style="width:15%" | L<sub>2</sub>
| + | | width="18%" | <math>f\!</math> |
− | ! style="width:15%" | L<sub>3</sub>
| + | | width="18%" | |
− | ! style="width:15%" | L<sub>4</sub>
| + | <p><math>\operatorname{T}_{11} f</math></p> |
− | ! style="width:15%" | L<sub>5</sub>
| + | <p><math>\operatorname{E}f|_{\operatorname{d}x~\operatorname{d}y}</math></p> |
− | ! style="width:15%" | L<sub>6</sub>
| + | | width="18%" | |
− | |- style="background:paleturquoise" | + | <p><math>\operatorname{T}_{10} f</math></p> |
− | | | + | <p><math>\operatorname{E}f|_{\operatorname{d}x(\operatorname{d}y)}</math></p> |
− | | align="right" | x : | + | | width="18%" | |
− | | 1 1 0 0 | + | <p><math>\operatorname{T}_{01} f</math></p> |
− | | | + | <p><math>\operatorname{E}f|_{(\operatorname{d}x)\operatorname{d}y}</math></p> |
− | | | + | | width="18%" | |
− | | | + | <p><math>\operatorname{T}_{00} f</math></p> |
− | |- style="background:paleturquoise" | + | <p><math>\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}</math></p> |
− | | | + | |- |
− | | align="right" | y : | + | | <math>f_0\!</math> |
− | | 1 0 1 0 | + | | <math>(~)</math> |
− | | | + | | <math>(~)</math> |
− | | | + | | <math>(~)</math> |
− | |
| + | | <math>(~)</math> |
| + | | <math>(~)</math> |
| + | |- |
| + | | |
| + | <math>\begin{matrix} |
| + | f_1 |
| + | \\[4pt] |
| + | f_2 |
| + | \\[4pt] |
| + | f_4 |
| + | \\[4pt] |
| + | f_8 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (x)(y) |
| + | \\[4pt] |
| + | (x)~y~ |
| + | \\[4pt] |
| + | ~x~(y) |
| + | \\[4pt] |
| + | ~x~~y~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~x~~y~ |
| + | \\[4pt] |
| + | ~x~(y) |
| + | \\[4pt] |
| + | (x)~y~ |
| + | \\[4pt] |
| + | (x)(y) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~x~(y) |
| + | \\[4pt] |
| + | ~x~~y~ |
| + | \\[4pt] |
| + | (x)(y) |
| + | \\[4pt] |
| + | (x)~y~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (x)~y~ |
| + | \\[4pt] |
| + | (x)(y) |
| + | \\[4pt] |
| + | ~x~~y~ |
| + | \\[4pt] |
| + | ~x~(y) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (x)(y) |
| + | \\[4pt] |
| + | (x)~y~ |
| + | \\[4pt] |
| + | ~x~(y) |
| + | \\[4pt] |
| + | ~x~~y~ |
| + | \end{matrix}</math> |
| |- | | |- |
− | | f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || ( ) || false || 0 | + | | |
− | |- | + | <math>\begin{matrix} |
− | | f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || ¬x ∧ ¬y
| + | f_3 |
− | |-
| + | \\[4pt] |
− | | f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || ¬x ∧ y
| + | f_{12} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (x) |
| + | \\[4pt] |
| + | ~x~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~x~ |
| + | \\[4pt] |
| + | (x) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~x~ |
| + | \\[4pt] |
| + | (x) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (x) |
| + | \\[4pt] |
| + | ~x~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (x) |
| + | \\[4pt] |
| + | ~x~ |
| + | \end{matrix}</math> |
| |- | | |- |
− | | f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || ¬x | + | | |
| + | <math>\begin{matrix} |
| + | f_6 |
| + | \\[4pt] |
| + | f_9 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~(x,~y)~ |
| + | \\[4pt] |
| + | ((x,~y)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~(x,~y)~ |
| + | \\[4pt] |
| + | ((x,~y)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((x,~y)) |
| + | \\[4pt] |
| + | ~(x,~y)~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((x,~y)) |
| + | \\[4pt] |
| + | ~(x,~y)~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~(x,~y)~ |
| + | \\[4pt] |
| + | ((x,~y)) |
| + | \end{matrix}</math> |
| |- | | |- |
− | | f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x ∧ ¬y | + | | |
| + | <math>\begin{matrix} |
| + | f_5 |
| + | \\[4pt] |
| + | f_{10} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (y) |
| + | \\[4pt] |
| + | ~y~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~y~ |
| + | \\[4pt] |
| + | (y) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (y) |
| + | \\[4pt] |
| + | ~y~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~y~ |
| + | \\[4pt] |
| + | (y) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (y) |
| + | \\[4pt] |
| + | ~y~ |
| + | \end{matrix}</math> |
| |- | | |- |
− | | f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || ¬y | + | | |
− | |-
| + | <math>\begin{matrix} |
− | | f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x ≠ y
| + | f_7 |
− | |-
| + | \\[4pt] |
− | | f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x y) || not both x and y || ¬x ∨ ¬y
| + | f_{11} |
| + | \\[4pt] |
| + | f_{13} |
| + | \\[4pt] |
| + | f_{14} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~x~~y~) |
| + | \\[4pt] |
| + | (~x~(y)) |
| + | \\[4pt] |
| + | ((x)~y~) |
| + | \\[4pt] |
| + | ((x)(y)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((x)(y)) |
| + | \\[4pt] |
| + | ((x)~y~) |
| + | \\[4pt] |
| + | (~x~(y)) |
| + | \\[4pt] |
| + | (~x~~y~) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((x)~y~) |
| + | \\[4pt] |
| + | ((x)(y)) |
| + | \\[4pt] |
| + | (~x~~y~) |
| + | \\[4pt] |
| + | (~x~(y)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~x~(y)) |
| + | \\[4pt] |
| + | (~x~~y~) |
| + | \\[4pt] |
| + | ((x)(y)) |
| + | \\[4pt] |
| + | ((x)~y~) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~x~~y~) |
| + | \\[4pt] |
| + | (~x~(y)) |
| + | \\[4pt] |
| + | ((x)~y~) |
| + | \\[4pt] |
| + | ((x)(y)) |
| + | \end{matrix}</math> |
| |- | | |- |
− | | f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x y || x and y || x ∧ y | + | | <math>f_{15}\!</math> |
| + | | <math>((~))</math> |
| + | | <math>((~))</math> |
| + | | <math>((~))</math> |
| + | | <math>((~))</math> |
| + | | <math>((~))</math> |
| + | |- style="background:#f0f0ff" |
| + | | colspan="2" | <math>\text{Fixed Point Total}\!</math> |
| + | | <math>4\!</math> |
| + | | <math>4\!</math> |
| + | | <math>4\!</math> |
| + | | <math>16\!</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
| + | |+ <math>\text{Table A4.}~~\operatorname{D}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}x, \operatorname{d}y \}</math> |
| + | |- style="background:#f0f0ff" |
| + | | width="10%" | |
| + | | width="18%" | <math>f\!</math> |
| + | | width="18%" | |
| + | <math>\operatorname{D}f|_{\operatorname{d}x~\operatorname{d}y}</math> |
| + | | width="18%" | |
| + | <math>\operatorname{D}f|_{\operatorname{d}x(\operatorname{d}y)}</math> |
| + | | width="18%" | |
| + | <math>\operatorname{D}f|_{(\operatorname{d}x)\operatorname{d}y}</math> |
| + | | width="18%" | |
| + | <math>\operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)}</math> |
| |- | | |- |
− | | f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y | + | | <math>f_0\!</math> |
| + | | <math>(~)</math> |
| + | | <math>(~)</math> |
| + | | <math>(~)</math> |
| + | | <math>(~)</math> |
| + | | <math>(~)</math> |
| |- | | |- |
− | | f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y | + | | |
| + | <math>\begin{matrix} |
| + | f_1 |
| + | \\[4pt] |
| + | f_2 |
| + | \\[4pt] |
| + | f_4 |
| + | \\[4pt] |
| + | f_8 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (x)(y) |
| + | \\[4pt] |
| + | (x)~y~ |
| + | \\[4pt] |
| + | ~x~(y) |
| + | \\[4pt] |
| + | ~x~~y~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((x,~y)) |
| + | \\[4pt] |
| + | ~(x,~y)~ |
| + | \\[4pt] |
| + | ~(x,~y)~ |
| + | \\[4pt] |
| + | ((x,~y)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (y) |
| + | \\[4pt] |
| + | ~y~ |
| + | \\[4pt] |
| + | (y) |
| + | \\[4pt] |
| + | ~y~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (x) |
| + | \\[4pt] |
| + | (x) |
| + | \\[4pt] |
| + | ~x~ |
| + | \\[4pt] |
| + | ~x~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~) |
| + | \\[4pt] |
| + | (~) |
| + | \\[4pt] |
| + | (~) |
| + | \\[4pt] |
| + | (~) |
| + | \end{matrix}</math> |
| |- | | |- |
− | | f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x → y | + | | |
| + | <math>\begin{matrix} |
| + | f_3 |
| + | \\[4pt] |
| + | f_{12} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (x) |
| + | \\[4pt] |
| + | ~x~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((~)) |
| + | \\[4pt] |
| + | ((~)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((~)) |
| + | \\[4pt] |
| + | ((~)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~) |
| + | \\[4pt] |
| + | (~) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~) |
| + | \\[4pt] |
| + | (~) |
| + | \end{matrix}</math> |
| |- | | |- |
− | | f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x | + | | |
| + | <math>\begin{matrix} |
| + | f_6 |
| + | \\[4pt] |
| + | f_9 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~(x,~y)~ |
| + | \\[4pt] |
| + | ((x,~y)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~) |
| + | \\[4pt] |
| + | (~) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((~)) |
| + | \\[4pt] |
| + | ((~)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((~)) |
| + | \\[4pt] |
| + | ((~)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~) |
| + | \\[4pt] |
| + | (~) |
| + | \end{matrix}</math> |
| |- | | |- |
− | | f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x ← y | + | | |
| + | <math>\begin{matrix} |
| + | f_5 |
| + | \\[4pt] |
| + | f_{10} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (y) |
| + | \\[4pt] |
| + | ~y~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((~)) |
| + | \\[4pt] |
| + | ((~)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~) |
| + | \\[4pt] |
| + | (~) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((~)) |
| + | \\[4pt] |
| + | ((~)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~) |
| + | \\[4pt] |
| + | (~) |
| + | \end{matrix}</math> |
| |- | | |- |
− | | f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y || x ∨ y | + | | |
| + | <math>\begin{matrix} |
| + | f_7 |
| + | \\[4pt] |
| + | f_{11} |
| + | \\[4pt] |
| + | f_{13} |
| + | \\[4pt] |
| + | f_{14} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~(x~~y)~ |
| + | \\[4pt] |
| + | ~(x~(y)) |
| + | \\[4pt] |
| + | ((x)~y)~ |
| + | \\[4pt] |
| + | ((x)(y)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((x,~y)) |
| + | \\[4pt] |
| + | ~(x,~y)~ |
| + | \\[4pt] |
| + | ~(x,~y)~ |
| + | \\[4pt] |
| + | ((x,~y)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~y~ |
| + | \\[4pt] |
| + | (y) |
| + | \\[4pt] |
| + | ~y~ |
| + | \\[4pt] |
| + | (y) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~x~ |
| + | \\[4pt] |
| + | ~x~ |
| + | \\[4pt] |
| + | (x) |
| + | \\[4pt] |
| + | (x) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~) |
| + | \\[4pt] |
| + | (~) |
| + | \\[4pt] |
| + | (~) |
| + | \\[4pt] |
| + | (~) |
| + | \end{matrix}</math> |
| |- | | |- |
− | | f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || (( )) || true || 1 | + | | <math>f_{15}\!</math> |
| + | | <math>((~))</math> |
| + | | <math>(~)</math> |
| + | | <math>(~)</math> |
| + | | <math>(~)</math> |
| + | | <math>(~)</math> |
| |} | | |} |
| + | |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:90%" | + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
− | |+ '''Table 1. Propositional Forms on Two Variables''' | + | |+ <math>\text{Table A5.}~~\operatorname{E}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math> |
− | |- style="background:aliceblue" | + | |- style="background:#f0f0ff" |
− | ! style="width:15%" | L<sub>1</sub>
| + | | width="10%" | |
− | ! style="width:15%" | L<sub>2</sub>
| + | | width="18%" | <math>f\!</math> |
− | ! style="width:15%" | L<sub>3</sub>
| + | | width="18%" | <math>\operatorname{E}f|_{xy}</math> |
− | ! style="width:15%" | L<sub>4</sub>
| + | | width="18%" | <math>\operatorname{E}f|_{x(y)}</math> |
− | ! style="width:15%" | L<sub>5</sub>
| + | | width="18%" | <math>\operatorname{E}f|_{(x)y}</math> |
− | ! style="width:15%" | L<sub>6</sub>
| + | | width="18%" | <math>\operatorname{E}f|_{(x)(y)}</math> |
− | |- style="background:aliceblue" | + | |- |
− | | | + | | <math>f_0\!</math> |
− | | align="right" | x : | + | | <math>(~)</math> |
− | | 1 1 0 0 | + | | <math>(~)</math> |
− | | | + | | <math>(~)</math> |
− | | | + | | <math>(~)</math> |
− | | | + | | <math>(~)</math> |
− | |- style="background:aliceblue"
| |
− | |
| |
− | | align="right" | y :
| |
− | | 1 0 1 0
| |
− | |
| |
− | |
| |
− | |
| |
| |- | | |- |
− | | f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || ( ) || false || 0 | + | | |
| + | <math>\begin{matrix} |
| + | f_1 |
| + | \\[4pt] |
| + | f_2 |
| + | \\[4pt] |
| + | f_4 |
| + | \\[4pt] |
| + | f_8 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (x)(y) |
| + | \\[4pt] |
| + | (x)~y~ |
| + | \\[4pt] |
| + | ~x~(y) |
| + | \\[4pt] |
| + | ~x~~y~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~\operatorname{d}x~~\operatorname{d}y~ |
| + | \\[4pt] |
| + | ~\operatorname{d}x~(\operatorname{d}y) |
| + | \\[4pt] |
| + | (\operatorname{d}x)~\operatorname{d}y~ |
| + | \\[4pt] |
| + | (\operatorname{d}x)(\operatorname{d}y) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~\operatorname{d}x~(\operatorname{d}y) |
| + | \\[4pt] |
| + | ~\operatorname{d}x~~\operatorname{d}y~ |
| + | \\[4pt] |
| + | (\operatorname{d}x)(\operatorname{d}y) |
| + | \\[4pt] |
| + | (\operatorname{d}x)~\operatorname{d}y~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (\operatorname{d}x)~\operatorname{d}y~ |
| + | \\[4pt] |
| + | (\operatorname{d}x)(\operatorname{d}y) |
| + | \\[4pt] |
| + | ~\operatorname{d}x~~\operatorname{d}y~ |
| + | \\[4pt] |
| + | ~\operatorname{d}x~(\operatorname{d}y) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (\operatorname{d}x)(\operatorname{d}y) |
| + | \\[4pt] |
| + | (\operatorname{d}x)~\operatorname{d}y~ |
| + | \\[4pt] |
| + | ~\operatorname{d}x~(\operatorname{d}y) |
| + | \\[4pt] |
| + | ~\operatorname{d}x~~\operatorname{d}y~ |
| + | \end{matrix}</math> |
| |- | | |- |
− | | f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || ¬x ∧ ¬y | + | | |
− | |- | + | <math>\begin{matrix} |
− | | f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || ¬x ∧ y
| + | f_3 |
| + | \\[4pt] |
| + | f_{12} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (x) |
| + | \\[4pt] |
| + | ~x~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~\operatorname{d}x~ |
| + | \\[4pt] |
| + | (\operatorname{d}x) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~\operatorname{d}x~ |
| + | \\[4pt] |
| + | (\operatorname{d}x) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (\operatorname{d}x) |
| + | \\[4pt] |
| + | ~\operatorname{d}x~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (\operatorname{d}x) |
| + | \\[4pt] |
| + | ~\operatorname{d}x~ |
| + | \end{matrix}</math> |
| |- | | |- |
− | | f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || ¬x | + | | |
| + | <math>\begin{matrix} |
| + | f_6 |
| + | \\[4pt] |
| + | f_9 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~(x,~y)~ |
| + | \\[4pt] |
| + | ((x,~y)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~(\operatorname{d}x,~\operatorname{d}y)~ |
| + | \\[4pt] |
| + | ((\operatorname{d}x,~\operatorname{d}y)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((\operatorname{d}x,~\operatorname{d}y)) |
| + | \\[4pt] |
| + | ~(\operatorname{d}x,~\operatorname{d}y)~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((\operatorname{d}x,~\operatorname{d}y)) |
| + | \\[4pt] |
| + | ~(\operatorname{d}x,~\operatorname{d}y)~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~(\operatorname{d}x,~\operatorname{d}y)~ |
| + | \\[4pt] |
| + | ((\operatorname{d}x,~\operatorname{d}y)) |
| + | \end{matrix}</math> |
| |- | | |- |
− | | f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x ∧ ¬y | + | | |
| + | <math>\begin{matrix} |
| + | f_5 |
| + | \\[4pt] |
| + | f_{10} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (y) |
| + | \\[4pt] |
| + | ~y~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~\operatorname{d}y~ |
| + | \\[4pt] |
| + | (\operatorname{d}y) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (\operatorname{d}y) |
| + | \\[4pt] |
| + | ~\operatorname{d}y~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~\operatorname{d}y~ |
| + | \\[4pt] |
| + | (\operatorname{d}y) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (\operatorname{d}y) |
| + | \\[4pt] |
| + | ~\operatorname{d}y~ |
| + | \end{matrix}</math> |
| |- | | |- |
− | | f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || ¬y | + | | |
− | |- | + | <math>\begin{matrix} |
− | | f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x ≠ y
| + | f_7 |
| + | \\[4pt] |
| + | f_{11} |
| + | \\[4pt] |
| + | f_{13} |
| + | \\[4pt] |
| + | f_{14} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~x~~y~) |
| + | \\[4pt] |
| + | (~x~(y)) |
| + | \\[4pt] |
| + | ((x)~y~) |
| + | \\[4pt] |
| + | ((x)(y)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((\operatorname{d}x)(\operatorname{d}y)) |
| + | \\[4pt] |
| + | ((\operatorname{d}x)~\operatorname{d}y~) |
| + | \\[4pt] |
| + | (~\operatorname{d}x~(\operatorname{d}y)) |
| + | \\[4pt] |
| + | (~\operatorname{d}x~~\operatorname{d}y~) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((\operatorname{d}x)~\operatorname{d}y~) |
| + | \\[4pt] |
| + | ((\operatorname{d}x)(\operatorname{d}y)) |
| + | \\[4pt] |
| + | (~\operatorname{d}x~~\operatorname{d}y~) |
| + | \\[4pt] |
| + | (~\operatorname{d}x~(\operatorname{d}y)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~\operatorname{d}x~(\operatorname{d}y)) |
| + | \\[4pt] |
| + | (~\operatorname{d}x~~\operatorname{d}y~) |
| + | \\[4pt] |
| + | ((\operatorname{d}x)(\operatorname{d}y)) |
| + | \\[4pt] |
| + | ((\operatorname{d}x)~\operatorname{d}y~) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~\operatorname{d}x~~\operatorname{d}y~) |
| + | \\[4pt] |
| + | (~\operatorname{d}x~(\operatorname{d}y)) |
| + | \\[4pt] |
| + | ((\operatorname{d}x)~\operatorname{d}y~) |
| + | \\[4pt] |
| + | ((\operatorname{d}x)(\operatorname{d}y)) |
| + | \end{matrix}</math> |
| |- | | |- |
− | | f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x y) || not both x and y || ¬x ∨ ¬y | + | | <math>f_{15}\!</math> |
− | |-
| + | | <math>((~))</math> |
− | | f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x y || x and y || x ∧ y
| + | | <math>((~))</math> |
− | |-
| + | | <math>((~))</math> |
− | | f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y | + | | <math>((~))</math> |
− | |-
| + | | <math>((~))</math> |
− | | f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y
| |
− | |-
| |
− | | f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x → y | |
− | |-
| |
− | | f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x
| |
− | |-
| |
− | | f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x ← y | |
− | |-
| |
− | | f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y || x ∨ y
| |
− | |-
| |
− | | f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || (( )) || true || 1
| |
| |} | | |} |
| + | |
| <br> | | <br> |
| | | |
− | ===Template Draft===
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
− | | + | |+ <math>\text{Table A6.}~~\operatorname{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math> |
− | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:98%" | + | |- style="background:#f0f0ff" |
− | |+ '''Propositional Forms on Two Variables''' | + | | width="10%" | |
− | |- style="background:aliceblue" | + | | width="18%" | <math>f\!</math> |
− | ! style="width:14%" | L<sub>1</sub>
| + | | width="18%" | <math>\operatorname{D}f|_{xy}</math> |
− | ! style="width:14%" | L<sub>2</sub>
| + | | width="18%" | <math>\operatorname{D}f|_{x(y)}</math> |
− | ! style="width:14%" | L<sub>3</sub>
| + | | width="18%" | <math>\operatorname{D}f|_{(x)y}</math> |
− | ! style="width:14%" | L<sub>4</sub>
| + | | width="18%" | <math>\operatorname{D}f|_{(x)(y)}</math> |
− | ! style="width:14%" | L<sub>5</sub>
| + | |- |
− | ! style="width:14%" | L<sub>6</sub> | + | | <math>f_0\!</math> |
− | ! style="width:14%" | Name
| + | | <math>(~)</math> |
− | |- style="background:aliceblue" | + | | <math>(~)</math> |
− | | | + | | <math>(~)</math> |
− | | align="right" | x : | + | | <math>(~)</math> |
− | | 1 1 0 0 | + | | <math>(~)</math> |
− | | | + | |- |
− | | | + | | |
− | | | + | <math>\begin{matrix} |
− | | | + | f_1 |
− | |- style="background:aliceblue" | + | \\[4pt] |
− | |
| + | f_2 |
− | | align="right" | y :
| + | \\[4pt] |
− | | 1 0 1 0
| + | f_4 |
− | |
| + | \\[4pt] |
− | |
| + | f_8 |
− | |
| + | \end{matrix}</math> |
− | |
| + | | |
| + | <math>\begin{matrix} |
| + | (x)(y) |
| + | \\[4pt] |
| + | (x)~y~ |
| + | \\[4pt] |
| + | ~x~(y) |
| + | \\[4pt] |
| + | ~x~~y~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~~\operatorname{d}x~~\operatorname{d}y~~ |
| + | \\[4pt] |
| + | ~~\operatorname{d}x~(\operatorname{d}y)~ |
| + | \\[4pt] |
| + | ~(\operatorname{d}x)~\operatorname{d}y~~ |
| + | \\[4pt] |
| + | ((\operatorname{d}x)(\operatorname{d}y)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~~\operatorname{d}x~(\operatorname{d}y)~ |
| + | \\[4pt] |
| + | ~~\operatorname{d}x~~\operatorname{d}y~~ |
| + | \\[4pt] |
| + | ((\operatorname{d}x)(\operatorname{d}y)) |
| + | \\[4pt] |
| + | ~(\operatorname{d}x)~\operatorname{d}y~~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~(\operatorname{d}x)~\operatorname{d}y~~ |
| + | \\[4pt] |
| + | ((\operatorname{d}x)(\operatorname{d}y)) |
| + | \\[4pt] |
| + | ~~\operatorname{d}x~~\operatorname{d}y~~ |
| + | \\[4pt] |
| + | ~~\operatorname{d}x~(\operatorname{d}y)~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((\operatorname{d}x)(\operatorname{d}y)) |
| + | \\[4pt] |
| + | ~(\operatorname{d}x)~\operatorname{d}y~~ |
| + | \\[4pt] |
| + | ~~\operatorname{d}x~(\operatorname{d}y)~ |
| + | \\[4pt] |
| + | ~~\operatorname{d}x~~\operatorname{d}y~~ |
| + | \end{matrix}</math> |
| |- | | |- |
− | | f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || ( ) || false || 0 || Falsity | + | | |
− | |-
| + | <math>\begin{matrix} |
− | | f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || ¬x ∧ ¬y || [[NNOR]]
| + | f_3 |
− | |-
| + | \\[4pt] |
− | | f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || ¬x ∧ y || Insuccede
| + | f_{12} |
− | |- | + | \end{matrix}</math> |
− | | f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || ¬x || Not One
| + | | |
| + | <math>\begin{matrix} |
| + | (x) |
| + | \\[4pt] |
| + | ~x~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{d}x |
| + | \\[4pt] |
| + | \operatorname{d}x |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{d}x |
| + | \\[4pt] |
| + | \operatorname{d}x |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{d}x |
| + | \\[4pt] |
| + | \operatorname{d}x |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{d}x |
| + | \\[4pt] |
| + | \operatorname{d}x |
| + | \end{matrix}</math> |
| |- | | |- |
− | | f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x ∧ ¬y || Imprecede | + | | |
| + | <math>\begin{matrix} |
| + | f_6 |
| + | \\[4pt] |
| + | f_9 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~(x,~y)~ |
| + | \\[4pt] |
| + | ((x,~y)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (\operatorname{d}x,~\operatorname{d}y) |
| + | \\[4pt] |
| + | (\operatorname{d}x,~\operatorname{d}y) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (\operatorname{d}x,~\operatorname{d}y) |
| + | \\[4pt] |
| + | (\operatorname{d}x,~\operatorname{d}y) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (\operatorname{d}x,~\operatorname{d}y) |
| + | \\[4pt] |
| + | (\operatorname{d}x,~\operatorname{d}y) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (\operatorname{d}x,~\operatorname{d}y) |
| + | \\[4pt] |
| + | (\operatorname{d}x,~\operatorname{d}y) |
| + | \end{matrix}</math> |
| |- | | |- |
− | | f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || ¬y || Not Two | + | | |
| + | <math>\begin{matrix} |
| + | f_5 |
| + | \\[4pt] |
| + | f_{10} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (y) |
| + | \\[4pt] |
| + | ~y~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{d}y |
| + | \\[4pt] |
| + | \operatorname{d}y |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{d}y |
| + | \\[4pt] |
| + | \operatorname{d}y |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{d}y |
| + | \\[4pt] |
| + | \operatorname{d}y |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{d}y |
| + | \\[4pt] |
| + | \operatorname{d}y |
| + | \end{matrix}</math> |
| |- | | |- |
− | | f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x ≠ y || Inequality | + | | |
− | |-
| + | <math>\begin{matrix} |
− | | f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x y) || not both x and y || ¬x ∨ ¬y || NAND
| + | f_7 |
− | |-
| + | \\[4pt] |
− | | f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x y || x and y || x ∧ y || [[Conjunction]]
| + | f_{11} |
| + | \\[4pt] |
| + | f_{13} |
| + | \\[4pt] |
| + | f_{14} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~x~~y~) |
| + | \\[4pt] |
| + | (~x~(y)) |
| + | \\[4pt] |
| + | ((x)~y~) |
| + | \\[4pt] |
| + | ((x)(y)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((\operatorname{d}x)(\operatorname{d}y)) |
| + | \\[4pt] |
| + | ~(\operatorname{d}x)~\operatorname{d}y~~ |
| + | \\[4pt] |
| + | ~~\operatorname{d}x~(\operatorname{d}y)~ |
| + | \\[4pt] |
| + | ~~\operatorname{d}x~~\operatorname{d}y~~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~(\operatorname{d}x)~\operatorname{d}y~~ |
| + | \\[4pt] |
| + | ((\operatorname{d}x)(\operatorname{d}y)) |
| + | \\[4pt] |
| + | ~~\operatorname{d}x~~\operatorname{d}y~~ |
| + | \\[4pt] |
| + | ~~\operatorname{d}x~(\operatorname{d}y)~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~~\operatorname{d}x~(\operatorname{d}y)~ |
| + | \\[4pt] |
| + | ~~\operatorname{d}x~~\operatorname{d}y~~ |
| + | \\[4pt] |
| + | ((\operatorname{d}x)(\operatorname{d}y)) |
| + | \\[4pt] |
| + | ~(\operatorname{d}x)~\operatorname{d}y~~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~~\operatorname{d}x~~\operatorname{d}y~~ |
| + | \\[4pt] |
| + | ~~\operatorname{d}x~(\operatorname{d}y)~ |
| + | \\[4pt] |
| + | ~(\operatorname{d}x)~\operatorname{d}y~~ |
| + | \\[4pt] |
| + | ((\operatorname{d}x)(\operatorname{d}y)) |
| + | \end{matrix}</math> |
| |- | | |- |
− | | f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y || Equality | + | | <math>f_{15}\!</math> |
− | |-
| + | | <math>((~))</math> |
− | | f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y || Two
| + | | <math>((~))</math> |
− | |-
| + | | <math>((~))</math> |
− | | f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x → y || [[Logical implcation|Implication]] | + | | <math>((~))</math> |
− | |-
| + | | <math>((~))</math> |
− | | f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x || One
| |
− | |- | |
− | | f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x ← y || [[Logical involution|Involution]]
| |
− | |-
| |
− | | f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y || x ∨ y || [[Disjunction]]
| |
− | |-
| |
− | | f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || (( )) || true || 1 || Tautology
| |
| |} | | |} |
| + | |
| <br> | | <br> |
| | | |
− | ===[[Truth Tables]]=== | + | ===Klein Four-Group V<sub>4</sub>=== |
| | | |
− | ====[[Logical negation]]====
| + | <br> |
| | | |
− | '''Logical negation''' is an [[logical operation|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.
| + | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%" |
| + | |- style="height:50px" |
| + | | width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math> |
| + | | width="22%" style="border-bottom:1px solid black" | |
| + | <math>\operatorname{T}_{00}</math> |
| + | | width="22%" style="border-bottom:1px solid black" | |
| + | <math>\operatorname{T}_{01}</math> |
| + | | width="22%" style="border-bottom:1px solid black" | |
| + | <math>\operatorname{T}_{10}</math> |
| + | | width="22%" style="border-bottom:1px solid black" | |
| + | <math>\operatorname{T}_{11}</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{T}_{00}</math> |
| + | | <math>\operatorname{T}_{00}</math> |
| + | | <math>\operatorname{T}_{01}</math> |
| + | | <math>\operatorname{T}_{10}</math> |
| + | | <math>\operatorname{T}_{11}</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{T}_{01}</math> |
| + | | <math>\operatorname{T}_{01}</math> |
| + | | <math>\operatorname{T}_{00}</math> |
| + | | <math>\operatorname{T}_{11}</math> |
| + | | <math>\operatorname{T}_{10}</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{T}_{10}</math> |
| + | | <math>\operatorname{T}_{10}</math> |
| + | | <math>\operatorname{T}_{11}</math> |
| + | | <math>\operatorname{T}_{00}</math> |
| + | | <math>\operatorname{T}_{01}</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{T}_{11}</math> |
| + | | <math>\operatorname{T}_{11}</math> |
| + | | <math>\operatorname{T}_{10}</math> |
| + | | <math>\operatorname{T}_{01}</math> |
| + | | <math>\operatorname{T}_{00}</math> |
| + | |} |
| | | |
− | The [[truth table]] of '''NOT p''' (also written as '''~p''' or '''¬p''') is as follows:
| + | <br> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:40%" | + | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%" |
− | |+ '''Logical Negation''' | + | |- style="height:50px" |
− | |- style="background:aliceblue" | + | | width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math> |
− | ! style="width:20%" | p
| + | | width="22%" style="border-bottom:1px solid black" | |
− | ! style="width:20%" | ¬p
| + | <math>\operatorname{e}</math> |
− | |- | + | | width="22%" style="border-bottom:1px solid black" | |
− | | F || T | + | <math>\operatorname{f}</math> |
− | |- | + | | width="22%" style="border-bottom:1px solid black" | |
− | | T || F | + | <math>\operatorname{g}</math> |
| + | | width="22%" style="border-bottom:1px solid black" | |
| + | <math>\operatorname{h}</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{e}</math> |
| + | | <math>\operatorname{e}</math> |
| + | | <math>\operatorname{f}</math> |
| + | | <math>\operatorname{g}</math> |
| + | | <math>\operatorname{h}</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{f}</math> |
| + | | <math>\operatorname{f}</math> |
| + | | <math>\operatorname{e}</math> |
| + | | <math>\operatorname{h}</math> |
| + | | <math>\operatorname{g}</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{g}</math> |
| + | | <math>\operatorname{g}</math> |
| + | | <math>\operatorname{h}</math> |
| + | | <math>\operatorname{e}</math> |
| + | | <math>\operatorname{f}</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{h}</math> |
| + | | <math>\operatorname{h}</math> |
| + | | <math>\operatorname{g}</math> |
| + | | <math>\operatorname{f}</math> |
| + | | <math>\operatorname{e}</math> |
| |} | | |} |
| + | |
| <br> | | <br> |
| | | |
− | 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:
| + | ===Symmetric Group S<sub>3</sub>=== |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; width:40%" | + | <br> |
− | |+ '''Variant Notations''' | + | |
− | |- style="background:aliceblue" | + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
− | ! style="text-align:center" | Notation
| + | |+ <math>\text{Permutation Substitutions in}~ \operatorname{Sym} \{ \mathrm{A}, \mathrm{B}, \mathrm{C} \}</math> |
− | ! Vocalization
| + | |- style="background:#f0f0ff" |
| + | | width="16%" | <math>\operatorname{e}</math> |
| + | | width="16%" | <math>\operatorname{f}</math> |
| + | | width="16%" | <math>\operatorname{g}</math> |
| + | | width="16%" | <math>\operatorname{h}</math> |
| + | | width="16%" | <math>\operatorname{i}</math> |
| + | | width="16%" | <math>\operatorname{j}</math> |
| |- | | |- |
− | | style="text-align:center" | <math>\bar{p}</math> | + | | |
− | | bar ''p'' | + | <math>\begin{matrix} |
− | |- | + | \mathrm{A} & \mathrm{B} & \mathrm{C} |
− | | style="text-align:center" | <math>p'\!</math>
| + | \\[3pt] |
− | | ''p'' prime,<p> ''p'' complement | + | \downarrow & \downarrow & \downarrow |
− | |- | + | \\[6pt] |
− | | style="text-align:center" | <math>!p\!</math>
| + | \mathrm{A} & \mathrm{B} & \mathrm{C} |
− | | bang ''p'' | + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \mathrm{A} & \mathrm{B} & \mathrm{C} |
| + | \\[3pt] |
| + | \downarrow & \downarrow & \downarrow |
| + | \\[6pt] |
| + | \mathrm{C} & \mathrm{A} & \mathrm{B} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \mathrm{A} & \mathrm{B} & \mathrm{C} |
| + | \\[3pt] |
| + | \downarrow & \downarrow & \downarrow |
| + | \\[6pt] |
| + | \mathrm{B} & \mathrm{C} & \mathrm{A} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \mathrm{A} & \mathrm{B} & \mathrm{C} |
| + | \\[3pt] |
| + | \downarrow & \downarrow & \downarrow |
| + | \\[6pt] |
| + | \mathrm{A} & \mathrm{C} & \mathrm{B} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \mathrm{A} & \mathrm{B} & \mathrm{C} |
| + | \\[3pt] |
| + | \downarrow & \downarrow & \downarrow |
| + | \\[6pt] |
| + | \mathrm{C} & \mathrm{B} & \mathrm{A} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \mathrm{A} & \mathrm{B} & \mathrm{C} |
| + | \\[3pt] |
| + | \downarrow & \downarrow & \downarrow |
| + | \\[6pt] |
| + | \mathrm{B} & \mathrm{A} & \mathrm{C} |
| + | \end{matrix}</math> |
| |} | | |} |
| + | |
| <br> | | <br> |
| | | |
− | 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''".
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
− | | + | |+ <math>\text{Matrix Representations of Permutations in}~ \operatorname{Sym}(3)</math> |
− | * 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''.
| + | |- style="background:#f0f0ff" |
− | | + | | width="16%" | <math>\operatorname{e}</math> |
− | * 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.
| + | | width="16%" | <math>\operatorname{f}</math> |
− | | + | | width="16%" | <math>\operatorname{g}</math> |
− | 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]].
| + | | width="16%" | <math>\operatorname{h}</math> |
| + | | width="16%" | <math>\operatorname{i}</math> |
| + | | width="16%" | <math>\operatorname{j}</math> |
| + | |- |
| + | | |
| + | <math>\begin{matrix} |
| + | 1 & 0 & 0 |
| + | \\ |
| + | 0 & 1 & 0 |
| + | \\ |
| + | 0 & 0 & 1 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | 0 & 0 & 1 |
| + | \\ |
| + | 1 & 0 & 0 |
| + | \\ |
| + | 0 & 1 & 0 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | 0 & 1 & 0 |
| + | \\ |
| + | 0 & 0 & 1 |
| + | \\ |
| + | 1 & 0 & 0 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | 1 & 0 & 0 |
| + | \\ |
| + | 0 & 0 & 1 |
| + | \\ |
| + | 0 & 1 & 0 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | 0 & 0 & 1 |
| + | \\ |
| + | 0 & 1 & 0 |
| + | \\ |
| + | 1 & 0 & 0 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | 0 & 1 & 0 |
| + | \\ |
| + | 1 & 0 & 0 |
| + | \\ |
| + | 0 & 0 & 1 |
| + | \end{matrix}</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | <pre> |
| + | 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 |
| + | </pre> |
| + | |
| + | <br> |
| + | |
| + | ===TeX Tables=== |
| + | |
| + | <pre> |
| + | \tableofcontents |
| + | |
| + | \subsection{Table A1. Propositional Forms on Two Variables} |
| | | |
− | Algebraically, logical negation corresponds to the ''complement'' in a [[Boolean algebra]] (for classical logic) or a [[Heyting algebra]] (for intuitionistic logic).
| + | Table A1 lists equivalent expressions for the Boolean functions of two variables in a number of different notational systems. |
| | | |
− | ====[[Logical conjunction]]==== | + | \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} |
| | | |
− | '''Logical conjunction''' is an [[logical operation|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 true.
| + | \subsection{Table A2. Propositional Forms on Two Variables} |
| | | |
− | The [[truth table]] of '''p AND q''' (also written as '''p ∧ q''', '''p & q''', or '''p<math>\cdot</math>q''') is as follows:
| + | Table A2 lists the sixteen Boolean functions of two variables in a different order, grouping them by structural similarity into seven natural classes. |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%" | + | \begin{quote}\begin{tabular}{|c|c|c|c|c|c|c|} |
− | |+ '''Logical Conjunction''' | + | \multicolumn{7}{c}{\textbf{Table A2. Propositional Forms on Two Variables}} \\ |
− | |- style="background:aliceblue" | + | \hline |
− | ! style="width:15%" | p
| + | $\mathcal{L}_1$ & |
− | ! style="width:15%" | q
| + | $\mathcal{L}_2$ && |
− | ! style="width:15%" | p ∧ q
| + | $\mathcal{L}_3$ & |
− | |-
| + | $\mathcal{L}_4$ & |
− | | F || F || F | + | $\mathcal{L}_5$ & |
− | |- | + | $\mathcal{L}_6$ \\ |
− | | F || T || F | + | \hline |
− | |- | + | & & $x =$ & 1 1 0 0 & & & \\ |
− | | T || F || F | + | & & $y =$ & 1 0 1 0 & & & \\ |
− | |- | + | \hline |
− | | T || T || T | + | $f_{0}$ & |
− | |} | + | $f_{0000}$ && |
− | <br>
| + | 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 $\{ x, y \}$} |
| | | |
− | ====[[Logical disjunction]]====
| + | \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} |
| + | </pre> |
| | | |
− | '''Logical disjunction''' is 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 both of its operands are false.
| + | ==Group Operation Tables== |
| | | |
− | The [[truth table]] of '''p OR q''' (also written as '''p ∨ q''') is as follows:
| + | <br> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%" | + | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:80%" |
− | |+ '''Logical Disjunction''' | + | |+ <math>\text{Table 32.1}~~\text{Scheme of a Group Operation Table}</math> |
− | |- style="background:aliceblue" | + | |- style="height:50px" |
− | ! style="width:15%" | p | + | | style="border-bottom:1px solid black; border-right:1px solid black" | <math>*\!</math> |
− | ! style="width:15%" | q | + | | style="border-bottom:1px solid black" | <math>x_0\!</math> |
− | ! style="width:15%" | p ∨ q
| + | | style="border-bottom:1px solid black" | <math>\cdots\!</math> |
− | |- | + | | style="border-bottom:1px solid black" | <math>x_j\!</math> |
− | | F || F || F | + | | style="border-bottom:1px solid black" | <math>\cdots\!</math> |
− | |- | + | |- style="height:50px" |
− | | F || T || T | + | | style="border-right:1px solid black" | <math>x_0\!</math> |
− | |- | + | | <math>x_0 * x_0\!</math> |
− | | T || F || T | + | | <math>\cdots\!</math> |
− | |- | + | | <math>x_0 * x_j\!</math> |
− | | T || T || T | + | | <math>\cdots\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\cdots\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>\cdots\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>x_i\!</math> |
| + | | <math>x_i * x_0\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>x_i * x_j\!</math> |
| + | | <math>\cdots\!</math> |
| + | |- style="height:50px" |
| + | | width="12%" style="border-right:1px solid black" | <math>\cdots\!</math> |
| + | | width="22%" | <math>\cdots\!</math> |
| + | | width="22%" | <math>\cdots\!</math> |
| + | | width="22%" | <math>\cdots\!</math> |
| + | | width="22%" | <math>\cdots\!</math> |
| |} | | |} |
| + | |
| <br> | | <br> |
| | | |
− | ====[[Logical equality]]==== | + | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:80%" |
| + | |+ <math>\text{Table 32.2}~~\text{Scheme of the Regular Ante-Representation}</math> |
| + | |- style="height:50px" |
| + | | style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math> |
| + | | colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>x_0\!</math> |
| + | | <math>\{\!</math> |
| + | | <math>(x_0 ~,~ x_0 * x_0),\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>(x_j ~,~ x_0 * x_j),\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\cdots\!</math> |
| + | | <math>\{\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>x_i\!</math> |
| + | | <math>\{\!</math> |
| + | | <math>(x_0 ~,~ x_i * x_0),\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>(x_j ~,~ x_i * x_j),\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | width="12%" style="border-right:1px solid black" | <math>\cdots\!</math> |
| + | | width="4%" | <math>\{\!</math> |
| + | | width="18%" | <math>\cdots\!</math> |
| + | | width="22%" | <math>\cdots\!</math> |
| + | | width="22%" | <math>\cdots\!</math> |
| + | | width="18%" | <math>\cdots\!</math> |
| + | | width="4%" | <math>\}\!</math> |
| + | |} |
| | | |
− | '''Logical equality''' is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' if and only if both operands are false or both operands are true.
| + | <br> |
| | | |
− | The [[truth table]] of '''p EQ q''' (also written as '''p = q''', '''p ↔ q''', or '''p ≡ q''') is as follows:
| + | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:80%" |
| + | |+ <math>\text{Table 32.3}~~\text{Scheme of the Regular Post-Representation}</math> |
| + | |- style="height:50px" |
| + | | style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math> |
| + | | colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>x_0\!</math> |
| + | | <math>\{\!</math> |
| + | | <math>(x_0 ~,~ x_0 * x_0),\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>(x_j ~,~ x_j * x_0),\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\cdots\!</math> |
| + | | <math>\{\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>x_i\!</math> |
| + | | <math>\{\!</math> |
| + | | <math>(x_0 ~,~ x_0 * x_i),\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>(x_j ~,~ x_j * x_i),\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | width="12%" style="border-right:1px solid black" | <math>\cdots\!</math> |
| + | | width="4%" | <math>\{\!</math> |
| + | | width="18%" | <math>\cdots\!</math> |
| + | | width="22%" | <math>\cdots\!</math> |
| + | | width="22%" | <math>\cdots\!</math> |
| + | | width="18%" | <math>\cdots\!</math> |
| + | | width="4%" | <math>\}\!</math> |
| + | |} |
| + | |
| + | <br> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%" | + | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%" |
− | |+ '''Logical Equality''' | + | |+ <math>\text{Table 33.1}~~\text{Multiplication Operation of the Group}~V_4</math> |
− | |- style="background:aliceblue" | + | |- style="height:50px" |
− | ! style="width:15%" | p | + | | width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math> |
− | ! style="width:15%" | q
| + | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{e}</math> |
− | ! style="width:15%" | p = q
| + | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{f}</math> |
− | |- | + | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{g}</math> |
− | | F || F || T | + | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{h}</math> |
− | |- | + | |- style="height:50px" |
− | | F || T || F | + | | style="border-right:1px solid black" | <math>\operatorname{e}</math> |
− | |- | + | | <math>\operatorname{e}</math> |
− | | T || F || F | + | | <math>\operatorname{f}</math> |
− | |- | + | | <math>\operatorname{g}</math> |
− | | T || T || T | + | | <math>\operatorname{h}</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{f}</math> |
| + | | <math>\operatorname{f}</math> |
| + | | <math>\operatorname{e}</math> |
| + | | <math>\operatorname{h}</math> |
| + | | <math>\operatorname{g}</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{g}</math> |
| + | | <math>\operatorname{g}</math> |
| + | | <math>\operatorname{h}</math> |
| + | | <math>\operatorname{e}</math> |
| + | | <math>\operatorname{f}</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{h}</math> |
| + | | <math>\operatorname{h}</math> |
| + | | <math>\operatorname{g}</math> |
| + | | <math>\operatorname{f}</math> |
| + | | <math>\operatorname{e}</math> |
| |} | | |} |
| + | |
| <br> | | <br> |
| | | |
− | ====[[Exclusive disjunction]]==== | + | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%" |
| + | |+ <math>\text{Table 33.2}~~\text{Regular Representation of the Group}~V_4</math> |
| + | |- style="height:50px" |
| + | | style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math> |
| + | | colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math> |
| + | |- style="height:50px" |
| + | | width="20%" style="border-right:1px solid black" | <math>\operatorname{e}</math> |
| + | | width="4%" | <math>\{\!</math> |
| + | | width="16%" | <math>(\operatorname{e}, \operatorname{e}),</math> |
| + | | width="20%" | <math>(\operatorname{f}, \operatorname{f}),</math> |
| + | | width="20%" | <math>(\operatorname{g}, \operatorname{g}),</math> |
| + | | width="16%" | <math>(\operatorname{h}, \operatorname{h})</math> |
| + | | width="4%" | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{f}</math> |
| + | | <math>\{\!</math> |
| + | | <math>(\operatorname{e}, \operatorname{f}),</math> |
| + | | <math>(\operatorname{f}, \operatorname{e}),</math> |
| + | | <math>(\operatorname{g}, \operatorname{h}),</math> |
| + | | <math>(\operatorname{h}, \operatorname{g})</math> |
| + | | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{g}</math> |
| + | | <math>\{\!</math> |
| + | | <math>(\operatorname{e}, \operatorname{g}),</math> |
| + | | <math>(\operatorname{f}, \operatorname{h}),</math> |
| + | | <math>(\operatorname{g}, \operatorname{e}),</math> |
| + | | <math>(\operatorname{h}, \operatorname{f})</math> |
| + | | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{h}</math> |
| + | | <math>\{\!</math> |
| + | | <math>(\operatorname{e}, \operatorname{h}),</math> |
| + | | <math>(\operatorname{f}, \operatorname{g}),</math> |
| + | | <math>(\operatorname{g}, \operatorname{f}),</math> |
| + | | <math>(\operatorname{h}, \operatorname{e})</math> |
| + | | <math>\}\!</math> |
| + | |} |
| | | |
− | '''Exclusive disjunction''' is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' just in case exactly one of its operands is true.
| + | <br> |
| | | |
− | The [[truth table]] of '''p XOR q''' (also written as '''p + q''', '''p ⊕ q''', or '''p ≠ q''') is as follows:
| + | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%" |
| + | |+ <math>\text{Table 33.3}~~\text{Regular Representation of the Group}~V_4</math> |
| + | |- style="height:50px" |
| + | | style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math> |
| + | | colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Symbols}\!</math> |
| + | |- style="height:50px" |
| + | | width="20%" style="border-right:1px solid black" | <math>\operatorname{e}</math> |
| + | | width="4%" | <math>\{\!</math> |
| + | | width="16%" | <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),</math> |
| + | | width="20%" | <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),</math> |
| + | | width="20%" | <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),</math> |
| + | | width="16%" | <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime})</math> |
| + | | width="4%" | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{f}</math> |
| + | | <math>\{\!</math> |
| + | | <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),</math> |
| + | | <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),</math> |
| + | | <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),</math> |
| + | | <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime})</math> |
| + | | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{g}</math> |
| + | | <math>\{\!</math> |
| + | | <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),</math> |
| + | | <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),</math> |
| + | | <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),</math> |
| + | | <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime})</math> |
| + | | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{h}</math> |
| + | | <math>\{\!</math> |
| + | | <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),</math> |
| + | | <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),</math> |
| + | | <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),</math> |
| + | | <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime})</math> |
| + | | <math>\}\!</math> |
| + | |} |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%"
| |
− | |+ '''Exclusive Disjunction'''
| |
− | |- style="background:aliceblue"
| |
− | ! style="width:15%" | p
| |
− | ! style="width:15%" | q
| |
− | ! style="width:15%" | p XOR q
| |
− | |-
| |
− | | F || F || F
| |
− | |-
| |
− | | F || T || T
| |
− | |-
| |
− | | T || F || T
| |
− | |-
| |
− | | T || T || F
| |
− | |}
| |
| <br> | | <br> |
| | | |
− | The following equivalents can then be deduced:
| + | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%" |
| + | |+ <math>\text{Table 34.1}~~\text{Multiplicative Presentation of the Group}~Z_4(\cdot)</math> |
| + | |- style="height:50px" |
| + | | width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math> |
| + | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{1}</math> |
| + | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{a}</math> |
| + | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{b}</math> |
| + | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{c}</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{1}</math> |
| + | | <math>\operatorname{1}</math> |
| + | | <math>\operatorname{a}</math> |
| + | | <math>\operatorname{b}</math> |
| + | | <math>\operatorname{c}</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{a}</math> |
| + | | <math>\operatorname{a}</math> |
| + | | <math>\operatorname{b}</math> |
| + | | <math>\operatorname{c}</math> |
| + | | <math>\operatorname{1}</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{b}</math> |
| + | | <math>\operatorname{b}</math> |
| + | | <math>\operatorname{c}</math> |
| + | | <math>\operatorname{1}</math> |
| + | | <math>\operatorname{a}</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{c}</math> |
| + | | <math>\operatorname{c}</math> |
| + | | <math>\operatorname{1}</math> |
| + | | <math>\operatorname{a}</math> |
| + | | <math>\operatorname{b}</math> |
| + | |} |
| + | |
| + | <br> |
| | | |
− | : <math>\begin{matrix} | + | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%" |
− | p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\
| + | |+ <math>\text{Table 34.2}~~\text{Regular Representation of the Group}~Z_4(\cdot)</math> |
− | \\ | + | |- style="height:50px" |
− | & = & (p \lor q) & \land & (\lnot p \lor \lnot q) \\
| + | | style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math> |
− | \\ | + | | colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math> |
− | & = & (p \lor q) & \land & \lnot (p \land q)
| + | |- style="height:50px" |
− | \end{matrix}</math> | + | | width="20%" style="border-right:1px solid black" | <math>\operatorname{1}</math> |
| + | | width="4%" | <math>\{\!</math> |
| + | | width="16%" | <math>(\operatorname{1}, \operatorname{1}),</math> |
| + | | width="20%" | <math>(\operatorname{a}, \operatorname{a}),</math> |
| + | | width="20%" | <math>(\operatorname{b}, \operatorname{b}),</math> |
| + | | width="16%" | <math>(\operatorname{c}, \operatorname{c})</math> |
| + | | width="4%" | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{a}</math> |
| + | | <math>\{\!</math> |
| + | | <math>(\operatorname{1}, \operatorname{a}),</math> |
| + | | <math>(\operatorname{a}, \operatorname{b}),</math> |
| + | | <math>(\operatorname{b}, \operatorname{c}),</math> |
| + | | <math>(\operatorname{c}, \operatorname{1})</math> |
| + | | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{b}</math> |
| + | | <math>\{\!</math> |
| + | | <math>(\operatorname{1}, \operatorname{b}),</math> |
| + | | <math>(\operatorname{a}, \operatorname{c}),</math> |
| + | | <math>(\operatorname{b}, \operatorname{1}),</math> |
| + | | <math>(\operatorname{c}, \operatorname{a})</math> |
| + | | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{c}</math> |
| + | | <math>\{\!</math> |
| + | | <math>(\operatorname{1}, \operatorname{c}),</math> |
| + | | <math>(\operatorname{a}, \operatorname{1}),</math> |
| + | | <math>(\operatorname{b}, \operatorname{a}),</math> |
| + | | <math>(\operatorname{c}, \operatorname{b})</math> |
| + | | <math>\}\!</math> |
| + | |} |
| | | |
− | '''Generalized''' or '''n-ary''' XOR is true when the number of 1-bits is odd.
| + | <br> |
| | | |
− | A + B = (A ∧ !B) ∨ (!A ∧ B)
| + | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%" |
− | = {(A ∧ !B) ∨ !A} ∧ {(A ∧ !B) ∨ B}
| + | |+ <math>\text{Table 35.1}~~\text{Additive Presentation of the Group}~Z_4(+)</math> |
− | = {(A ∨ !A) ∧ (!B ∨ !A)} ∧ {(A ∨ B) ∧ (!B ∨ B)}
| + | |- style="height:50px" |
− | = (!A ∨ !B) ∧ (A ∨ B)
| + | | width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>+\!</math> |
− | = !(A ∧ B) ∧ (A ∨ B)
| + | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{0}</math> |
| + | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{1}</math> |
| + | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{2}</math> |
| + | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{3}</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{0}</math> |
| + | | <math>\operatorname{0}</math> |
| + | | <math>\operatorname{1}</math> |
| + | | <math>\operatorname{2}</math> |
| + | | <math>\operatorname{3}</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{1}</math> |
| + | | <math>\operatorname{1}</math> |
| + | | <math>\operatorname{2}</math> |
| + | | <math>\operatorname{3}</math> |
| + | | <math>\operatorname{0}</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{2}</math> |
| + | | <math>\operatorname{2}</math> |
| + | | <math>\operatorname{3}</math> |
| + | | <math>\operatorname{0}</math> |
| + | | <math>\operatorname{1}</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{3}</math> |
| + | | <math>\operatorname{3}</math> |
| + | | <math>\operatorname{0}</math> |
| + | | <math>\operatorname{1}</math> |
| + | | <math>\operatorname{2}</math> |
| + | |} |
| | | |
| + | <br> |
| | | |
− | p + q = (p ∧ !q) ∨ (!p ∧ B)
| + | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%" |
− | | + | |+ <math>\text{Table 35.2}~~\text{Regular Representation of the Group}~Z_4(+)</math> |
− | = {(p ∧ !q) ∨ !p} ∧ {(p ∧ !q) ∨ q}
| + | |- style="height:50px" |
− |
| + | | style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math> |
− | = {(p ∨ !q) ∧ (!q ∨ !p)} ∧ {(p ∨ q) ∧ (!q ∨ q)}
| + | | colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math> |
− |
| + | |- style="height:50px" |
− | = (!p ∨ !q) ∧ (p ∨ q)
| + | | width="20%" style="border-right:1px solid black" | <math>\operatorname{0}</math> |
− |
| + | | width="4%" | <math>\{\!</math> |
− | = !(p ∧ q) ∧ (p ∨ q)
| + | | width="16%" | <math>(\operatorname{0}, \operatorname{0}),</math> |
| + | | width="20%" | <math>(\operatorname{1}, \operatorname{1}),</math> |
| + | | width="20%" | <math>(\operatorname{2}, \operatorname{2}),</math> |
| + | | width="16%" | <math>(\operatorname{3}, \operatorname{3})</math> |
| + | | width="4%" | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{1}</math> |
| + | | <math>\{\!</math> |
| + | | <math>(\operatorname{0}, \operatorname{1}),</math> |
| + | | <math>(\operatorname{1}, \operatorname{2}),</math> |
| + | | <math>(\operatorname{2}, \operatorname{3}),</math> |
| + | | <math>(\operatorname{3}, \operatorname{0})</math> |
| + | | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{2}</math> |
| + | | <math>\{\!</math> |
| + | | <math>(\operatorname{0}, \operatorname{2}),</math> |
| + | | <math>(\operatorname{1}, \operatorname{3}),</math> |
| + | | <math>(\operatorname{2}, \operatorname{0}),</math> |
| + | | <math>(\operatorname{3}, \operatorname{1})</math> |
| + | | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{3}</math> |
| + | | <math>\{\!</math> |
| + | | <math>(\operatorname{0}, \operatorname{3}),</math> |
| + | | <math>(\operatorname{1}, \operatorname{0}),</math> |
| + | | <math>(\operatorname{2}, \operatorname{1}),</math> |
| + | | <math>(\operatorname{3}, \operatorname{2})</math> |
| + | | <math>\}\!</math> |
| + | |} |
| | | |
| + | <br> |
| | | |
− | p + q = (p ∧ ~q) ∨ (~p ∧ q)
| + | ==Higher Order Propositions== |
− |
| |
− | = ((p ∧ ~q) ∨ ~p) ∧ ((p ∧ ~q) ∨ q)
| |
− |
| |
− | = ((p ∨ ~q) ∧ (~q ∨ ~p)) ∧ ((p ∨ q) ∧ (~q ∨ q))
| |
− |
| |
− | = (~p ∨ ~q) ∧ (p ∨ q)
| |
− |
| |
− | = ~(p ∧ q) ∧ (p ∨ q)
| |
| | | |
− | : <math>\begin{matrix} | + | <br> |
− | 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) \\
| + | <table align="center" cellpadding="4" cellspacing="0" style="text-align:center; width:90%"> |
− | & = & ((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) \\
| + | <caption><font size="+2"><math>\text{Table 1.} ~~ \text{Higher Order Propositions} ~ (n = 1)</math></font></caption> |
− | & = & \lnot (p \land q) & \land & (p \lor q)
| + | |
− | \end{matrix}</math>
| + | <tr> |
| + | <td style="border-bottom:2px solid black" align="right"><math>x:</math></td> |
| + | <td style="border-bottom:2px solid black"><math>1 ~ 0</math></td> |
| + | <td style="border-bottom:2px solid black; border-right:2px solid black"><math>f</math></td> |
| + | <td style="border-bottom:2px solid black"><math>m_{0}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>m_{1}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>m_{2}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>m_{3}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>m_{4}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>m_{5}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>m_{6}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>m_{7}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>m_{8}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>m_{9}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>m_{10}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>m_{11}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>m_{12}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>m_{13}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>m_{14}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>m_{15}</math></td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{0}</math></td> |
| + | <td><math>0 ~ 0</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(~)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{1}</math></td> |
| + | <td><math>0 ~ 1</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} x \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| | | |
− | ====[[Logical implication]]==== | + | <tr> |
| + | <td><math>f_{2}</math></td> |
| + | <td><math>1 ~ 0</math></td> |
| + | <td style="border-right:2px solid black"><math>x</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| | | |
− | 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.
| + | <tr> |
| + | <td><math>f_{3}</math></td> |
| + | <td><math>1 ~ 1</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{((~))}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| | | |
− | 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:
| + | </table> |
| | | |
− | {| 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> | | <br> |
| | | |
− | ====[[Logical NAND]]==== | + | <table align="center" border="1" cellpadding="4" cellspacing="0" style="text-align:center; width:90%"> |
| + | |
| + | <caption><font size="+2"><math>\text{Table 2.} ~~ \text{Interpretive Categories for Higher Order Propositions} ~ (n = 1)</math></font></caption> |
| + | |
| + | <tr> |
| + | <td style="border-bottom:2px solid black; border-right:2px solid black">Measure</td> |
| + | <td style="border-bottom:2px solid black">Happening</td> |
| + | <td style="border-bottom:2px solid black">Exactness</td> |
| + | <td style="border-bottom:2px solid black">Existence</td> |
| + | <td style="border-bottom:2px solid black">Linearity</td> |
| + | <td style="border-bottom:2px solid black">Uniformity</td> |
| + | <td style="border-bottom:2px solid black">Information</td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-right:2px solid black"><math>m_{0}</math></td> |
| + | <td>Nothing happens</td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-right:2px solid black"><math>m_{1}</math></td> |
| + | <td> </td> |
| + | <td>Just false</td> |
| + | <td>Nothing exists</td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td></tr> |
| | | |
− | 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.
| + | <tr> |
| + | <td style="border-right:2px solid black"><math>m_{2}</math></td> |
| + | <td> </td> |
| + | <td>Just not <math>x</math></td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td></tr> |
| | | |
− | The [[truth table]] of '''p NAND q''' (also written as '''p | q''' or '''p ↑ q''') is as follows:
| + | <tr> |
| + | <td style="border-right:2px solid black"><math>m_{3}</math></td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td>Nothing is <math>x</math></td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td></tr> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%"
| + | <tr> |
− | |+ '''Logical NAND'''
| + | <td style="border-right:2px solid black"><math>m_{4}</math></td> |
− | |- style="background:aliceblue"
| + | <td> </td> |
− | ! style="width:15%" | p
| + | <td>Just <math>x</math></td> |
− | ! style="width:15%" | q
| + | <td> </td> |
− | ! style="width:15%" | p ↑ q
| + | <td> </td> |
− | |-
| + | <td> </td> |
− | | F || F || T
| + | <td> </td></tr> |
− | |-
| + | |
− | | F || T || T
| + | <tr> |
− | |-
| + | <td style="border-right:2px solid black"><math>m_{5}</math></td> |
− | | T || F || T
| + | <td> </td> |
− | |-
| + | <td> </td> |
− | | T || T || F
| + | <td>Everything is <math>x</math></td> |
− | |}
| + | <td><math>f</math> is linear</td> |
− | <br> | + | <td> </td> |
| + | <td> </td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-right:2px solid black"><math>m_{6}</math></td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td><math>f</math> is not uniform</td> |
| + | <td><math>f</math> is informed</td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-right:2px solid black"><math>m_{7}</math></td> |
| + | <td> </td> |
| + | <td>Not just true</td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-right:2px solid black"><math>m_{8}</math></td> |
| + | <td> </td> |
| + | <td>Just true</td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-right:2px solid black"><math>m_{9}</math></td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td><math>f</math> is uniform</td> |
| + | <td><math>f</math> is not informed</td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-right:2px solid black"><math>m_{10}</math></td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td>Something is not <math>x</math></td> |
| + | <td><math>f</math> is not linear</td> |
| + | <td> </td> |
| + | <td> </td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-right:2px solid black"><math>m_{11}</math></td> |
| + | <td> </td> |
| + | <td>Not just <math>x</math></td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-right:2px solid black"><math>m_{12}</math></td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td>Something is <math>x</math></td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-right:2px solid black"><math>m_{13}</math></td> |
| + | <td> </td> |
| + | <td>Not just not <math>x</math></td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td></tr> |
| | | |
− | ====[[Logical NNOR]]==== | + | <tr> |
| + | <td style="border-right:2px solid black"><math>m_{14}</math></td> |
| + | <td> </td> |
| + | <td>Not just false</td> |
| + | <td>Something exists</td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td></tr> |
| | | |
− | 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.
| + | <tr> |
| + | <td style="border-right:2px solid black"><math>m_{15}</math></td> |
| + | <td>Anything happens</td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td></tr> |
| | | |
− | The [[truth table]] of '''p NNOR q''' (also written as '''p ⊥ q''' or '''p ↓ q''') is as follows:
| + | </table> |
| | | |
− | {| 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> | | <br> |
| | | |
− | ==Relational Tables== | + | <table align="center" cellpadding="1" cellspacing="0" style="background:white; color:black; text-align:center; width:90%"> |
| | | |
− | ===Sign Relations=== | + | <caption><font size="+2"><math>\text{Table 3.} ~~ \text{Higher Order Propositions} ~ (n = 2)</math></font></caption> |
| + | |
| + | <tr> |
| + | <td style="border-bottom:2px solid black" align="right"><math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td> |
| + | <td style="border-bottom:2px solid black"> |
| + | <math>\begin{matrix}1100\\1010\end{matrix}</math></td> |
| + | <td style="border-bottom:2px solid black; border-right:2px solid black"><math>f</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{0}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{1}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{2}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{3}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{4}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{5}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{6}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{7}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{8}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{9}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{10}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{11}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{12}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{13}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{14}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{15}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{16}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{17}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{18}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{19}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{20}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{21}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{22}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{23}{m}</math></td> |
| + | </tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{0}</math></td> |
| + | <td><math>0000</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(~)}</math></td> |
| + | <td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{1}</math></td> |
| + | <td><math>0001</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td> |
| + | <td>0</td><td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td><td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td><td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td><td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td><td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td><td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{2}</math></td> |
| + | <td><math>0010</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{3}</math></td> |
| + | <td><math>0011</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)}</math></td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{4}</math></td> |
| + | <td><math>0100</math></td> |
| + | <td style="border-right:2px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{5}</math></td> |
| + | <td><math>0101</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} v \texttt{)}</math></td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{6}</math></td> |
| + | <td><math>0110</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{7}</math></td> |
| + | <td><math>0111</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{8}</math></td> |
| + | <td><math>1000</math></td> |
| + | <td style="border-right:2px solid black"><math>u ~ v</math></td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{9}</math></td> |
| + | <td><math>1001</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{10}</math></td> |
| + | <td><math>1010</math></td> |
| + | <td style="border-right:2px solid black"><math>v</math></td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{11}</math></td> |
| + | <td><math>1011</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{12}</math></td> |
| + | <td><math>1100</math></td> |
| + | <td style="border-right:2px solid black"><math>u</math></td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{13}</math></td> |
| + | <td><math>1101</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{14}</math></td> |
| + | <td><math>1110</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{15}</math></td> |
| + | <td><math>1111</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{((~))}</math></td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td></tr> |
| + | |
| + | </table> |
| | | |
− | {| cellpadding="4"
| |
− | | width="20px" |
| |
− | | align="center" | '''O''' || = || Object Domain
| |
− | |-
| |
− | | width="20px" |
| |
− | | align="center" | '''S''' || = || Sign Domain
| |
− | |-
| |
− | | width="20px" |
| |
− | | align="center" | '''I''' || = || Interpretant Domain
| |
− | |}
| |
| <br> | | <br> |
| | | |
− | {| cellpadding="4" | + | <table align="center" cellpadding="1" cellspacing="0" style="text-align:center; width:90%"> |
− | | width="20px" |
| + | |
− | | align="center" | '''O'''
| + | <caption><font size="+2"><math>\text{Table 4.} ~~ \text{Qualifiers of the Implication Ordering:} ~ \alpha_{i} f = \Upsilon (f_{i}, f) = \Upsilon (f_{i} \Rightarrow f)</math></font></caption> |
− | | =
| + | |
− | | {Ann, Bob}
| + | <tr> |
− | | =
| + | <td style="border-bottom:2px solid black" align="right"> |
− | | {A, B}
| + | <math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td> |
− | |-
| + | <td style="border-bottom:2px solid black"> |
− | | width="20px" |
| + | <math>\begin{matrix}1100\\1010\end{matrix}</math></td> |
− | | align="center" | '''S'''
| + | <td style="border-bottom:2px solid black; border-right:2px solid black"><math>f</math></td> |
− | | =
| + | <td style="border-bottom:2px solid black"><math>\alpha_{15}</math></td> |
− | | {"Ann", "Bob", "I", "You"}
| + | <td style="border-bottom:2px solid black"><math>\alpha_{14}</math></td> |
− | | =
| + | <td style="border-bottom:2px solid black"><math>\alpha_{13}</math></td> |
− | | {"A", "B", "i", "u"}
| + | <td style="border-bottom:2px solid black"><math>\alpha_{12}</math></td> |
− | |-
| + | <td style="border-bottom:2px solid black"><math>\alpha_{11}</math></td> |
− | | width="20px" |
| + | <td style="border-bottom:2px solid black"><math>\alpha_{10}</math></td> |
− | | align="center" | '''I'''
| + | <td style="border-bottom:2px solid black"><math>\alpha_{9}</math></td> |
− | | =
| + | <td style="border-bottom:2px solid black"><math>\alpha_{8}</math></td> |
− | | {"Ann", "Bob", "I", "You"}
| + | <td style="border-bottom:2px solid black"><math>\alpha_{7}</math></td> |
− | | =
| + | <td style="border-bottom:2px solid black"><math>\alpha_{6}</math></td> |
− | | {"A", "B", "i", "u"}
| + | <td style="border-bottom:2px solid black"><math>\alpha_{5}</math></td> |
− | |}
| + | <td style="border-bottom:2px solid black"><math>\alpha_{4}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\alpha_{3}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\alpha_{2}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\alpha_{1}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\alpha_{0}</math></td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{0}</math></td> |
| + | <td><math>0000</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(~)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{1}</math></td> |
| + | <td><math>0001</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{2}</math></td> |
| + | <td><math>0010</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{3}</math></td> |
| + | <td><math>0011</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{4}</math></td> |
| + | <td><math>0100</math></td> |
| + | <td style="border-right:2px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{5}</math></td> |
| + | <td><math>0101</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} v \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{6}</math></td> |
| + | <td><math>0110</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{7}</math></td> |
| + | <td><math>0111</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{8}</math></td> |
| + | <td><math>1000</math></td> |
| + | <td style="border-right:2px solid black"><math>u ~ v</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{9}</math></td> |
| + | <td><math>1001</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{10}</math></td> |
| + | <td><math>1010</math></td> |
| + | <td style="border-right:2px solid black"><math>v</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{11}</math></td> |
| + | <td><math>1011</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{12}</math></td> |
| + | <td><math>1100</math></td> |
| + | <td style="border-right:2px solid black"><math>u</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{13}</math></td> |
| + | <td><math>1101</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{14}</math></td> |
| + | <td><math>1110</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{15}</math></td> |
| + | <td><math>1111</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{((~))}</math></td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | </table> |
| + | |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" | + | <table align="center" cellpadding="1" cellspacing="0" style="text-align:center; width:90%"> |
− | |+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A
| + | |
− | |- style="background:paleturquoise"
| + | <caption><font size="+2"><math>\text{Table 5.} ~~ \text{Qualifiers of the Implication Ordering:} ~ \beta_i f = \Upsilon (f, f_i) = \Upsilon (f \Rightarrow f_i)</math></font></caption> |
− | ! style="width:20%" | Object
| + | |
− | ! style="width:20%" | Sign
| + | <tr> |
− | ! style="width:20%" | Interpretant
| + | <td style="border-bottom:2px solid black" align="right"> |
− | |-
| + | <math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td> |
− | | '''A''' || '''"A"''' || '''"A"'''
| + | <td style="border-bottom:2px solid black"> |
− | |-
| + | <math>\begin{matrix}1100\\1010\end{matrix}</math></td> |
− | | '''A''' || '''"A"''' || '''"i"'''
| + | |
− | |-
| + | <td style="border-bottom:2px solid black; border-right:2px solid black"><math>f</math></td> |
− | | '''A''' || '''"i"''' || '''"A"'''
| + | <td style="border-bottom:2px solid black"><math>\beta_{0}</math></td> |
− | |-
| + | <td style="border-bottom:2px solid black"><math>\beta_{1}</math></td> |
− | | '''A''' || '''"i"''' || '''"i"'''
| + | <td style="border-bottom:2px solid black"><math>\beta_{2}</math></td> |
− | |-
| + | <td style="border-bottom:2px solid black"><math>\beta_{3}</math></td> |
− | | '''B''' || '''"B"''' || '''"B"'''
| + | <td style="border-bottom:2px solid black"><math>\beta_{4}</math></td> |
− | |-
| + | <td style="border-bottom:2px solid black"><math>\beta_{5}</math></td> |
− | | '''B''' || '''"B"''' || '''"u"'''
| + | <td style="border-bottom:2px solid black"><math>\beta_{6}</math></td> |
− | |-
| + | <td style="border-bottom:2px solid black"><math>\beta_{7}</math></td> |
− | | '''B''' || '''"u"''' || '''"B"'''
| + | <td style="border-bottom:2px solid black"><math>\beta_{8}</math></td> |
− | |-
| + | <td style="border-bottom:2px solid black"><math>\beta_{9}</math></td> |
− | | '''B''' || '''"u"''' || '''"u"''' | + | <td style="border-bottom:2px solid black"><math>\beta_{10}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\beta_{11}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\beta_{12}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\beta_{13}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\beta_{14}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\beta_{15}</math></td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{0}</math></td> |
| + | <td><math>0000</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(~)}</math></td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{1}</math></td> |
| + | <td><math>0001</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{2}</math></td> |
| + | <td><math>0010</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{3}</math></td> |
| + | <td><math>0011</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{4}</math></td> |
| + | <td><math>0100</math></td> |
| + | <td style="border-right:2px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{5}</math></td> |
| + | <td><math>0101</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} v \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{6}</math></td> |
| + | <td><math>0110</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{7}</math></td> |
| + | <td><math>0111</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{8}</math></td> |
| + | <td><math>1000</math></td> |
| + | <td style="border-right:2px solid black"><math>u ~ v</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{9}</math></td> |
| + | <td><math>1001</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{10}</math></td> |
| + | <td><math>1010</math></td> |
| + | <td style="border-right:2px solid black"><math>v</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{11}</math></td> |
| + | <td><math>1011</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{12}</math></td> |
| + | <td><math>1100</math></td> |
| + | <td style="border-right:2px solid black"><math>u</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{13}</math></td> |
| + | <td><math>1101</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{14}</math></td> |
| + | <td><math>1110</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{15}</math></td> |
| + | <td><math>1111</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{((~))}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | </table> |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
| + | |+ <math>\text{Table 7.} ~~ \text{Syllogistic Premisses as Higher Order Indicator Functions}</math> |
| + | | |
| + | <math>\begin{array}{clcl} |
| + | \mathrm{A} |
| + | & \mathrm{Universal~Affirmative} |
| + | & \mathrm{All} ~ u ~ \mathrm{is} ~ v |
| + | & \mathrm{Indicator~of} ~ u \texttt{(} v \texttt{)} = 0 |
| + | \\ |
| + | \mathrm{E} |
| + | & \mathrm{Universal~Negative} |
| + | & \mathrm{All} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)} |
| + | & \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} ~ \texttt{(} v \texttt{)} |
| + | & \mathrm{Indicator~of} ~ u \texttt{(} v \texttt{)} = 1 |
| + | \end{array}</math> |
| |} | | |} |
| + | |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
| + | <table align="center" cellpadding="4" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:90%"> |
− | |+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B
| + | |
− | |- style="background:paleturquoise"
| + | <caption><font size="+2"><math>\text{Table 8.} ~~ \text{Simple Qualifiers of Propositions (Version 1)}</math></font></caption> |
− | ! style="width:20%" | Object | + | |
− | ! style="width:20%" | Sign
| + | <tr> |
− | ! style="width:20%" | Interpretant
| + | <td width="4%" style="border-bottom:1px solid black" align="right"> |
− | |-
| + | <math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td> |
− | | '''A''' || '''"A"''' || '''"A"'''
| + | <td width="6%" style="border-bottom:1px solid black"> |
− | |-
| + | <math>\begin{matrix}1100\\1010\end{matrix}</math></td> |
− | | '''A''' || '''"A"''' || '''"u"'''
| + | <td width="10%" style="border-bottom:1px solid black; border-right:1px solid black"> |
− | |-
| + | <math>f</math></td> |
− | | '''A''' || '''"u"''' || '''"A"'''
| + | <td width="10%" style="border-bottom:1px solid black"> |
− | |-
| + | <math>\begin{smallmatrix} |
− | | '''A''' || '''"u"''' || '''"u"'''
| + | \texttt{(} \ell_{11} \texttt{)} |
− | |-
| + | \\ |
− | | '''B''' || '''"B"''' || '''"B"'''
| + | \mathrm{No} ~ u |
− | |-
| + | \\ |
− | | '''B''' || '''"B"''' || '''"i"'''
| + | \mathrm{is} ~ v |
− | |-
| + | \end{smallmatrix}</math></td> |
− | | '''B''' || '''"i"''' || '''"B"'''
| + | <td width="10%" style="border-bottom:1px solid black"> |
− | |-
| + | <math>\begin{smallmatrix} |
− | | '''B''' || '''"i"''' || '''"i"'''
| + | \texttt{(} \ell_{10} \texttt{)} |
− | |}
| + | \\ |
− | <br> | + | \mathrm{No} ~ u |
| + | \\ |
| + | \mathrm{is} ~ \texttt{(} v \texttt{)} |
| + | \end{smallmatrix}</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black"> |
| + | <math>\begin{smallmatrix} |
| + | \texttt{(} \ell_{01} \texttt{)} |
| + | \\ |
| + | \mathrm{No} ~ \texttt{(} u \texttt{)} |
| + | \\ |
| + | \mathrm{is} ~ v |
| + | \end{smallmatrix}</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black"> |
| + | <math>\begin{smallmatrix} |
| + | \texttt{(} \ell_{00} \texttt{)} |
| + | \\ |
| + | \mathrm{No} ~ \texttt{(} u \texttt{)} |
| + | \\ |
| + | \mathrm{is} ~ \texttt{(} v \texttt{)} |
| + | \end{smallmatrix}</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black"> |
| + | <math>\begin{smallmatrix} |
| + | \ell_{00} |
| + | \\ |
| + | \mathrm{Some} ~ \texttt{(} u \texttt{)} |
| + | \\ |
| + | \mathrm{is} ~ \texttt{(} v \texttt{)} |
| + | \end{smallmatrix}</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black"> |
| + | <math>\begin{smallmatrix} |
| + | \ell_{01} |
| + | \\ |
| + | \mathrm{Some} ~ \texttt{(} u \texttt{)} |
| + | \\ |
| + | \mathrm{is} ~ v |
| + | \end{smallmatrix}</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black"> |
| + | <math>\begin{smallmatrix} |
| + | \ell_{10} |
| + | \\ |
| + | \mathrm{Some} ~ u |
| + | \\ |
| + | \mathrm{is} ~ \texttt{(} v \texttt{)} |
| + | \end{smallmatrix}</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black"> |
| + | <math>\begin{smallmatrix} |
| + | \ell_{11} |
| + | \\ |
| + | \mathrm{Some} ~ u |
| + | \\ |
| + | \mathrm{is} ~ v |
| + | \end{smallmatrix}</math></td></tr> |
| | | |
− | ===Triadic Relations=== | + | <tr> |
| + | <td><math>f_{0}</math></td> |
| + | <td><math>0000</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{(~)}</math></td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td></tr> |
| | | |
− | ====Algebraic Examples==== | + | <tr> |
| + | <td><math>f_{1}</math></td> |
| + | <td><math>0001</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td></tr> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" | + | <tr> |
− | |+ '''L'''<sub>0</sub> = {(''x'', ''y'', ''z'') ∈ '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 0}
| + | <td><math>f_{2}</math></td> |
− | |- style="background:paleturquoise"
| + | <td><math>0010</math></td> |
− | ! X !! Y !! Z
| + | <td style="border-right:1px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td> |
− | |-
| + | <td style="background:black; color:white">1</td> |
− | | '''0''' || '''0''' || '''0'''
| + | <td style="background:black; color:white">1</td> |
− | |-
| + | <td style="background:white; color:black">0</td> |
− | | '''0''' || '''1''' || '''1'''
| + | <td style="background:black; color:white">1</td> |
− | |-
| + | <td style="background:white; color:black">0</td> |
− | | '''1''' || '''0''' || '''1'''
| + | <td style="background:black; color:white">1</td> |
− | |-
| + | <td style="background:white; color:black">0</td> |
− | | '''1''' || '''1''' || '''0'''
| + | <td style="background:white; color:black">0</td></tr> |
− | |}
| + | |
− | <br> | + | <tr> |
| + | <td><math>f_{3}</math></td> |
| + | <td><math>0011</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{(} u \texttt{)}</math></td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{4}</math></td> |
| + | <td><math>0100</math></td> |
| + | <td style="border-right:1px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{5}</math></td> |
| + | <td><math>0101</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{(} v \texttt{)}</math></td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{6}</math></td> |
| + | <td><math>0110</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{7}</math></td> |
| + | <td><math>0111</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{8}</math></td> |
| + | <td><math>1000</math></td> |
| + | <td style="border-right:1px solid black"><math>u ~ v</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{9}</math></td> |
| + | <td><math>1001</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" | + | <tr> |
− | |+ '''L'''<sub>1</sub> = {(''x'', ''y'', ''z'') ∈ '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 1}
| + | <td><math>f_{10}</math></td> |
− | |- style="background:paleturquoise"
| + | <td><math>1010</math></td> |
− | ! X !! Y !! Z
| + | <td style="border-right:1px solid black"><math>v</math></td> |
− | |-
| + | <td style="background:white; color:black">0</td> |
− | | '''0''' || '''0''' || '''1'''
| + | <td style="background:black; color:white">1</td> |
− | |-
| + | <td style="background:white; color:black">0</td> |
− | | '''0''' || '''1''' || '''0'''
| + | <td style="background:black; color:white">1</td> |
− | |-
| + | <td style="background:white; color:black">0</td> |
− | | '''1''' || '''0''' || '''0'''
| + | <td style="background:black; color:white">1</td> |
− | |-
| + | <td style="background:white; color:black">0</td> |
− | | '''1''' || '''1''' || '''1'''
| + | <td style="background:black; color:white">1</td></tr> |
− | |}
| |
− | <br> | |
| | | |
− | ====Semiotic Examples==== | + | <tr> |
| + | <td><math>f_{11}</math></td> |
| + | <td><math>1011</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{12}</math></td> |
| + | <td><math>1100</math></td> |
| + | <td style="border-right:1px solid black"><math>u</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{13}</math></td> |
| + | <td><math>1101</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{14}</math></td> |
| + | <td><math>1110</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{15}</math></td> |
| + | <td><math>1111</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{((~))}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | </table> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
| |
− | |+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A
| |
− | |- style="background:paleturquoise"
| |
− | ! style="width:20%" | Object
| |
− | ! style="width:20%" | Sign
| |
− | ! style="width:20%" | Interpretant
| |
− | |-
| |
− | | '''A''' || '''"A"''' || '''"A"'''
| |
− | |-
| |
− | | '''A''' || '''"A"''' || '''"i"'''
| |
− | |-
| |
− | | '''A''' || '''"i"''' || '''"A"'''
| |
− | |-
| |
− | | '''A''' || '''"i"''' || '''"i"'''
| |
− | |-
| |
− | | '''B''' || '''"B"''' || '''"B"'''
| |
− | |-
| |
− | | '''B''' || '''"B"''' || '''"u"'''
| |
− | |-
| |
− | | '''B''' || '''"u"''' || '''"B"'''
| |
− | |-
| |
− | | '''B''' || '''"u"''' || '''"u"'''
| |
− | |}
| |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
| + | <table align="center" cellpadding="4" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:90%"> |
− | |+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B
| + | |
− | |- style="background:paleturquoise"
| + | <caption><font size="+2"><math>\text{Table 9.} ~~ \text{Simple Qualifiers of Propositions (Version 2)}</math></font></caption> |
− | ! style="width:20%" | Object
| + | |
− | ! style="width:20%" | Sign
| + | <tr> |
− | ! style="width:20%" | Interpretant
| + | <td width="4%" style="border-bottom:1px solid black" align="right"> |
− | |-
| + | <math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td> |
− | | '''A''' || '''"A"''' || '''"A"'''
| + | <td width="6%" style="border-bottom:1px solid black"> |
− | |-
| + | <math>\begin{matrix}1100\\1010\end{matrix}</math></td> |
− | | '''A''' || '''"A"''' || '''"u"'''
| + | <td width="10%" style="border-bottom:1px solid black; border-right:1px solid black"> |
− | |-
| + | <math>f</math></td> |
− | | '''A''' || '''"u"''' || '''"A"'''
| + | <td width="10%" style="border-bottom:1px solid black"> |
− | |-
| + | <math>\begin{smallmatrix} |
− | | '''A''' || '''"u"''' || '''"u"'''
| + | \texttt{(} \ell_{11} \texttt{)} |
− | |-
| + | \\ |
− | | '''B''' || '''"B"''' || '''"B"'''
| + | \mathrm{No} ~ u |
− | |-
| + | \\ |
− | | '''B''' || '''"B"''' || '''"i"'''
| + | \mathrm{is} ~ v |
− | |-
| + | \end{smallmatrix}</math></td> |
− | | '''B''' || '''"i"''' || '''"B"'''
| + | <td width="10%" style="border-bottom:1px solid black"> |
− | |-
| + | <math>\begin{smallmatrix} |
− | | '''B''' || '''"i"''' || '''"i"'''
| + | \texttt{(} \ell_{10} \texttt{)} |
− | |}
| + | \\ |
− | <br> | + | \mathrm{No} ~ u |
| + | \\ |
| + | \mathrm{is} ~ \texttt{(} v \texttt{)} |
| + | \end{smallmatrix}</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black"> |
| + | <math>\begin{smallmatrix} |
| + | \texttt{(} \ell_{01} \texttt{)} |
| + | \\ |
| + | \mathrm{No} ~ \texttt{(} u \texttt{)} |
| + | \\ |
| + | \mathrm{is} ~ v |
| + | \end{smallmatrix}</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black"> |
| + | <math>\begin{smallmatrix} |
| + | \texttt{(} \ell_{00} \texttt{)} |
| + | \\ |
| + | \mathrm{No} ~ \texttt{(} u \texttt{)} |
| + | \\ |
| + | \mathrm{is} ~ \texttt{(} v \texttt{)} |
| + | \end{smallmatrix}</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black"> |
| + | <math>\begin{smallmatrix} |
| + | \ell_{00} |
| + | \\ |
| + | \mathrm{Some} ~ \texttt{(} u \texttt{)} |
| + | \\ |
| + | \mathrm{is} ~ \texttt{(} v \texttt{)} |
| + | \end{smallmatrix}</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black"> |
| + | <math>\begin{smallmatrix} |
| + | \ell_{01} |
| + | \\ |
| + | \mathrm{Some} ~ \texttt{(} u \texttt{)} |
| + | \\ |
| + | \mathrm{is} ~ v |
| + | \end{smallmatrix}</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black"> |
| + | <math>\begin{smallmatrix} |
| + | \ell_{10} |
| + | \\ |
| + | \mathrm{Some} ~ u |
| + | \\ |
| + | \mathrm{is} ~ \texttt{(} v \texttt{)} |
| + | \end{smallmatrix}</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black"> |
| + | <math>\begin{smallmatrix} |
| + | \ell_{11} |
| + | \\ |
| + | \mathrm{Some} ~ u |
| + | \\ |
| + | \mathrm{is} ~ v |
| + | \end{smallmatrix}</math></td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-bottom:1px solid black"><math>f_{0}</math></td> |
| + | <td style="border-bottom:1px solid black"><math>0000</math></td> |
| + | <td style="border-bottom:1px solid black; border-right:1px solid black"><math>\texttt{(~)}</math></td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{1}</math></td> |
| + | <td><math>0001</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{2}</math></td> |
| + | <td><math>0010</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{4}</math></td> |
| + | <td><math>0100</math></td> |
| + | <td style="border-right:1px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-bottom:1px solid black"><math>f_{8}</math></td> |
| + | <td style="border-bottom:1px solid black"><math>1000</math></td> |
| + | <td style="border-bottom:1px solid black; border-right:1px solid black"><math>u ~ v</math></td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{3}</math></td> |
| + | <td><math>0011</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{(} u \texttt{)}</math></td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-bottom:1px solid black"><math>f_{12}</math></td> |
| + | <td style="border-bottom:1px solid black"><math>1100</math></td> |
| + | <td style="border-bottom:1px solid black; border-right:1px solid black"><math>u</math></td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{6}</math></td> |
| + | <td><math>0110</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-bottom:1px solid black"><math>f_{9}</math></td> |
| + | <td style="border-bottom:1px solid black"><math>1001</math></td> |
| + | <td style="border-bottom:1px solid black; border-right:1px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td></tr> |
| | | |
− | ===Dyadic Projections=== | + | <tr> |
| + | <td><math>f_{5}</math></td> |
| + | <td><math>0101</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{(} v \texttt{)}</math></td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td></tr> |
| | | |
− | {| cellpadding="4"
| + | <tr> |
− | | width="20px" |
| + | <td style="border-bottom:1px solid black"><math>f_{10}</math></td> |
− | | '''L'''<sub>OS</sub>
| + | <td style="border-bottom:1px solid black"><math>1010</math></td> |
− | | =
| + | <td style="border-bottom:1px solid black; border-right:1px solid black"><math>v</math></td> |
− | | ''proj''<sub>OS</sub>('''L''')
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
− | | =
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
− | | { (''o'', ''s'') ∈ '''O''' × '''S''' : (''o'', ''s'', ''i'') ∈ '''L''' for some ''i'' ∈ '''I''' }
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
− | |-
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
− | | width="20px" |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
− | | '''L'''<sub>SO</sub>
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
− | | =
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
− | | ''proj''<sub>SO</sub>('''L''')
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td></tr> |
− | | =
| + | |
− | | { (''s'', ''o'') ∈ '''S''' × '''O''' : (''o'', ''s'', ''i'') ∈ '''L''' for some ''i'' ∈ '''I''' }
| + | <tr> |
− | |-
| + | <td><math>f_{7}</math></td> |
− | | width="20px" |
| + | <td><math>0111</math></td> |
− | | '''L'''<sub>IS</sub>
| + | <td style="border-right:1px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td> |
− | | =
| + | <td style="background:black; color:white">1</td> |
− | | ''proj''<sub>IS</sub>('''L''')
| + | <td style="background:white; color:black">0</td> |
− | | =
| + | <td style="background:white; color:black">0</td> |
− | | { (''i'', ''s'') ∈ '''I''' × '''S''' : (''o'', ''s'', ''i'') ∈ '''L''' for some ''o'' ∈ '''O''' }
| + | <td style="background:white; color:black">0</td> |
− | |-
| + | <td style="background:black; color:white">1</td> |
− | | width="20px" |
| + | <td style="background:black; color:white">1</td> |
− | | '''L'''<sub>SI</sub>
| + | <td style="background:black; color:white">1</td> |
− | | =
| + | <td style="background:white; color:black">0</td></tr> |
− | | ''proj''<sub>SI</sub>('''L''')
| + | |
− | | =
| + | <tr> |
− | | { (''s'', ''i'') ∈ '''S''' × '''I''' : (''o'', ''s'', ''i'') ∈ '''L''' for some ''o'' ∈ '''O''' }
| + | <td><math>f_{11}</math></td> |
− | |-
| + | <td><math>1011</math></td> |
− | | width="20px" |
| + | <td style="border-right:1px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td> |
− | | '''L'''<sub>OI</sub>
| + | <td style="background:white; color:black">0</td> |
− | | =
| + | <td style="background:black; color:white">1</td> |
− | | ''proj''<sub>OI</sub>('''L''')
| + | <td style="background:white; color:black">0</td> |
− | | =
| + | <td style="background:white; color:black">0</td> |
− | | { (''o'', ''i'') ∈ '''O''' × '''I''' : (''o'', ''s'', ''i'') ∈ '''L''' for some ''s'' ∈ '''S''' }
| + | <td style="background:black; color:white">1</td> |
− | |-
| + | <td style="background:black; color:white">1</td> |
− | | width="20px" |
| + | <td style="background:white; color:black">0</td> |
− | | '''L'''<sub>IO</sub>
| + | <td style="background:black; color:white">1</td></tr> |
− | | =
| |
− | | ''proj''<sub>IO</sub>('''L''')
| |
− | | =
| |
− | | { (''i'', ''o'') ∈ '''I''' × '''O''' : (''o'', ''s'', ''i'') ∈ '''L''' for some ''s'' ∈ '''S''' }
| |
− | |}
| |
− | <br> | |
| | | |
− | ====Method 1 : Subtitles as Captions==== | + | <tr> |
| + | <td><math>f_{13}</math></td> |
| + | <td><math>1101</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-bottom:1px solid black"><math>f_{14}</math></td> |
| + | <td style="border-bottom:1px solid black"><math>1110</math></td> |
| + | <td style="border-bottom:1px solid black; border-right:1px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{15}</math></td> |
| + | <td><math>1111</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{((~))}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | </table> |
| | | |
− | {| align="center" style="width:90%"
| |
− | |
| |
− | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
| |
− | |+ ''proj''<sub>OS</sub>('''L'''<sub>A</sub>)
| |
− | |- style="background:paleturquoise"
| |
− | ! style="width:50%" | Object
| |
− | ! style="width:50%" | Sign
| |
− | |-
| |
− | | '''A''' || '''"A"'''
| |
− | |-
| |
− | | '''A''' || '''"i"'''
| |
− | |-
| |
− | | '''B''' || '''"B"'''
| |
− | |-
| |
− | | '''B''' || '''"u"'''
| |
− | |}
| |
− | |
| |
− | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
| |
− | |+ ''proj''<sub>OS</sub>('''L'''<sub>B</sub>)
| |
− | |- style="background:paleturquoise"
| |
− | ! style="width:50%" | Object
| |
− | ! style="width:50%" | Sign
| |
− | |-
| |
− | | '''A''' || '''"A"'''
| |
− | |-
| |
− | | '''A''' || '''"u"'''
| |
− | |-
| |
− | | '''B''' || '''"B"'''
| |
− | |-
| |
− | | '''B''' || '''"i"'''
| |
− | |}
| |
− | |}
| |
| <br> | | <br> |
| | | |
− | {| align="center" style="width:90%"
| + | <table align="center" cellpadding="4" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:90%"> |
− | | | + | |
− | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" | + | <caption><font size="+2"><math>\text{Table 10.} ~~ \text{Relation of Quantifiers to Higher Order Propositions}</math></font></caption> |
− | |+ ''proj''<sub>SI</sub>('''L'''<sub>A</sub>) | + | |
| + | <tr> |
| + | <td style="border-bottom:1px solid black"><math>\mathrm{Mnemonic}</math></td> |
| + | <td style="border-bottom:1px solid black"><math>\mathrm{Category}</math></td> |
| + | <td style="border-bottom:1px solid black"><math>\mathrm{Classical~Form}</math></td> |
| + | <td style="border-bottom:1px solid black"><math>\mathrm{Alternate~Form}</math></td> |
| + | <td style="border-bottom:1px solid black"><math>\mathrm{Symmetric~Form}</math></td> |
| + | <td style="border-bottom:1px solid black"><math>\mathrm{Operator}</math></td></tr> |
| + | |
| + | <tr> |
| + | <td><math>\begin{matrix} |
| + | \mathrm{E} |
| + | \\ |
| + | \mathrm{Exclusive} |
| + | \end{matrix}</math></td> |
| + | <td><math>\begin{matrix} |
| + | \mathrm{Universal} |
| + | \\ |
| + | \mathrm{Negative} |
| + | \end{matrix}</math></td> |
| + | <td><math>\mathrm{All} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td> |
| + | <td> </td> |
| + | <td><math>\mathrm{No} ~ u ~ \mathrm{is} ~ v</math></td> |
| + | <td><math>\texttt{(} \ell_{11} \texttt{)}</math></td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-bottom:1px solid black"> |
| + | <math>\begin{matrix} |
| + | \mathrm{A} |
| + | \\ |
| + | \mathrm{Absolute} |
| + | \end{matrix}</math></td> |
| + | <td style="border-bottom:1px solid black"> |
| + | <math>\begin{matrix} |
| + | \mathrm{Universal} |
| + | \\ |
| + | \mathrm{Affirmative} |
| + | \end{matrix}</math></td> |
| + | <td style="border-bottom:1px solid black"><math>\mathrm{All} ~ u ~ \mathrm{is} ~ v</math></td> |
| + | <td style="border-bottom:1px solid black"> </td> |
| + | <td style="border-bottom:1px solid black"><math>\mathrm{No} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td> |
| + | <td style="border-bottom:1px solid black"><math>\texttt{(} \ell_{10} \texttt{)}</math></td></tr> |
| + | |
| + | <tr> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td><math>\mathrm{All} ~ v ~ \mathrm{is} ~ u</math></td> |
| + | <td><math>\mathrm{No} ~ v ~ \mathrm{is} ~ \texttt{(} u \texttt{)}</math></td> |
| + | <td><math>\mathrm{No} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td> |
| + | <td><math>\texttt{(} \ell_{01} \texttt{)}</math></td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-bottom:1px solid black"> </td> |
| + | <td style="border-bottom:1px solid black"> </td> |
| + | <td style="border-bottom:1px solid black"><math>\mathrm{All} ~ \texttt{(} v \texttt{)} ~ \mathrm{is} ~ u</math></td> |
| + | <td style="border-bottom:1px solid black"><math>\mathrm{No} ~ \texttt{(} v \texttt{)} ~ \mathrm{is} ~ \texttt{(} u \texttt{)}</math></td> |
| + | <td style="border-bottom:1px solid black"><math>\mathrm{No} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td> |
| + | <td style="border-bottom:1px solid black"><math>\texttt{(} \ell_{00} \texttt{)}</math></td></tr> |
| + | |
| + | <tr> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td> |
| + | <td> </td> |
| + | <td><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td> |
| + | <td><math>\ell_{00}</math></td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-bottom:1px solid black"> </td> |
| + | <td style="border-bottom:1px solid black"> </td> |
| + | <td style="border-bottom:1px solid black"><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td> |
| + | <td style="border-bottom:1px solid black"> </td> |
| + | <td style="border-bottom:1px solid black"><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td> |
| + | <td style="border-bottom:1px solid black"><math>\ell_{01}</math></td></tr> |
| + | |
| + | <tr> |
| + | <td><math>\begin{matrix} |
| + | \mathrm{O} |
| + | \\ |
| + | \mathrm{Obtrusive} |
| + | \end{matrix}</math></td> |
| + | <td><math>\begin{matrix} |
| + | \mathrm{Particular} |
| + | \\ |
| + | \mathrm{Negative} |
| + | \end{matrix}</math></td> |
| + | <td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td> |
| + | <td> </td> |
| + | <td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td> |
| + | <td><math>\ell_{10}</math></td></tr> |
| + | |
| + | <tr> |
| + | <td><math>\begin{matrix} |
| + | \mathrm{I} |
| + | \\ |
| + | \mathrm{Indefinite} |
| + | \end{matrix}</math></td> |
| + | <td><math>\begin{matrix} |
| + | \mathrm{Particular} |
| + | \\ |
| + | \mathrm{Affirmative} |
| + | \end{matrix}</math></td> |
| + | <td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ v</math></td> |
| + | <td> </td> |
| + | <td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ v</math></td> |
| + | <td><math>\ell_{11}</math></td></tr> |
| + | |
| + | </table> |
| + | |
| + | <br> |
| + | |
| + | ==Inquiry Driven Systems== |
| + | |
| + | ===Table 1. Sign Relation of Interpreter ''A''=== |
| + | |
| + | <pre> |
| + | 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 |
| + | </pre> |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |
| + | |+ Table 1. Sign Relation of Interpreter ''A'' |
| |- style="background:paleturquoise" | | |- style="background:paleturquoise" |
− | ! style="width:50%" | Sign | + | ! style="width:20%" | Object |
− | ! style="width:50%" | Interpretant | + | ! style="width:20%" | Sign |
| + | ! style="width:20%" | Interpretant |
| |- | | |- |
− | | '''"A"''' || '''"A"''' | + | | ''A'' || "A" || "A" |
| |- | | |- |
− | | '''"A"''' || '''"i"''' | + | | ''A'' || "A" || "i" |
| |- | | |- |
− | | '''"i"''' || '''"A"''' | + | | ''A'' || "i" || "A" |
| |- | | |- |
− | | '''"i"''' || '''"i"''' | + | | ''A'' || "i" || "i" |
| |- | | |- |
− | | '''"B"''' || '''"B"''' | + | | ''B'' || "B" || "B" |
| |- | | |- |
− | | '''"B"''' || '''"u"''' | + | | ''B'' || "B" || "u" |
| |- | | |- |
− | | '''"u"''' || '''"B"''' | + | | ''B'' || "u" || "B" |
| |- | | |- |
− | | '''"u"''' || '''"u"''' | + | | ''B'' || "u" || "u" |
| |} | | |} |
− | | | + | <br> |
− | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" | + | |
− | |+ ''proj''<sub>SI</sub>('''L'''<sub>B</sub>) | + | ===Table 2. Sign Relation of Interpreter ''B''=== |
| + | |
| + | <pre> |
| + | 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 |
| + | </pre> |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |
| + | |+ Table 2. Sign Relation of Interpreter ''B'' |
| |- style="background:paleturquoise" | | |- style="background:paleturquoise" |
− | ! style="width:50%" | Sign | + | ! style="width:20%" | Object |
− | ! style="width:50%" | Interpretant | + | ! style="width:20%" | Sign |
| + | ! style="width:20%" | Interpretant |
| |- | | |- |
− | | '''"A"''' || '''"A"''' | + | | ''A'' || "A" || "A" |
| |- | | |- |
− | | '''"A"''' || '''"u"''' | + | | ''A'' || "A" || "u" |
| |- | | |- |
− | | '''"u"''' || '''"A"''' | + | | ''A'' || "u" || "A" |
| |- | | |- |
− | | '''"u"''' || '''"u"''' | + | | ''A'' || "u" || "u" |
| |- | | |- |
− | | '''"B"''' || '''"B"''' | + | | ''B'' || "B" || "B" |
| |- | | |- |
− | | '''"B"''' || '''"i"''' | + | | ''B'' || "B" || "i" |
| |- | | |- |
− | | '''"i"''' || '''"B"''' | + | | ''B'' || "i" || "B" |
| |- | | |- |
− | | '''"i"''' || '''"i"''' | + | | ''B'' || "i" || "i" |
− | |}
| |
| |} | | |} |
| <br> | | <br> |
| | | |
− | {| align="center" style="width:90%" | + | ===Table 3. Semiotic Partition of Interpreter ''A''=== |
| + | |
| + | <pre> |
| + | Table 3. A's Semiotic Partition |
| + | o-------------------------------o |
| + | | "A" "i" | |
| + | o-------------------------------o |
| + | | "u" "B" | |
| + | o-------------------------------o |
| + | </pre> |
| + | |
| + | {| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |
| + | |+ Table 3. Semiotic Partition of Interpreter ''A'' |
| | | | | |
− | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" | + | {| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
− | |+ ''proj''<sub>OI</sub>('''L'''<sub>A</sub>) | + | | width="50%" | "A" |
− | |- style="background:paleturquoise"
| + | | width="50%" | "i" |
− | ! style="width:50%" | Object
| + | |} |
− | ! style="width:50%" | Interpretant
| |
− | |-
| |
− | | '''A''' || '''"A"'''
| |
− | |- | |
− | | '''A''' || '''"i"'''
| |
| |- | | |- |
− | | '''B''' || '''"B"'''
| |
− | |-
| |
− | | '''B''' || '''"u"'''
| |
− | |}
| |
| | | | | |
− | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" | + | {| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
− | |+ ''proj''<sub>OI</sub>('''L'''<sub>B</sub>) | + | | width="50%" | "u" |
− | |- style="background:paleturquoise"
| + | | width="50%" | "B" |
− | ! style="width:50%" | Object
| |
− | ! style="width:50%" | Interpretant
| |
− | |-
| |
− | | '''A''' || '''"A"'''
| |
− | |- | |
− | | '''A''' || '''"u"'''
| |
− | |- | |
− | | '''B''' || '''"B"'''
| |
− | |-
| |
− | | '''B''' || '''"i"'''
| |
| |} | | |} |
| |} | | |} |
| <br> | | <br> |
| | | |
− | ====Method 2 : Subtitles as Top Rows==== | + | ===Table 4. Semiotic Partition of Interpreter ''B''=== |
| + | |
| + | <pre> |
| + | Table 4. B's Semiotic Partition |
| + | o---------------o---------------o |
| + | | "A" | "i" | |
| + | | | | |
| + | | "u" | "B" | |
| + | o---------------o---------------o |
| + | </pre> |
| | | |
− | {| align="center" style="width:90%" | + | {| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |
− | | align="center" style="width:45%" | ''proj''<sub>OS</sub>('''L'''<sub>A</sub>)
| + | |+ Table 4. Semiotic Partition of Interpreter ''B'' |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" | + | | |
− | |- style="background:paleturquoise" | + | {| align="center" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:50%" |
− | ! style="width:50%" | Object
| + | | "A" |
− | ! style="width:50%" | Sign
| |
| |- | | |- |
− | | '''A''' || '''"A"''' | + | | "u" |
− | |-
| |
− | | '''A''' || '''"i"'''
| |
− | |-
| |
− | | '''B''' || '''"B"'''
| |
− | |-
| |
− | | '''B''' || '''"u"'''
| |
| |} | | |} |
− | | align="center" style="width:45%" | ''proj''<sub>OS</sub>('''L'''<sub>B</sub>) | + | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" | + | {| align="center" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:50%" |
− | |- style="background:paleturquoise" | + | | "i" |
− | ! style="width:50%" | Object
| |
− | ! style="width:50%" | Sign
| |
| |- | | |- |
− | | '''A''' || '''"A"''' | + | | "B" |
− | |-
| |
− | | '''A''' || '''"u"'''
| |
− | |-
| |
− | | '''B''' || '''"B"'''
| |
− | |-
| |
− | | '''B''' || '''"i"'''
| |
| |} | | |} |
| |} | | |} |
| <br> | | <br> |
| | | |
− | {| align="center" style="width:90%"
| + | ===Table 5. Alignments of Capacities=== |
− | | align="center" style="width:45%" | ''proj''<sub>SI</sub>('''L'''<sub>A</sub>) | + | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" | + | <pre> |
− | |- style="background:paleturquoise" | + | Table 5. Alignments of Capacities |
− | ! style="width:50%" | Sign
| + | o-------------------o-----------------------------o |
− | ! style="width:50%" | Interpretant
| + | | Formal | Formative | |
− | |- | + | o-------------------o-----------------------------o |
− | | '''"A"''' || '''"A"'''
| + | | Objective | Instrumental | |
| + | | Passive | Active | |
| + | o-------------------o--------------o--------------o |
| + | | Afforded | Possessed | Exercised | |
| + | o-------------------o--------------o--------------o |
| + | </pre> |
| + | |
| + | ===Table 6. Alignments of Capacities in Aristotle=== |
| + | |
| + | <pre> |
| + | 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 |
| + | </pre> |
| + | |
| + | ===Table 7. Synthesis of Alignments=== |
| + | |
| + | <pre> |
| + | 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 |
| + | </pre> |
| + | |
| + | ===Table 8. Boolean Product=== |
| + | |
| + | <pre> |
| + | 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 |
| + | </pre> |
| + | |
| + | ===Table 9. Boolean Sum=== |
| + | |
| + | <pre> |
| + | 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 |
| + | </pre> |
| + | |
| + | ==Logical Tables== |
| + | |
| + | ===Table Templates=== |
| + | |
| + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%" |
| + | |+ Table 1. Two Variable Template |
| + | |- style="background:paleturquoise" |
| + | | |
| + | {| align="right" style="background:paleturquoise; text-align:right" |
| + | | u : |
| |- | | |- |
− | | '''"A"''' || '''"i"''' | + | | v : |
| + | |} |
| + | | |
| + | {| style="background:paleturquoise" |
| + | | 1 1 0 0 |
| + | |- |
| + | | 1 0 1 0 |
| + | |} |
| + | | |
| + | {| style="background:paleturquoise" |
| + | | f |
| + | |- |
| + | | |
| + | |} |
| + | | |
| + | {| style="background:paleturquoise" |
| + | | f |
| |- | | |- |
− | | '''"i"''' || '''"A"''' | + | | |
| + | |} |
| + | | |
| + | {| style="background:paleturquoise" |
| + | | f |
| |- | | |- |
− | | '''"i"''' || '''"i"''' | + | | |
| + | |} |
| |- | | |- |
− | | '''"B"''' || '''"B"''' | + | | |
| + | {| cellpadding="2" style="background:lightcyan" |
| + | | f<sub>0</sub> |
| |- | | |- |
− | | '''"B"''' || '''"u"''' | + | | f<sub>1</sub> |
| |- | | |- |
− | | '''"u"''' || '''"B"''' | + | | f<sub>2</sub> |
| |- | | |- |
− | | '''"u"''' || '''"u"''' | + | | f<sub>3</sub> |
− | |}
| + | |- |
− | | align="center" style="width:45%" | ''proj''<sub>SI</sub>('''L'''<sub>B</sub>)
| + | | f<sub>4</sub> |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
| |
− | |- style="background:paleturquoise"
| |
− | ! style="width:50%" | Sign
| |
− | ! style="width:50%" | Interpretant
| |
| |- | | |- |
− | | '''"A"''' || '''"A"''' | + | | f<sub>5</sub> |
| |- | | |- |
− | | '''"A"''' || '''"u"''' | + | | f<sub>6</sub> |
| |- | | |- |
− | | '''"u"''' || '''"A"''' | + | | f<sub>7</sub> |
| + | |} |
| + | | |
| + | {| cellpadding="2" style="background:lightcyan" |
| + | | 0 0 0 0 |
| |- | | |- |
− | | '''"u"''' || '''"u"''' | + | | 0 0 0 1 |
| |- | | |- |
− | | '''"B"''' || '''"B"''' | + | | 0 0 1 0 |
| |- | | |- |
− | | '''"B"''' || '''"i"''' | + | | 0 0 1 1 |
| |- | | |- |
− | | '''"i"''' || '''"B"''' | + | | 0 1 0 0 |
| |- | | |- |
− | | '''"i"''' || '''"i"''' | + | | 0 1 0 1 |
− | |}
| |
− | |}
| |
− | <br>
| |
− | | |
− | {| align="center" style="width:90%"
| |
− | | align="center" style="width:45%" | ''proj''<sub>OI</sub>('''L'''<sub>A</sub>)
| |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
| |
− | |- style="background:paleturquoise"
| |
− | ! style="width:50%" | Object
| |
− | ! style="width:50%" | Interpretant
| |
| |- | | |- |
− | | '''A''' || '''"A"''' | + | | 0 1 1 0 |
| |- | | |- |
− | | '''A''' || '''"i"''' | + | | 0 1 1 1 |
| + | |} |
| + | | |
| + | {| cellpadding="2" style="background:lightcyan" |
| + | | f<sub>0</sub> |
| |- | | |- |
− | | '''B''' || '''"B"''' | + | | f<sub>1</sub> |
| |- | | |- |
− | | '''B''' || '''"u"''' | + | | f<sub>2</sub> |
− | |}
| |
− | | align="center" style="width:45%" | ''proj''<sub>OI</sub>('''L'''<sub>B</sub>)
| |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
| |
− | |- style="background:paleturquoise"
| |
− | ! style="width:50%" | Object
| |
− | ! style="width:50%" | Interpretant
| |
| |- | | |- |
− | | '''A''' || '''"A"''' | + | | f<sub>3</sub> |
| |- | | |- |
− | | '''A''' || '''"u"''' | + | | f<sub>4</sub> |
| |- | | |- |
− | | '''B''' || '''"B"''' | + | | f<sub>5</sub> |
| |- | | |- |
− | | '''B''' || '''"i"''' | + | | f<sub>6</sub> |
| + | |- |
| + | | f<sub>7</sub> |
| |} | | |} |
− | |} | + | | |
− | <br>
| + | {| cellpadding="2" style="background:lightcyan" |
− | | + | | f<sub>0</sub> |
− | ===Relation Reduction===
| |
− | | |
− | ====Method 1 : Subtitles as Captions====
| |
− | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" | |
− | |+ '''L'''<sub>0</sub> = {(''x'', ''y'', ''z'') ∈ '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 0} | |
− | |- style="background:paleturquoise"
| |
− | ! X !! Y !! Z
| |
| |- | | |- |
− | | '''0''' || '''0''' || '''0''' | + | | f<sub>1</sub> |
| |- | | |- |
− | | '''0''' || '''1''' || '''1''' | + | | f<sub>2</sub> |
| |- | | |- |
− | | '''1''' || '''0''' || '''1''' | + | | f<sub>3</sub> |
| |- | | |- |
− | | '''1''' || '''1''' || '''0''' | + | | f<sub>4</sub> |
− | |}
| |
− | <br>
| |
− | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
| |
− | |+ '''L'''<sub>1</sub> = {(''x'', ''y'', ''z'') ∈ '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 1}
| |
− | |- style="background:paleturquoise"
| |
− | ! X !! Y !! Z
| |
− | |-
| |
− | | '''0''' || '''0''' || '''1'''
| |
| |- | | |- |
− | | '''0''' || '''1''' || '''0''' | + | | f<sub>5</sub> |
| |- | | |- |
− | | '''1''' || '''0''' || '''0''' | + | | f<sub>6</sub> |
| |- | | |- |
− | | '''1''' || '''1''' || '''1''' | + | | f<sub>7</sub> |
| |} | | |} |
− | <br>
| |
− |
| |
− | {| align="center" style="width:90%"
| |
| | | | | |
− | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" | + | {| cellpadding="2" style="background:lightcyan" |
− | |+ proj<sub>''XY''</sub>('''L'''<sub>0</sub>) | + | | f<sub>0</sub> |
− | |- style="background:paleturquoise"
| + | |- |
− | ! X !! Y
| + | | f<sub>1</sub> |
| |- | | |- |
− | | '''0''' || '''0''' | + | | f<sub>2</sub> |
| |- | | |- |
− | | '''0''' || '''1''' | + | | f<sub>3</sub> |
| |- | | |- |
− | | '''1''' || '''0''' | + | | f<sub>4</sub> |
| |- | | |- |
− | | '''1''' || '''1''' | + | | f<sub>5</sub> |
| + | |- |
| + | | f<sub>6</sub> |
| + | |- |
| + | | f<sub>7</sub> |
| |} | | |} |
| + | |- |
| | | | | |
− | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" | + | {| cellpadding="2" style="background:lightcyan" |
− | |+ proj<sub>''XZ''</sub>('''L'''<sub>0</sub>) | + | | f<sub>8</sub> |
− | |- style="background:paleturquoise" | + | |- |
− | ! X !! Z
| + | | f<sub>9</sub> |
| + | |- |
| + | | f<sub>10</sub> |
| + | |- |
| + | | f<sub>11</sub> |
| |- | | |- |
− | | '''0''' || '''0''' | + | | f<sub>12</sub> |
| |- | | |- |
− | | '''0''' || '''1''' | + | | f<sub>13</sub> |
| |- | | |- |
− | | '''1''' || '''1''' | + | | f<sub>14</sub> |
| |- | | |- |
− | | '''1''' || '''0''' | + | | f<sub>15</sub> |
| |} | | |} |
| | | | | |
− | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" | + | {| cellpadding="2" style="background:lightcyan" |
− | |+ proj<sub>''YZ''</sub>('''L'''<sub>0</sub>) | + | | 1 0 0 0 |
− | |- style="background:paleturquoise" | + | |- |
− | ! Y !! Z
| + | | 1 0 0 1 |
| + | |- |
| + | | 1 0 1 0 |
| + | |- |
| + | | 1 0 1 1 |
| |- | | |- |
− | | '''0''' || '''0''' | + | | 1 1 0 0 |
| |- | | |- |
− | | '''1''' || '''1''' | + | | 1 1 0 1 |
| |- | | |- |
− | | '''0''' || '''1''' | + | | 1 1 1 0 |
| |- | | |- |
− | | '''1''' || '''0''' | + | | 1 1 1 1 |
| |} | | |} |
− | |}
| |
− | <br>
| |
− |
| |
− | {| align="center" style="width:90%"
| |
| | | | | |
− | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" | + | {| cellpadding="2" style="background:lightcyan" |
− | |+ proj<sub>''XY''</sub>('''L'''<sub>1</sub>) | + | | f<sub>8</sub> |
− | |- style="background:paleturquoise" | + | |- |
− | ! X !! Y
| + | | f<sub>9</sub> |
| + | |- |
| + | | f<sub>10</sub> |
| |- | | |- |
− | | '''0''' || '''0''' | + | | f<sub>11</sub> |
| |- | | |- |
− | | '''0''' || '''1''' | + | | f<sub>12</sub> |
| |- | | |- |
− | | '''1''' || '''0''' | + | | f<sub>13</sub> |
| |- | | |- |
− | | '''1''' || '''1''' | + | | f<sub>14</sub> |
| + | |- |
| + | | f<sub>15</sub> |
| |} | | |} |
| | | | | |
− | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" | + | {| cellpadding="2" style="background:lightcyan" |
− | |+ proj<sub>''XZ''</sub>('''L'''<sub>1</sub>) | + | | f<sub>8</sub> |
− | |- style="background:paleturquoise" | + | |- |
− | ! X !! Z
| + | | f<sub>9</sub> |
| + | |- |
| + | | f<sub>10</sub> |
| + | |- |
| + | | f<sub>11</sub> |
| |- | | |- |
− | | '''0''' || '''1''' | + | | f<sub>12</sub> |
| |- | | |- |
− | | '''0''' || '''0''' | + | | f<sub>13</sub> |
| |- | | |- |
− | | '''1''' || '''0''' | + | | f<sub>14</sub> |
| |- | | |- |
− | | '''1''' || '''1''' | + | | f<sub>15</sub> |
| |} | | |} |
| | | | | |
− | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" | + | {| cellpadding="2" style="background:lightcyan" |
− | |+ proj<sub>''YZ''</sub>('''L'''<sub>1</sub>) | + | | f<sub>8</sub> |
− | |- style="background:paleturquoise" | + | |- |
− | ! Y !! Z
| + | | f<sub>9</sub> |
| + | |- |
| + | | f<sub>10</sub> |
| + | |- |
| + | | f<sub>11</sub> |
| |- | | |- |
− | | '''0''' || '''1''' | + | | f<sub>12</sub> |
| |- | | |- |
− | | '''1''' || '''0''' | + | | f<sub>13</sub> |
| |- | | |- |
− | | '''0''' || '''0''' | + | | f<sub>14</sub> |
| |- | | |- |
− | | '''1''' || '''1''' | + | | f<sub>15</sub> |
| |} | | |} |
| |} | | |} |
| <br> | | <br> |
| | | |
− | {| align="center" cellpadding="4" style="text-align:center; width:90%"
| + | <font face="courier new"> |
− | | proj<sub>''XY''</sub>('''L'''<sub>0</sub>) = proj<sub>''XY''</sub>('''L'''<sub>1</sub>)
| + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%" |
− | | proj<sub>''XZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''XZ''</sub>('''L'''<sub>1</sub>)
| + | |+ Table 2. Two Variable Template |
− | | proj<sub>''YZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''YZ''</sub>('''L'''<sub>1</sub>)
| |
− | |}
| |
− | <br>
| |
− | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" | |
− | |+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A | |
| |- style="background:paleturquoise" | | |- style="background:paleturquoise" |
− | ! style="width:20%" | Object
| + | | |
− | ! style="width:20%" | Sign
| + | {| align="right" style="background:paleturquoise; text-align:right" |
− | ! style="width:20%" | Interpretant
| + | | u : |
| |- | | |- |
− | | '''A''' || '''"A"''' || '''"A"''' | + | | v : |
| + | |} |
| + | | |
| + | {| style="background:paleturquoise" |
| + | | 1100 |
| |- | | |- |
− | | '''A''' || '''"A"''' || '''"i"''' | + | | 1010 |
| + | |} |
| + | | |
| + | {| style="background:paleturquoise" |
| + | | f |
| + | |- |
| + | | |
| + | |} |
| + | | |
| + | {| style="background:paleturquoise" |
| + | | f |
| |- | | |- |
− | | '''A''' || '''"i"''' || '''"A"''' | + | | |
| + | |} |
| + | | |
| + | {| style="background:paleturquoise" |
| + | | f |
| + | |- |
| + | | |
| + | |} |
| + | |- |
| + | | |
| + | {| cellpadding="2" style="background:lightcyan" |
| + | | f<sub>0</sub> |
| + | |- |
| + | | f<sub>1</sub> |
| + | |- |
| + | | f<sub>2</sub> |
| |- | | |- |
− | | '''A''' || '''"i"''' || '''"i"''' | + | | f<sub>3</sub> |
| |- | | |- |
− | | '''B''' || '''"B"''' || '''"B"''' | + | | f<sub>4</sub> |
| |- | | |- |
− | | '''B''' || '''"B"''' || '''"u"''' | + | | f<sub>5</sub> |
| |- | | |- |
− | | '''B''' || '''"u"''' || '''"B"''' | + | | f<sub>6</sub> |
| |- | | |- |
− | | '''B''' || '''"u"''' || '''"u"''' | + | | f<sub>7</sub> |
| |} | | |} |
− | <br>
| + | | |
− | | + | {| cellpadding="2" style="background:lightcyan" |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" | + | | 0000 |
− | |+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B
| |
− | |- style="background:paleturquoise"
| |
− | ! style="width:20%" | Object
| |
− | ! style="width:20%" | Sign
| |
− | ! style="width:20%" | Interpretant
| |
− | |- | |
− | | '''A''' || '''"A"''' || '''"A"'''
| |
| |- | | |- |
− | | '''A''' || '''"A"''' || '''"u"''' | + | | 0001 |
| |- | | |- |
− | | '''A''' || '''"u"''' || '''"A"''' | + | | 0010 |
| |- | | |- |
− | | '''A''' || '''"u"''' || '''"u"''' | + | | 0011 |
| |- | | |- |
− | | '''B''' || '''"B"''' || '''"B"''' | + | | 0100 |
| |- | | |- |
− | | '''B''' || '''"B"''' || '''"i"''' | + | | 0101 |
| |- | | |- |
− | | '''B''' || '''"i"''' || '''"B"''' | + | | 0110 |
| |- | | |- |
− | | '''B''' || '''"i"''' || '''"i"''' | + | | 0111 |
| |} | | |} |
− | <br>
| + | | |
− | | + | {| cellpadding="2" style="background:lightcyan" |
− | {| align="center" style="width:90%" | + | | () |
− | | | + | |- |
− | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
| + | | (u)(v) |
− | |+ proj<sub>''XY''</sub>('''L'''<sub>A</sub>) | + | |- |
− | |- style="background:paleturquoise" | + | | (u) v |
− | ! style="width:50%" | Object
| + | |- |
− | ! style="width:50%" | Sign
| + | | (u) |
| |- | | |- |
− | | '''A''' || '''"A"''' | + | | u (v) |
| |- | | |- |
− | | '''A''' || '''"i"''' | + | | (v) |
| |- | | |- |
− | | '''B''' || '''"B"''' | + | | (u, v) |
| |- | | |- |
− | | '''B''' || '''"u"''' | + | | (u v) |
| |} | | |} |
| | | | | |
− | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" | + | {| cellpadding="2" style="background:lightcyan" |
− | |+ proj<sub>''XZ''</sub>('''L'''<sub>A</sub>) | + | | f<sub>0</sub> |
− | |- style="background:paleturquoise" | + | |- |
− | ! style="width:50%" | Object
| + | | f<sub>1</sub> |
− | ! style="width:50%" | Interpretant
| + | |- |
| + | | f<sub>2</sub> |
| + | |- |
| + | | f<sub>3</sub> |
| |- | | |- |
− | | '''A''' || '''"A"''' | + | | f<sub>4</sub> |
| |- | | |- |
− | | '''A''' || '''"i"''' | + | | f<sub>5</sub> |
| |- | | |- |
− | | '''B''' || '''"B"''' | + | | f<sub>6</sub> |
| |- | | |- |
− | | '''B''' || '''"u"''' | + | | f<sub>7</sub> |
| |} | | |} |
| | | | | |
− | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" | + | {| cellpadding="2" style="background:lightcyan" |
− | |+ proj<sub>''YZ''</sub>('''L'''<sub>A</sub>) | + | | f<sub>0</sub> |
− | |- style="background:paleturquoise"
| |
− | ! style="width:50%" | Sign
| |
− | ! style="width:50%" | Interpretant
| |
| |- | | |- |
− | | '''"A"''' || '''"A"''' | + | | f<sub>1</sub> |
| |- | | |- |
− | | '''"A"''' || '''"i"''' | + | | f<sub>2</sub> |
| |- | | |- |
− | | '''"i"''' || '''"A"''' | + | | f<sub>3</sub> |
| |- | | |- |
− | | '''"i"''' || '''"i"''' | + | | f<sub>4</sub> |
| |- | | |- |
− | | '''"B"''' || '''"B"''' | + | | f<sub>5</sub> |
| |- | | |- |
− | | '''"B"''' || '''"u"''' | + | | f<sub>6</sub> |
| |- | | |- |
− | | '''"u"''' || '''"B"''' | + | | f<sub>7</sub> |
| + | |} |
| |- | | |- |
− | | '''"u"''' || '''"u"'''
| |
− | |}
| |
− | |}
| |
− | <br>
| |
− |
| |
− | {| align="center" style="width:90%"
| |
| | | | | |
− | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" | + | {| cellpadding="2" style="background:lightcyan" |
− | |+ proj<sub>''XY''</sub>('''L'''<sub>B</sub>) | + | | f<sub>8</sub> |
− | |- style="background:paleturquoise" | + | |- |
− | ! style="width:50%" | Object
| + | | f<sub>9</sub> |
− | ! style="width:50%" | Sign
| + | |- |
| + | | f<sub>10</sub> |
| + | |- |
| + | | f<sub>11</sub> |
| |- | | |- |
− | | '''A''' || '''"A"''' | + | | f<sub>12</sub> |
| |- | | |- |
− | | '''A''' || '''"u"''' | + | | f<sub>13</sub> |
| |- | | |- |
− | | '''B''' || '''"B"''' | + | | f<sub>14</sub> |
| |- | | |- |
− | | '''B''' || '''"i"''' | + | | f<sub>15</sub> |
| |} | | |} |
| | | | | |
− | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" | + | {| cellpadding="2" style="background:lightcyan" |
− | |+ proj<sub>''XZ''</sub>('''L'''<sub>B</sub>) | + | | 1000 |
− | |- style="background:paleturquoise" | + | |- |
− | ! style="width:50%" | Object
| + | | 1001 |
− | ! style="width:50%" | Interpretant
| + | |- |
| + | | 1010 |
| + | |- |
| + | | 1011 |
| |- | | |- |
− | | '''A''' || '''"A"''' | + | | 1100 |
| |- | | |- |
− | | '''A''' || '''"u"''' | + | | 1101 |
| |- | | |- |
− | | '''B''' || '''"B"''' | + | | 1110 |
| |- | | |- |
− | | '''B''' || '''"i"''' | + | | 1111 |
| |} | | |} |
| | | | | |
− | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" | + | {| cellpadding="2" style="background:lightcyan" |
− | |+ proj<sub>''YZ''</sub>('''L'''<sub>B</sub>)
| + | | u v |
− | |- style="background:paleturquoise"
| |
− | ! style="width:50%" | Sign
| |
− | ! style="width:50%" | Interpretant
| |
| |- | | |- |
− | | '''"A"''' || '''"A"''' | + | | ((u, v)) |
| |- | | |- |
− | | '''"A"''' || '''"u"''' | + | | v |
| |- | | |- |
− | | '''"u"''' || '''"A"''' | + | | (u (v)) |
| |- | | |- |
− | | '''"u"''' || '''"u"''' | + | | u |
| |- | | |- |
− | | '''"B"''' || '''"B"''' | + | | ((u) v) |
| |- | | |- |
− | | '''"B"''' || '''"i"''' | + | | ((u)(v)) |
| |- | | |- |
− | | '''"i"''' || '''"B"''' | + | | (()) |
| + | |} |
| + | | |
| + | {| cellpadding="2" style="background:lightcyan" |
| + | | f<sub>8</sub> |
| + | |- |
| + | | f<sub>9</sub> |
| + | |- |
| + | | f<sub>10</sub> |
| |- | | |- |
− | | '''"i"''' || '''"i"''' | + | | f<sub>11</sub> |
− | |}
| |
− | |}
| |
− | <br>
| |
− | | |
− | {| align="center" cellpadding="4" style="text-align:center; width:90%"
| |
− | | proj<sub>''XY''</sub>('''L'''<sub>A</sub>) ≠ proj<sub>''XY''</sub>('''L'''<sub>B</sub>)
| |
− | | proj<sub>''XZ''</sub>('''L'''<sub>A</sub>) ≠ proj<sub>''XZ''</sub>('''L'''<sub>B</sub>)
| |
− | | proj<sub>''YZ''</sub>('''L'''<sub>A</sub>) ≠ proj<sub>''YZ''</sub>('''L'''<sub>B</sub>)
| |
− | |}
| |
− | <br>
| |
− | | |
− | ====Method 2 : Subtitles as Top Rows====
| |
− | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
| |
− | |+ '''L'''<sub>0</sub> = {(''x'', ''y'', ''z'') ∈ '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 0}
| |
− | |- style="background:paleturquoise"
| |
− | ! X !! Y !! Z
| |
| |- | | |- |
− | | '''0''' || '''0''' || '''0''' | + | | f<sub>12</sub> |
| |- | | |- |
− | | '''0''' || '''1''' || '''1''' | + | | f<sub>13</sub> |
| |- | | |- |
− | | '''1''' || '''0''' || '''1''' | + | | f<sub>14</sub> |
| |- | | |- |
− | | '''1''' || '''1''' || '''0''' | + | | f<sub>15</sub> |
| |} | | |} |
− | <br>
| + | | |
− | | + | {| cellpadding="2" style="background:lightcyan" |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" | + | | f<sub>8</sub> |
− | |+ '''L'''<sub>1</sub> = {(''x'', ''y'', ''z'') ∈ '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 1} | + | |- |
− | |- style="background:paleturquoise" | + | | f<sub>9</sub> |
− | ! X !! Y !! Z
| + | |- |
| + | | f<sub>10</sub> |
| + | |- |
| + | | f<sub>11</sub> |
| |- | | |- |
− | | '''0''' || '''0''' || '''1''' | + | | f<sub>12</sub> |
| |- | | |- |
− | | '''0''' || '''1''' || '''0''' | + | | f<sub>13</sub> |
| |- | | |- |
− | | '''1''' || '''0''' || '''0''' | + | | f<sub>14</sub> |
| |- | | |- |
− | | '''1''' || '''1''' || '''1''' | + | | f<sub>15</sub> |
| + | |} |
| |} | | |} |
− | <br> | + | </font><br> |
| + | |
| + | ===Higher Order Propositions=== |
| | | |
− | {| align="center" style="width:90%" | + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
− | | align="center" | proj<sub>''XY''</sub>('''L'''<sub>0</sub>)
| + | |+ '''Table 7. Higher Order Propositions (n = 1)''' |
− | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
| + | |- style="background:paleturquoise" |
| + | | \ ''x'' || 1 0 || ''F'' |
| + | |''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m'' |
| + | |''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m'' |
| |- style="background:paleturquoise" | | |- style="background:paleturquoise" |
− | ! X !! Y
| + | | ''F'' \ || || |
| + | |00||01||02||03||04||05||06||07||08||09||10||11||12||13||14||15 |
| |- | | |- |
− | | '''0''' || '''0''' | + | | ''F<sub>0</sub> || 0 0 || 0 ||0||1||0||1||0||1||0||1||0||1||0||1||0||1||0||1 |
| |- | | |- |
− | | '''0''' || '''1''' | + | | ''F<sub>1</sub> || 0 1 || (x) ||0||0||1||1||0||0||1||1||0||0||1||1||0||0||1||1 |
| |- | | |- |
− | | '''1''' || '''0''' | + | | ''F<sub>2</sub> || 1 0 || x ||0||0||0||0||1||1||1||1||0||0||0||0||1||1||1||1 |
| |- | | |- |
− | | '''1''' || '''1''' | + | | ''F<sub>3</sub> || 1 1 || 1 ||0||0||0||0||0||0||0||0||1||1||1||1||1||1||1||1 |
| |} | | |} |
− | | align="center" | proj<sub>''XZ''</sub>('''L'''<sub>0</sub>) | + | <br> |
− | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
| + | |
| + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |+ '''Table 8. Interpretive Categories for Higher Order Propositions (n = 1)''' |
| |- style="background:paleturquoise" | | |- style="background:paleturquoise" |
− | ! X !! Z
| + | |Measure||Happening||Exactness||Existence||Linearity||Uniformity||Information |
| + | |- |
| + | |''m''<sub>0</sub>||nothing happens|| || || || || |
| |- | | |- |
− | | '''0''' || '''0''' | + | |''m''<sub>1</sub>|| ||just false||nothing exists|| || || |
| |- | | |- |
− | | '''0''' || '''1''' | + | |''m''<sub>2</sub>|| ||just not x|| || || || |
| |- | | |- |
− | | '''1''' || '''1''' | + | |''m''<sub>3</sub>|| || ||nothing is x|| || || |
| |- | | |- |
− | | '''1''' || '''0''' | + | |''m''<sub>4</sub>|| ||just x|| || || || |
− | |}
| |
− | | align="center" | proj<sub>''YZ''</sub>('''L'''<sub>0</sub>)
| |
− | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
| |
− | |- style="background:paleturquoise" | |
− | ! Y !! Z
| |
| |- | | |- |
− | | '''0''' || '''0''' | + | |''m''<sub>5</sub>|| || ||everything is x||F is linear|| || |
| |- | | |- |
− | | '''1''' || '''1''' | + | |''m''<sub>6</sub>|| || || || ||F is not uniform||F is informed |
| |- | | |- |
− | | '''0''' || '''1''' | + | |''m''<sub>7</sub>|| ||not just true|| || || || |
| |- | | |- |
− | | '''1''' || '''0''' | + | |''m''<sub>8</sub>|| ||just true|| || || || |
− | |}
| |
− | |}
| |
− | <br>
| |
− | | |
− | {| align="center" style="width:90%"
| |
− | | align="center" | proj<sub>''XY''</sub>('''L'''<sub>1</sub>)
| |
− | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
| |
− | |- style="background:paleturquoise" | |
− | ! X !! Y
| |
| |- | | |- |
− | | '''0''' || '''0''' | + | |''m''<sub>9</sub>|| || || || ||F is uniform||F is not informed |
| |- | | |- |
− | | '''0''' || '''1''' | + | |''m''<sub>10</sub>|| || ||something is not x||F is not linear|| || |
| |- | | |- |
− | | '''1''' || '''0''' | + | |''m''<sub>11</sub>|| ||not just x|| || || || |
| |- | | |- |
− | | '''1''' || '''1''' | + | |''m''<sub>12</sub>|| || ||something is x|| || || |
− | |}
| |
− | | align="center" | proj<sub>''XZ''</sub>('''L'''<sub>1</sub>)
| |
− | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
| |
− | |- style="background:paleturquoise" | |
− | ! X !! Z
| |
| |- | | |- |
− | | '''0''' || '''1''' | + | |''m''<sub>13</sub>|| ||not just not x|| || || || |
| |- | | |- |
− | | '''0''' || '''0''' | + | |''m''<sub>14</sub>|| ||not just false||something exists|| || || |
| |- | | |- |
− | | '''1''' || '''0''' | + | |''m''<sub>15</sub>||anything happens|| || || || || |
− | |- | |
− | | '''1''' || '''1''' | |
| |} | | |} |
− | | align="center" | proj<sub>''YZ''</sub>('''L'''<sub>1</sub>) | + | <br> |
− | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
| + | |
| + | {| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |+ '''Table 9. Higher Order Propositions (n = 2)''' |
| + | |- style="background:paleturquoise" |
| + | | align=right | ''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'' |
| |- style="background:paleturquoise" | | |- style="background:paleturquoise" |
− | ! Y !! Z
| + | | align=right | ''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 |
| + | |- |
| + | | ''f<sub>0</sub> || 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 |
| |- | | |- |
− | | '''0''' || '''1''' | + | | ''f<sub>1</sub> || 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 |
| |- | | |- |
− | | '''1''' || '''0''' | + | | ''f<sub>2</sub> || 0010 || (x) y |
| + | | || || || || 1 || 1 || 1 || 1 |
| + | | 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 |
| + | | 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 |
| |- | | |- |
− | | '''0''' || '''0''' | + | | ''f<sub>3</sub> || 0011 || (x) |
| + | | || || || || || || || |
| + | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| |- | | |- |
− | | '''1''' || '''1''' | + | | ''f<sub>4</sub> || 0100 || x (y) |
− | |}
| + | | || || || || || || || |
− | |}
| + | | || || || || || || || |
− | <br> | + | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 |
− | | |
− | {| align="center" cellpadding="4" style="text-align:center; width:90%"
| |
− | | proj<sub>''XY''</sub>('''L'''<sub>0</sub>) = proj<sub>''XY''</sub>('''L'''<sub>1</sub>)
| |
− | | proj<sub>''XZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''XZ''</sub>('''L'''<sub>1</sub>) | |
− | | proj<sub>''YZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''YZ''</sub>('''L'''<sub>1</sub>) | |
− | |} | |
− | <br>
| |
− | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
| |
− | |+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A | |
− | |- style="background:paleturquoise" | |
− | ! style="width:20%" | Object
| |
− | ! style="width:20%" | Sign
| |
− | ! style="width:20%" | Interpretant
| |
| |- | | |- |
− | | '''A''' || '''"A"''' || '''"A"''' | + | | ''f<sub>5</sub> || 0101 || (y) |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| |- | | |- |
− | | '''A''' || '''"A"''' || '''"i"''' | + | | ''f<sub>6</sub> || 0110 || (x, y) |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| |- | | |- |
− | | '''A''' || '''"i"''' || '''"A"''' | + | | ''f<sub>7</sub> || 0111 || (x y) |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| |- | | |- |
− | | '''A''' || '''"i"''' || '''"i"''' | + | | ''f<sub>8</sub> || 1000 || x y |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| |- | | |- |
− | | '''B''' || '''"B"''' || '''"B"''' | + | | ''f<sub>9</sub> || 1001 || ((x, y)) |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| |- | | |- |
− | | '''B''' || '''"B"''' || '''"u"''' | + | | ''f<sub>10</sub> || 1010 || y |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | |- |
| + | | ''f<sub>11</sub> || 1011 || (x (y)) |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | |- |
| + | | ''f<sub>12</sub> || 1100 || x |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | |- |
| + | | ''f<sub>13</sub> || 1101 || ((x) y) |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| |- | | |- |
− | | '''B''' || '''"u"''' || '''"B"''' | + | | ''f<sub>14</sub> || 1110 || ((x)(y)) |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| |- | | |- |
− | | '''B''' || '''"u"''' || '''"u"''' | + | | ''f<sub>15</sub> || 1111 || (( )) |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| |} | | |} |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" | + | {| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
− | |+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B | + | |+ '''Table 10. Qualifiers of Implication Ordering: α<sub>''i'' </sub>''f'' = Υ(''f''<sub>''i''</sub> ⇒ ''f'')''' |
| |- style="background:paleturquoise" | | |- style="background:paleturquoise" |
− | ! style="width:20%" | Object
| + | | align=right | ''x'' : || 1100 || ''f'' |
− | ! style="width:20%" | Sign
| + | |α||α||α||α||α||α||α||α |
− | ! style="width:20%" | Interpretant
| + | |α||α||α||α||α||α||α||α |
| + | |- style="background:paleturquoise" |
| + | | align=right | ''y'' : || 1010 || |
| + | |15||14||13||12||11||10||9||8||7||6||5||4||3||2||1||0 |
| + | |- |
| + | | ''f<sub>0</sub> || 0000 || ( ) |
| + | | || || || || || || || |
| + | | || || || || || || || 1 |
| |- | | |- |
− | | '''A''' || '''"A"''' || '''"A"''' | + | | ''f<sub>1</sub> || 0001 || (x)(y) |
| + | | || || || || || || || |
| + | | || || || || || || 1 || 1 |
| |- | | |- |
− | | '''A''' || '''"A"''' || '''"u"''' | + | | ''f<sub>2</sub> || 0010 || (x) y |
| + | | || || || || || || || |
| + | | || || || || || 1 || || 1 |
| |- | | |- |
− | | '''A''' || '''"u"''' || '''"A"''' | + | | ''f<sub>3</sub> || 0011 || (x) |
| + | | || || || || || || || |
| + | | || || || || 1 || 1 || 1 || 1 |
| |- | | |- |
− | | '''A''' || '''"u"''' || '''"u"''' | + | | ''f<sub>4</sub> || 0100 || x (y) |
| + | | || || || || || || || |
| + | | || || || 1 || || || || 1 |
| |- | | |- |
− | | '''B''' || '''"B"''' || '''"B"''' | + | | ''f<sub>5</sub> || 0101 || (y) |
| + | | || || || || || || || |
| + | | || || 1 || 1 || || || 1 || 1 |
| |- | | |- |
− | | '''B''' || '''"B"''' || '''"i"''' | + | | ''f<sub>6</sub> || 0110 || (x, y) |
| + | | || || || || || || || |
| + | | || 1 || || 1 || || 1 || || 1 |
| |- | | |- |
− | | '''B''' || '''"i"''' || '''"B"''' | + | | ''f<sub>7</sub> || 0111 || (x y) |
| + | | || || || || || || || |
| + | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 |
| |- | | |- |
− | | '''B''' || '''"i"''' || '''"i"''' | + | | ''f<sub>8</sub> || 1000 || x y |
− | |} | + | | || || || || || || || 1 |
− | <br>
| + | | || || || || || || || 1 |
− | | |
− | {| align="center" style="width:90%"
| |
− | | align="center" style="width:30%" | proj<sub>''XY''</sub>('''L'''<sub>A</sub>) | |
− | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
| |
− | |- style="background:paleturquoise" | |
− | ! style="width:50%" | Object
| |
− | ! style="width:50%" | Sign
| |
| |- | | |- |
− | | '''A''' || '''"A"''' | + | | ''f<sub>9</sub> || 1001 || ((x, y)) |
| + | | || || || || || || 1 || 1 |
| + | | || || || || || || 1 || 1 |
| |- | | |- |
− | | '''A''' || '''"i"''' | + | | ''f<sub>10</sub> || 1010 || y |
| + | | || || || || || 1 || || 1 |
| + | | || || || || || 1 || || 1 |
| |- | | |- |
− | | '''B''' || '''"B"''' | + | | ''f<sub>11</sub> || 1011 || (x (y)) |
| + | | || || || || 1 || 1 || 1 || 1 |
| + | | || || || || 1 || 1 || 1 || 1 |
| |- | | |- |
− | | '''B''' || '''"u"''' | + | | ''f<sub>12</sub> || 1100 || x |
− | |}
| + | | || || || 1 || || || || 1 |
− | | align="center" style="width:30%" | proj<sub>''XZ''</sub>('''L'''<sub>A</sub>)
| + | | || || || 1 || || || || 1 |
− | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
| |
− | |- style="background:paleturquoise" | |
− | ! style="width:50%" | Object
| |
− | ! style="width:50%" | Interpretant
| |
| |- | | |- |
− | | '''A''' || '''"A"''' | + | | ''f<sub>13</sub> || 1101 || ((x) y) |
| + | | || || 1 || 1 || || || 1 || 1 |
| + | | || || 1 || 1 || || || 1 || 1 |
| |- | | |- |
− | | '''A''' || '''"i"''' | + | | ''f<sub>14</sub> || 1110 || ((x)(y)) |
| + | | || 1 || || 1 || || 1 || || 1 |
| + | | || 1 || || 1 || || 1 || || 1 |
| |- | | |- |
− | | '''B''' || '''"B"''' | + | | ''f<sub>15</sub> || 1111 || (( )) |
− | |- | + | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 |
− | | '''B''' || '''"u"''' | + | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 |
| |} | | |} |
− | | align="center" style="width:30%" | proj<sub>''YZ''</sub>('''L'''<sub>A</sub>) | + | <br> |
− | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
| + | |
| + | {| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |+ '''Table 11. Qualifiers of Implication Ordering: β<sub>''i'' </sub>''f'' = Υ(''f'' ⇒ ''f''<sub>''i''</sub>)''' |
| + | |- style="background:paleturquoise" |
| + | | align=right | ''x'' : || 1100 || ''f'' |
| + | |β||β||β||β||β||β||β||β |
| + | |β||β||β||β||β||β||β||β |
| |- style="background:paleturquoise" | | |- style="background:paleturquoise" |
− | ! style="width:50%" | Sign
| + | | align=right | ''y'' : || 1010 || |
− | ! style="width:50%" | Interpretant
| + | |0||1||2||3||4||5||6||7||8||9||10||11||12||13||14||15 |
| |- | | |- |
− | | '''"A"''' || '''"A"''' | + | | ''f<sub>0</sub> || 0000 || ( ) |
| + | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 |
| + | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 |
| |- | | |- |
− | | '''"A"''' || '''"i"''' | + | | ''f<sub>1</sub> || 0001 || (x)(y) |
| + | | || 1 || || 1 || || 1 || || 1 |
| + | | || 1 || || 1 || || 1 || || 1 |
| |- | | |- |
− | | '''"i"''' || '''"A"''' | + | | ''f<sub>2</sub> || 0010 || (x) y |
| + | | || || 1 || 1 || || || 1 || 1 |
| + | | || || 1 || 1 || || || 1 || 1 |
| |- | | |- |
− | | '''"i"''' || '''"i"''' | + | | ''f<sub>3</sub> || 0011 || (x) |
| + | | || || || 1 || || || || 1 |
| + | | || || || 1 || || || || 1 |
| |- | | |- |
− | | '''"B"''' || '''"B"''' | + | | ''f<sub>4</sub> || 0100 || x (y) |
| + | | || || || || 1 || 1 || 1 || 1 |
| + | | || || || || 1 || 1 || 1 || 1 |
| |- | | |- |
− | | '''"B"''' || '''"u"''' | + | | ''f<sub>5</sub> || 0101 || (y) |
| + | | || || || || || 1 || || 1 |
| + | | || || || || || 1 || || 1 |
| |- | | |- |
− | | '''"u"''' || '''"B"''' | + | | ''f<sub>6</sub> || 0110 || (x, y) |
| + | | || || || || || || 1 || 1 |
| + | | || || || || || || 1 || 1 |
| |- | | |- |
− | | '''"u"''' || '''"u"''' | + | | ''f<sub>7</sub> || 0111 || (x y) |
− | |}
| + | | || || || || || || || 1 |
− | |}
| + | | || || || || || || || 1 |
− | <br>
| |
− | | |
− | {| align="center" style="width:90%"
| |
− | | align="center" style="width:30%" | proj<sub>''XY''</sub>('''L'''<sub>B</sub>)
| |
− | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
| |
− | |- style="background:paleturquoise" | |
− | ! style="width:50%" | Object
| |
− | ! style="width:50%" | Sign
| |
| |- | | |- |
− | | '''A''' || '''"A"''' | + | | ''f<sub>8</sub> || 1000 || x y |
| + | | || || || || || || || |
| + | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 |
| |- | | |- |
− | | '''A''' || '''"u"''' | + | | ''f<sub>9</sub> || 1001 || ((x, y)) |
| + | | || || || || || || || |
| + | | || 1 || || 1 || || 1 || || 1 |
| |- | | |- |
− | | '''B''' || '''"B"''' | + | | ''f<sub>10</sub> || 1010 || y |
| + | | || || || || || || || |
| + | | || || 1 || 1 || || || 1 || 1 |
| |- | | |- |
− | | '''B''' || '''"i"''' | + | | ''f<sub>11</sub> || 1011 || (x (y)) |
− | |}
| + | | || || || || || || || |
− | | align="center" style="width:30%" | proj<sub>''XZ''</sub>('''L'''<sub>B</sub>)
| + | | || || || 1 || || || || 1 |
− | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
| |
− | |- style="background:paleturquoise" | |
− | ! style="width:50%" | Object
| |
− | ! style="width:50%" | Interpretant
| |
| |- | | |- |
− | | '''A''' || '''"A"''' | + | | ''f<sub>12</sub> || 1100 || x |
| + | | || || || || || || || |
| + | | || || || || 1 || 1 || 1 || 1 |
| |- | | |- |
− | | '''A''' || '''"u"''' | + | | ''f<sub>13</sub> || 1101 || ((x) y) |
| + | | || || || || || || || |
| + | | || || || || || 1 || || 1 |
| |- | | |- |
− | | '''B''' || '''"B"''' | + | | ''f<sub>14</sub> || 1110 || ((x)(y)) |
| + | | || || || || || || || |
| + | | || || || || || || 1 || 1 |
| |- | | |- |
− | | '''B''' || '''"i"''' | + | | ''f<sub>15</sub> || 1111 || (( )) |
| + | | || || || || || || || |
| + | | || || || || || || || 1 |
| |} | | |} |
− | | align="center" style="width:30%" | proj<sub>''YZ''</sub>('''L'''<sub>B</sub>) | + | <br> |
− | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
| + | |
− | |- style="background:paleturquoise" | + | {| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
− | ! style="width:50%" | Sign
| + | |+ '''Table 13. Syllogistic Premisses as Higher Order Indicator Functions''' |
− | ! style="width:50%" | Interpretant
| + | | A |
| + | | align=left | Universal Affirmative |
| + | | align=left | All |
| + | | x || is || y |
| + | | align=left | Indicator of " x (y)" = 0 |
| |- | | |- |
− | | '''"A"''' || '''"A"''' | + | | E |
| + | | align=left | Universal Negative |
| + | | align=left | All |
| + | | x || is || (y) |
| + | | align=left | Indicator of " x y " = 0 |
| |- | | |- |
− | | '''"A"''' || '''"u"''' | + | | I |
| + | | align=left | Particular Affirmative |
| + | | align=left | Some |
| + | | x || is || y |
| + | | align=left | Indicator of " x y " = 1 |
| |- | | |- |
− | | '''"u"''' || '''"A"''' | + | | O |
| + | | align=left | Particular Negative |
| + | | align=left | Some |
| + | | x || is || (y) |
| + | | align=left | Indicator of " x (y)" = 1 |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |+ '''Table 14. Relation of Quantifiers to Higher Order Propositions''' |
| + | |- style="background:paleturquoise" |
| + | |Mnemonic||Category||Classical Form||Alternate Form||Symmetric Form||Operator |
| |- | | |- |
− | | '''"u"''' || '''"u"''' | + | | E<br>Exclusive |
| + | | Universal<br>Negative |
| + | | align=left | All x is (y) |
| + | | align=left | |
| + | | align=left | No x is y |
| + | | (''L''<sub>11</sub>) |
| |- | | |- |
− | | '''"B"''' || '''"B"''' | + | | A<br>Absolute |
| + | | Universal<br>Affirmative |
| + | | align=left | All x is y |
| + | | align=left | |
| + | | align=left | No x is (y) |
| + | | (''L''<sub>10</sub>) |
| |- | | |- |
− | | '''"B"''' || '''"i"''' | + | | |
| + | | |
| + | | align=left | All y is x |
| + | | align=left | No y is (x) |
| + | | align=left | No (x) is y |
| + | | (''L''<sub>01</sub>) |
| |- | | |- |
− | | '''"i"''' || '''"B"''' | + | | |
| + | | |
| + | | align=left | All (y) is x |
| + | | align=left | No (y) is (x) |
| + | | align=left | No (x) is (y) |
| + | | (''L''<sub>00</sub>) |
| |- | | |- |
− | | '''"i"''' || '''"i"''' | + | | |
− | |}
| + | | |
| + | | align=left | Some (x) is (y) |
| + | | align=left | |
| + | | align=left | Some (x) is (y) |
| + | | ''L''<sub>00</sub> |
| + | |- |
| + | | |
| + | | |
| + | | align=left | Some (x) is y |
| + | | align=left | |
| + | | align=left | Some (x) is y |
| + | | ''L''<sub>01</sub> |
| + | |- |
| + | | O<br>Obtrusive |
| + | | Particular<br>Negative |
| + | | align=left | Some x is (y) |
| + | | align=left | |
| + | | align=left | Some x is (y) |
| + | | ''L''<sub>10</sub> |
| + | |- |
| + | | I<br>Indefinite |
| + | | Particular<br>Affirmative |
| + | | align=left | Some x is y |
| + | | align=left | |
| + | | align=left | Some x is y |
| + | | ''L''<sub>11</sub> |
| |} | | |} |
| <br> | | <br> |
| | | |
− | {| align="center" cellpadding="4" style="text-align:center; width:90%" | + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
− | | proj<sub>''XY''</sub>('''L'''<sub>A</sub>) ≠ proj<sub>''XY''</sub>('''L'''<sub>B</sub>) | + | |+ '''Table 15. Simple Qualifiers of Propositions (n = 2)''' |
− | | proj<sub>''XZ''</sub>('''L'''<sub>A</sub>) ≠ proj<sub>''XZ''</sub>('''L'''<sub>B</sub>) | + | |- style="background:paleturquoise" |
− | | proj<sub>''YZ''</sub>('''L'''<sub>A</sub>) ≠ proj<sub>''YZ''</sub>('''L'''<sub>B</sub>) | + | | align=right | ''x'' : || 1100 || ''f'' |
− | |} | + | | (''L''<sub>11</sub>) |
− | <br>
| + | | (''L''<sub>10</sub>) |
− | | + | | (''L''<sub>01</sub>) |
− | ===Formatted Text Display===
| + | | (''L''<sub>00</sub>) |
− | | + | | ''L''<sub>00</sub> |
− | : So in a triadic fact, say, the example <br>
| + | | ''L''<sub>01</sub> |
− | {| align="center" cellspacing="8" style="width:72%"
| + | | ''L''<sub>10</sub> |
− | | align="center" | ''A'' gives ''B'' to ''C'' | + | | ''L''<sub>11</sub> |
− | |} | + | |- style="background:paleturquoise" |
− | : we make no distinction in the ordinary logic of relations between the ''[[subject (grammar)|subject]] [[nominative]]'', the ''[[direct object]]'', and the ''[[indirect object]]''. We say that the proposition has three ''logical subjects''. We regard it as a mere affair of English grammar that there are six ways of expressing this: <br>
| + | | align=right | ''y'' : || 1010 || |
− | {| align="center" cellspacing="8" style="width:72%"
| + | | align=left | no x <br> is y |
− | | style="width:36%" | ''A'' gives ''B'' to ''C'' | + | | align=left | no x <br> is (y) |
− | | style="width:36%" | ''A'' benefits ''C'' with ''B'' | + | | align=left | no (x) <br> is y |
| + | | align=left | no (x) <br> is (y) |
| + | | align=left | some (x) <br> is (y) |
| + | | align=left | some (x) <br> is y |
| + | | align=left | some x <br> is (y) |
| + | | align=left | some x <br> is y |
| |- | | |- |
− | | ''B'' enriches ''C'' at expense of ''A'' | + | | ''f<sub>0</sub> || 0000 || ( ) |
− | | ''C'' receives ''B'' from ''A'' | + | | 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0 |
| |- | | |- |
− | | ''C'' thanks ''A'' for ''B'' | + | | ''f<sub>1</sub> || 0001 || (x)(y) |
− | | ''B'' leaves ''A'' for ''C'' | + | | 1 || 1 || 1 || 0 || 1 || 0 || 0 || 0 |
− | |} | |
− | : 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==
| |
− |
| |
− | {| border="1" cellspacing="0" cellpadding="0" style="text-align:center"
| |
− | |+ Binary Operations
| |
| |- | | |- |
− | ! style="width:2em" | x<sub>0</sub>
| + | | ''f<sub>2</sub> || 0010 || (x) y |
− | ! style="width:2em" | x<sub>1</sub>
| + | | 1 || 1 || 0 || 1 || 0 || 1 || 0 || 0 |
− | | style="width:2em" | <sup>2</sup>f<sub>0</sub> | + | |- |
− | | style="width:2em" | <sup>2</sup>f<sub>1</sub> | + | | ''f<sub>3</sub> || 0011 || (x) |
− | | style="width:2em" | <sup>2</sup>f<sub>2</sub> | + | | 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0 |
− | | style="width:2em" | <sup>2</sup>f<sub>3</sub> | + | |- |
− | | style="width:2em" | <sup>2</sup>f<sub>4</sub> | + | | ''f<sub>4</sub> || 0100 || x (y) |
− | | style="width:2em" | <sup>2</sup>f<sub>5</sub> | + | | 1 || 0 || 1 || 1 || 0 || 0 || 1 || 0 |
− | | style="width:2em" | <sup>2</sup>f<sub>6</sub> | |
− | | style="width:2em" | <sup>2</sup>f<sub>7</sub> | |
− | | style="width:2em" | <sup>2</sup>f<sub>8</sub> | |
− | | style="width:2em" | <sup>2</sup>f<sub>9</sub> | |
− | | style="width:2em" | <sup>2</sup>f<sub>10</sub> | |
− | | style="width:2em" | <sup>2</sup>f<sub>11</sub> | |
− | | style="width:2em" | <sup>2</sup>f<sub>12</sub> | |
− | | style="width:2em" | <sup>2</sup>f<sub>13</sub> | |
− | | style="width:2em" | <sup>2</sup>f<sub>14</sub> | |
− | | style="width:2em" | <sup>2</sup>f<sub>15</sub> | |
| |- | | |- |
− | | 0 || 0 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 | + | | ''f<sub>5</sub> || 0101 || (y) |
| + | | 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 |
| |- | | |- |
− | | 1 || 0 || 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 | + | | ''f<sub>6</sub> || 0110 || (x, y) |
| + | | 1 || 0 || 0 || 1 || 0 || 1 || 1 || 0 |
| |- | | |- |
− | | 0 || 1 || 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 | + | | ''f<sub>7</sub> || 0111 || (x y) |
| + | | 1 || 0 || 0 || 0 || 1 || 1 || 1 || 0 |
| |- | | |- |
− | | 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 | + | | ''f<sub>8</sub> || 1000 || x y |
− | |} | + | | 0 || 1 || 1 || 1 || 0 || 0 || 0 || 1 |
− | <br>
| + | |- |
− | | + | | ''f<sub>9</sub> || 1001 || ((x, y)) |
− | ===Draft 1===
| + | | 0 || 1 || 1 || 0 || 1 || 0 || 0 || 1 |
− | | + | |- |
− | <center><table>
| + | | ''f<sub>10</sub> || 1010 || y |
− | <caption>TRUTH TABLES FOR THE BOOLEAN OPERATIONS OF ARITY UP TO 2</caption>
| + | | 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 |
− | <tr valign="top">
| + | |- |
− | <td><table border=5 cellspacing=0>
| + | | ''f<sub>11</sub> || 1011 || (x (y)) |
− | <caption>Constants</caption>
| + | | 0 || 1 || 0 || 0 || 1 || 1 || 0 || 1 |
− | <tr><td></td>
| + | |- |
− | <td><sup>0</sup>f<sub>0</sub></td> <td><sup>0</sup>f<sub>1</sub></td>
| + | | ''f<sub>12</sub> || 1100 || x |
− | </tr> <tr><td></td>
| + | | 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 |
− | <td align=center>0</td> <td align=center>1</td>
| + | |- |
− | </tr></table></td><td> </td>
| + | | ''f<sub>13</sub> || 1101 || ((x) y) |
− | <td><table border=5 cellspacing=0><caption>Unary Operations</caption><tr>
| + | | 0 || 0 || 1 || 0 || 1 || 0 || 1 || 1 |
− | <td>x<sub>0</sub></td> <td></td>
| + | |- |
− | <td><sup>1</sup>f<sub>0 </sub></td> <td><sup>1</sup>f<sub>1 </sub></td>
| + | | ''f<sub>14</sub> || 1110 || ((x)(y)) |
− | <td><sup>1</sup>f<sub>2 </sub></td> <td><sup>1</sup>f<sub>3 </sub></td>
| + | | 0 || 0 || 0 || 1 || 0 || 1 || 1 || 1 |
− | </tr><tr> <td align=center>0</td> <td></td>
| + | |- |
− | <td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
| + | | ''f<sub>15</sub> || 1111 || (( )) |
− | </tr> <tr> <td align=center>1</td> <td></td>
| + | | 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 |
− | <td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
| + | |} |
− | </tr></table></td><td> </td>
| + | <br> |
− | <td><table border=5 cellspacing=0><caption>Binary Operations</caption><tr>
| |
− | <td>x<sub>0</sub></td> <td>x<sub>1</sub></td>
| |
− | <td></td>
| |
− | <td><sup>2</sup>f<sub>0</sub></td> <td><sup>2</sup>f<sub>1 </sub></td>
| |
− | <td><sup> 2</sup>f<sub>2 </sub></td> <td><sup>2</sup>f<sub>3 </sub></td>
| |
− | <td><sup>2</sup>f<sub>4 </sub></td> <td><sup>2</sup>f<sub>5 </sub></td>
| |
− | <td><sup>2</sup>f<sub>6 </sub></td> <td><sup>2</sup>f<sub>7 </sub></td>
| |
− | <td><sup>2</sup>f<sub>8 </sub></td> <td><sup>2</sup>f<sub>9 </sub></td>
| |
− | <td><sup>2</sup>f<sub>10</sub></td> <td><sup>2</sup>f<sub>11</sub></td>
| |
− | <td><sup>2</sup>f<sub>12</sub></td> <td><sup>2</sup>f<sub>13</sub></td>
| |
− | <td><sup>2</sup>f<sub>14</sub></td> <td><sup>2</sup>f<sub>15</sub></td>
| |
− | </tr><tr> <td align=center>0</td> <td align=center>0</td> <td></td>
| |
− | <td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
| |
− | <td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
| |
− | <td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
| |
− | <td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
| |
− | </tr> <tr> <td align=center>1</td> <td align=center>0</td> <td></td> | |
− | <td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
| |
− | <td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
| |
− | <td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
| |
− | <td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
| |
− | </tr> <tr> <td align=center>0</td> <td align=center>1</td> <td></td> | |
− | <td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td>
| |
− | <td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td>
| |
− | <td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td>
| |
− | <td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td>
| |
− | </tr> <tr> <td align=center>1</td> <td align=center>1</td> <td></td>
| |
− | <td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td> | |
− | <td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td>
| |
− | <td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td>
| |
− | <td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td>
| |
− | </tr> </table></td> | |
− | </table></center>
| |
| | | |
− | ===Draft 2=== | + | 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 |
− | <center><table> | + | | \ x | 1 0 | F |m |m |m |m |m |m |m |m |m |m |m |m |m |m |m |m | |
− | <caption>TRUTH TABLES FOR THE BOOLEAN OPERATIONS OF ARITY UP TO 2</caption> | + | | F \ | | |00|01|02|03|04|05|06|07|08|09|10|11|12|13|14|15 | |
− | <tr valign="top"> | + | o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o |
− | <td><table border=5 cellspacing=0> | + | | | | | | |
− | <caption>Constants</caption> | + | | F_0 | 0 0 | 0 | 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 | |
− | <tr><td></td> | + | | | | | | |
− | <td><sup>0</sup>f<sub>0</sub></td> <td><sup>0</sup>f<sub>1</sub></td> | + | | F_1 | 0 1 | (x) | 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 | |
− | </tr> <tr><td></td> | + | | | | | | |
− | <td align=center>0</td> <td align=center>1</td> | + | | F_2 | 1 0 | x | 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 | |
− | </tr></table></td><td> </td> | + | | | | | | |
− | <td><table border=5 cellspacing=0><caption>Unary Operations</caption><tr> | + | | F_3 | 1 1 | 1 | 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 | |
− | <td>x<sub>0</sub></td> <td></td> | + | | | | | | |
− | <td><sup>1</sup>f<sub>0 </sub></td> <td><sup>1</sup>f<sub>1 </sub></td> | + | o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o |
− | <td><sup>1</sup>f<sub>2 </sub></td> <td><sup>1</sup>f<sub>3 </sub></td> | + | <br> |
− | </tr><tr> <td align=center>0</td> <td></td> | + | |
− | <td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td> | + | Table 8. Interpretive Categories for Higher Order Propositions (n = 1) |
− | </tr> <tr> <td align=center>1</td> <td></td> | + | o-------o----------o------------o------------o----------o----------o-----------o |
− | <td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td> | + | |Measure| Happening| Exactness | Existence | Linearity|Uniformity|Information| |
− | </tr></table></td><td> </td> | + | o-------o----------o------------o------------o----------o----------o-----------o |
− | <td><table border=5 cellspacing=0><caption>Binary Operations</caption><tr> | + | | m_0 | nothing | | | | | | |
− | <td>x<sub>0</sub></td> <td>x<sub>1</sub></td> | + | | | happens | | | | | | |
− | <td></td> | + | o-------o----------o------------o------------o----------o----------o-----------o |
− | <td><sup>2</sup>f<sub>0</sub></td> <td><sup>2</sup>f<sub>1 </sub></td> | + | | m_1 | | | nothing | | | | |
− | <td><sup> 2</sup>f<sub>2 </sub></td> <td><sup>2</sup>f<sub>3 </sub></td> | + | | | | just false | exists | | | | |
− | <td><sup>2</sup>f<sub>4 </sub></td> <td><sup>2</sup>f<sub>5 </sub></td> | + | o-------o----------o------------o------------o----------o----------o-----------o |
− | <td><sup>2</sup>f<sub>6 </sub></td> <td><sup>2</sup>f<sub>7 </sub></td> | + | | m_2 | | | | | | | |
− | <td><sup>2</sup>f<sub>8 </sub></td> <td><sup>2</sup>f<sub>9 </sub></td> | + | | | | just not x | | | | | |
− | <td><sup>2</sup>f<sub>10</sub></td> <td><sup>2</sup>f<sub>11</sub></td> | + | o-------o----------o------------o------------o----------o----------o-----------o |
− | <td><sup>2</sup>f<sub>12</sub></td> <td><sup>2</sup>f<sub>13</sub></td> | + | | m_3 | | | nothing | | | | |
− | <td><sup>2</sup>f<sub>14</sub></td> <td><sup>2</sup>f<sub>15</sub></td> | + | | | | | is x | | | | |
− | </tr><tr> <td align=center>0</td> <td align=center>0</td> <td></td> | + | o-------o----------o------------o------------o----------o----------o-----------o |
− | <td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td> | + | | m_4 | | | | | | | |
− | <td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td> | + | | | | just x | | | | | |
− | <td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td> | + | o-------o----------o------------o------------o----------o----------o-----------o |
− | <td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td> | + | | m_5 | | | everything | F is | | | |
− | </tr> <tr> <td align=center>1</td> <td align=center>0</td> <td></td> | + | | | | | is x | linear | | | |
− | <td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td> | + | o-------o----------o------------o------------o----------o----------o-----------o |
− | <td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td> | + | | m_6 | | | | | F is not | F is | |
− | <td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td> | + | | | | | | | uniform | informed | |
− | <td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td> | + | o-------o----------o------------o------------o----------o----------o-----------o |
− | </tr> <tr> <td align=center>0</td> <td align=center>1</td> <td></td> | + | | m_7 | | not | | | | | |
− | <td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td> | + | | | | just true | | | | | |
− | <td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td> | + | o-------o----------o------------o------------o----------o----------o-----------o |
− | <td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td> | + | | m_8 | | | | | | | |
− | <td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td> | + | | | | just true | | | | | |
− | </tr> <tr> <td align=center>1</td> <td align=center>1</td> <td></td> | + | o-------o----------o------------o------------o----------o----------o-----------o |
− | <td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td> | + | | m_9 | | | | | F is | F is not | |
− | <td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td> | + | | | | | | | uniform | informed | |
− | <td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td> | + | o-------o----------o------------o------------o----------o----------o-----------o |
− | <td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td> | + | | m_10 | | | something | F is not | | | |
− | </tr> </table></td> | + | | | | | 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 |
| + | <br> |
| + | |
| + | 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 |
| + | <br> |
| + | |
| + | 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 |
| + | <br> |
| + | |
| + | 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 |
| + | <br> |
| + | |
| + | 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 |
| + | <br> |
| + | |
| + | 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 |
| + | <br> |
| + | |
| + | 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 |
| + | <br> |
| + | |
| + | ===[[Zeroth Order Logic]]=== |
| + | |
| + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |+ '''Table 1. Propositional Forms on Two Variables''' |
| + | |- style="background:paleturquoise" |
| + | ! style="width:15%" | L<sub>1</sub> |
| + | ! style="width:15%" | L<sub>2</sub> |
| + | ! style="width:15%" | L<sub>3</sub> |
| + | ! style="width:15%" | L<sub>4</sub> |
| + | ! style="width:15%" | L<sub>5</sub> |
| + | ! style="width:15%" | L<sub>6</sub> |
| + | |- style="background:paleturquoise" |
| + | | |
| + | | align="right" | x : |
| + | | 1 1 0 0 |
| + | | |
| + | | |
| + | | |
| + | |- style="background:paleturquoise" |
| + | | |
| + | | align="right" | y : |
| + | | 1 0 1 0 |
| + | | |
| + | | |
| + | | |
| + | |- |
| + | | f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || ( ) || false || 0 |
| + | |- |
| + | | f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || ¬x ∧ ¬y |
| + | |- |
| + | | f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || ¬x ∧ y |
| + | |- |
| + | | f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || ¬x |
| + | |- |
| + | | f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x ∧ ¬y |
| + | |- |
| + | | f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || ¬y |
| + | |- |
| + | | f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x ≠ y |
| + | |- |
| + | | f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x y) || not both x and y || ¬x ∨ ¬y |
| + | |- |
| + | | f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x y || x and y || x ∧ y |
| + | |- |
| + | | f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y |
| + | |- |
| + | | f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y |
| + | |- |
| + | | f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x → y |
| + | |- |
| + | | f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x |
| + | |- |
| + | | f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x ← y |
| + | |- |
| + | | f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y || x ∨ y |
| + | |- |
| + | | f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || (( )) || true || 1 |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:90%" |
| + | |+ '''Table 1. Propositional Forms on Two Variables''' |
| + | |- style="background:aliceblue" |
| + | ! style="width:15%" | L<sub>1</sub> |
| + | ! style="width:15%" | L<sub>2</sub> |
| + | ! style="width:15%" | L<sub>3</sub> |
| + | ! style="width:15%" | L<sub>4</sub> |
| + | ! style="width:15%" | L<sub>5</sub> |
| + | ! style="width:15%" | L<sub>6</sub> |
| + | |- style="background:aliceblue" |
| + | | |
| + | | align="right" | x : |
| + | | 1 1 0 0 |
| + | | |
| + | | |
| + | | |
| + | |- style="background:aliceblue" |
| + | | |
| + | | align="right" | y : |
| + | | 1 0 1 0 |
| + | | |
| + | | |
| + | | |
| + | |- |
| + | | f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || ( ) || false || 0 |
| + | |- |
| + | | f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || ¬x ∧ ¬y |
| + | |- |
| + | | f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || ¬x ∧ y |
| + | |- |
| + | | f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || ¬x |
| + | |- |
| + | | f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x ∧ ¬y |
| + | |- |
| + | | f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || ¬y |
| + | |- |
| + | | f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x ≠ y |
| + | |- |
| + | | f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x y) || not both x and y || ¬x ∨ ¬y |
| + | |- |
| + | | f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x y || x and y || x ∧ y |
| + | |- |
| + | | f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y |
| + | |- |
| + | | f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y |
| + | |- |
| + | | f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x → y |
| + | |- |
| + | | f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x |
| + | |- |
| + | | f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x ← y |
| + | |- |
| + | | f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y || x ∨ y |
| + | |- |
| + | | f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || (( )) || true || 1 |
| + | |} |
| + | <br> |
| + | |
| + | ===Template Draft=== |
| + | |
| + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:98%" |
| + | |+ '''Propositional Forms on Two Variables''' |
| + | |- style="background:aliceblue" |
| + | ! style="width:14%" | L<sub>1</sub> |
| + | ! style="width:14%" | L<sub>2</sub> |
| + | ! style="width:14%" | L<sub>3</sub> |
| + | ! style="width:14%" | L<sub>4</sub> |
| + | ! style="width:14%" | L<sub>5</sub> |
| + | ! style="width:14%" | L<sub>6</sub> |
| + | ! style="width:14%" | Name |
| + | |- style="background:aliceblue" |
| + | | |
| + | | align="right" | x : |
| + | | 1 1 0 0 |
| + | | |
| + | | |
| + | | |
| + | | |
| + | |- style="background:aliceblue" |
| + | | |
| + | | align="right" | y : |
| + | | 1 0 1 0 |
| + | | |
| + | | |
| + | | |
| + | | |
| + | |- |
| + | | f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || ( ) || false || 0 || Falsity |
| + | |- |
| + | | f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || ¬x ∧ ¬y || [[NNOR]] |
| + | |- |
| + | | f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || ¬x ∧ y || Insuccede |
| + | |- |
| + | | f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || ¬x || Not One |
| + | |- |
| + | | f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x ∧ ¬y || Imprecede |
| + | |- |
| + | | f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || ¬y || Not Two |
| + | |- |
| + | | f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x ≠ y || Inequality |
| + | |- |
| + | | f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x y) || not both x and y || ¬x ∨ ¬y || NAND |
| + | |- |
| + | | f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x y || x and y || x ∧ y || [[Conjunction]] |
| + | |- |
| + | | f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y || Equality |
| + | |- |
| + | | f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y || Two |
| + | |- |
| + | | f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x → y || [[Logical implcation|Implication]] |
| + | |- |
| + | | f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x || One |
| + | |- |
| + | | f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x ← y || [[Logical involution|Involution]] |
| + | |- |
| + | | f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y || x ∨ y || [[Disjunction]] |
| + | |- |
| + | | f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || (( )) || true || 1 || Tautology |
| + | |} |
| + | <br> |
| + | |
| + | ===[[Truth Tables]]=== |
| + | |
| + | ====[[Logical negation]]==== |
| + | |
| + | '''Logical negation''' is an [[logical operation|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: |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:40%" |
| + | |+ '''Logical Negation''' |
| + | |- style="background:aliceblue" |
| + | ! style="width:20%" | p |
| + | ! style="width:20%" | ¬p |
| + | |- |
| + | | F || T |
| + | |- |
| + | | T || F |
| + | |} |
| + | <br> |
| + | |
| + | 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: |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; width:40%" |
| + | |+ '''Variant Notations''' |
| + | |- style="background:aliceblue" |
| + | ! style="text-align:center" | Notation |
| + | ! Vocalization |
| + | |- |
| + | | style="text-align:center" | <math>\bar{p}</math> |
| + | | bar ''p'' |
| + | |- |
| + | | style="text-align:center" | <math>p'\!</math> |
| + | | ''p'' prime,<p> ''p'' complement |
| + | |- |
| + | | style="text-align:center" | <math>!p\!</math> |
| + | | bang ''p'' |
| + | |} |
| + | <br> |
| + | |
| + | 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 [[logical operation|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 true. |
| + | |
| + | The [[truth table]] of '''p AND q''' (also written as '''p ∧ q''', '''p & q''', or '''p<math>\cdot</math>q''') is as follows: |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%" |
| + | |+ '''Logical Conjunction''' |
| + | |- style="background:aliceblue" |
| + | ! style="width:15%" | p |
| + | ! style="width:15%" | q |
| + | ! style="width:15%" | p ∧ q |
| + | |- |
| + | | F || F || F |
| + | |- |
| + | | F || T || F |
| + | |- |
| + | | T || F || F |
| + | |- |
| + | | T || T || T |
| + | |} |
| + | <br> |
| + | |
| + | ====[[Logical disjunction]]==== |
| + | |
| + | '''Logical disjunction''' is 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 both of its operands are false. |
| + | |
| + | The [[truth table]] of '''p OR q''' (also written 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 Disjunction''' |
| + | |- style="background:aliceblue" |
| + | ! style="width:15%" | p |
| + | ! style="width:15%" | q |
| + | ! style="width:15%" | p ∨ q |
| + | |- |
| + | | F || F || F |
| + | |- |
| + | | F || T || T |
| + | |- |
| + | | T || F || T |
| + | |- |
| + | | T || T || T |
| + | |} |
| + | <br> |
| + | |
| + | ====[[Logical equality]]==== |
| + | |
| + | '''Logical equality''' is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, 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: |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%" |
| + | |+ '''Logical Equality''' |
| + | |- 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 || T |
| + | |} |
| + | <br> |
| + | |
| + | ====[[Exclusive disjunction]]==== |
| + | |
| + | '''Exclusive disjunction''' is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, 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: |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%" |
| + | |+ '''Exclusive Disjunction''' |
| + | |- style="background:aliceblue" |
| + | ! style="width:15%" | p |
| + | ! style="width:15%" | q |
| + | ! style="width:15%" | p XOR q |
| + | |- |
| + | | F || F || F |
| + | |- |
| + | | F || T || T |
| + | |- |
| + | | T || F || T |
| + | |- |
| + | | T || T || F |
| + | |} |
| + | <br> |
| + | |
| + | The following equivalents can then be deduced: |
| + | |
| + | : <math>\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}</math> |
| + | |
| + | '''Generalized''' or '''n-ary''' XOR is true when the number of 1-bits is odd. |
| + | |
| + | <pre> |
| + | 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) |
| + | </pre> |
| + | |
| + | <pre> |
| + | 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) |
| + | </pre> |
| + | |
| + | <pre> |
| + | 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) |
| + | </pre> |
| + | |
| + | : <math>\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}</math> |
| + | |
| + | ====[[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> |
| + | |
| + | ====[[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> |
| + | |
| + | ====[[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> |
| + | |
| + | ==Relational Tables== |
| + | |
| + | ===Factorization=== |
| + | |
| + | {| align="center" style="text-align:center; width:60%" |
| + | | |
| + | {| align="center" style="text-align:center; width:100%" |
| + | | <math>\text{Table 7. Plural Denotation}\!</math> |
| + | |} |
| + | |- |
| + | | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:100%" |
| + | |- style="background:#f0f0ff" |
| + | | width="33%" | <math>\text{Object}\!</math> |
| + | | width="33%" | <math>\text{Sign}\!</math> |
| + | | width="33%" | <math>\text{Interpretant}\!</math> |
| + | |- |
| + | | |
| + | <math>\begin{matrix} |
| + | o_1 \\ o_2 \\ o_3 \\ \ldots \\ o_k \\ \ldots |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | s \\ s \\ s \\ \ldots \\ s \\ \ldots |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \ldots \\ \ldots \\ \ldots \\ \ldots \\ \ldots \\ \ldots |
| + | \end{matrix}</math> |
| + | |} |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | {| align="center" style="text-align:center; width:60%" |
| + | | |
| + | {| align="center" style="text-align:center; width:100%" |
| + | | <math>\text{Table 8. Sign Relation}~ L</math> |
| + | |} |
| + | |- |
| + | | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:100%" |
| + | |- style="background:#f0f0ff" |
| + | | width="33%" | <math>\text{Object}\!</math> |
| + | | width="33%" | <math>\text{Sign}\!</math> |
| + | | width="33%" | <math>\text{Interpretant}\!</math> |
| + | |- |
| + | | |
| + | <math>\begin{matrix} |
| + | o_1 \\ o_2 \\ o_3 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | s \\ s \\ s |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \ldots \\ \ldots \\ \ldots |
| + | \end{matrix}</math> |
| + | |} |
| + | |} |
| + | |
| + | ===Sign Relations=== |
| + | |
| + | {| cellpadding="4" |
| + | | width="20px" | |
| + | | align="center" | '''O''' || = || Object Domain |
| + | |- |
| + | | width="20px" | |
| + | | align="center" | '''S''' || = || Sign Domain |
| + | |- |
| + | | width="20px" | |
| + | | align="center" | '''I''' || = || Interpretant Domain |
| + | |} |
| + | <br> |
| + | |
| + | {| cellpadding="4" |
| + | | width="20px" | |
| + | | align="center" | '''O''' |
| + | | = |
| + | | {Ann, Bob} |
| + | | = |
| + | | {A, B} |
| + | |- |
| + | | width="20px" | |
| + | | align="center" | '''S''' |
| + | | = |
| + | | {"Ann", "Bob", "I", "You"} |
| + | | = |
| + | | {"A", "B", "i", "u"} |
| + | |- |
| + | | width="20px" | |
| + | | align="center" | '''I''' |
| + | | = |
| + | | {"Ann", "Bob", "I", "You"} |
| + | | = |
| + | | {"A", "B", "i", "u"} |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |
| + | |+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A |
| + | |- style="background:paleturquoise" |
| + | ! style="width:20%" | Object |
| + | ! style="width:20%" | Sign |
| + | ! style="width:20%" | Interpretant |
| + | |- |
| + | | '''A''' || '''"A"''' || '''"A"''' |
| + | |- |
| + | | '''A''' || '''"A"''' || '''"i"''' |
| + | |- |
| + | | '''A''' || '''"i"''' || '''"A"''' |
| + | |- |
| + | | '''A''' || '''"i"''' || '''"i"''' |
| + | |- |
| + | | '''B''' || '''"B"''' || '''"B"''' |
| + | |- |
| + | | '''B''' || '''"B"''' || '''"u"''' |
| + | |- |
| + | | '''B''' || '''"u"''' || '''"B"''' |
| + | |- |
| + | | '''B''' || '''"u"''' || '''"u"''' |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |
| + | |+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B |
| + | |- style="background:paleturquoise" |
| + | ! style="width:20%" | Object |
| + | ! style="width:20%" | Sign |
| + | ! style="width:20%" | Interpretant |
| + | |- |
| + | | '''A''' || '''"A"''' || '''"A"''' |
| + | |- |
| + | | '''A''' || '''"A"''' || '''"u"''' |
| + | |- |
| + | | '''A''' || '''"u"''' || '''"A"''' |
| + | |- |
| + | | '''A''' || '''"u"''' || '''"u"''' |
| + | |- |
| + | | '''B''' || '''"B"''' || '''"B"''' |
| + | |- |
| + | | '''B''' || '''"B"''' || '''"i"''' |
| + | |- |
| + | | '''B''' || '''"i"''' || '''"B"''' |
| + | |- |
| + | | '''B''' || '''"i"''' || '''"i"''' |
| + | |} |
| + | <br> |
| + | |
| + | ===Triadic Relations=== |
| + | |
| + | ====Algebraic Examples==== |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |
| + | |+ '''L'''<sub>0</sub> = {(''x'', ''y'', ''z'') ∈ '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 0} |
| + | |- style="background:paleturquoise" |
| + | ! X !! Y !! Z |
| + | |- |
| + | | '''0''' || '''0''' || '''0''' |
| + | |- |
| + | | '''0''' || '''1''' || '''1''' |
| + | |- |
| + | | '''1''' || '''0''' || '''1''' |
| + | |- |
| + | | '''1''' || '''1''' || '''0''' |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |
| + | |+ '''L'''<sub>1</sub> = {(''x'', ''y'', ''z'') ∈ '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 1} |
| + | |- style="background:paleturquoise" |
| + | ! X !! Y !! Z |
| + | |- |
| + | | '''0''' || '''0''' || '''1''' |
| + | |- |
| + | | '''0''' || '''1''' || '''0''' |
| + | |- |
| + | | '''1''' || '''0''' || '''0''' |
| + | |- |
| + | | '''1''' || '''1''' || '''1''' |
| + | |} |
| + | <br> |
| + | |
| + | ====Semiotic Examples==== |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |
| + | |+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A |
| + | |- style="background:paleturquoise" |
| + | ! style="width:20%" | Object |
| + | ! style="width:20%" | Sign |
| + | ! style="width:20%" | Interpretant |
| + | |- |
| + | | '''A''' || '''"A"''' || '''"A"''' |
| + | |- |
| + | | '''A''' || '''"A"''' || '''"i"''' |
| + | |- |
| + | | '''A''' || '''"i"''' || '''"A"''' |
| + | |- |
| + | | '''A''' || '''"i"''' || '''"i"''' |
| + | |- |
| + | | '''B''' || '''"B"''' || '''"B"''' |
| + | |- |
| + | | '''B''' || '''"B"''' || '''"u"''' |
| + | |- |
| + | | '''B''' || '''"u"''' || '''"B"''' |
| + | |- |
| + | | '''B''' || '''"u"''' || '''"u"''' |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |
| + | |+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B |
| + | |- style="background:paleturquoise" |
| + | ! style="width:20%" | Object |
| + | ! style="width:20%" | Sign |
| + | ! style="width:20%" | Interpretant |
| + | |- |
| + | | '''A''' || '''"A"''' || '''"A"''' |
| + | |- |
| + | | '''A''' || '''"A"''' || '''"u"''' |
| + | |- |
| + | | '''A''' || '''"u"''' || '''"A"''' |
| + | |- |
| + | | '''A''' || '''"u"''' || '''"u"''' |
| + | |- |
| + | | '''B''' || '''"B"''' || '''"B"''' |
| + | |- |
| + | | '''B''' || '''"B"''' || '''"i"''' |
| + | |- |
| + | | '''B''' || '''"i"''' || '''"B"''' |
| + | |- |
| + | | '''B''' || '''"i"''' || '''"i"''' |
| + | |} |
| + | <br> |
| + | |
| + | ===Dyadic Projections=== |
| + | |
| + | {| cellpadding="4" |
| + | | width="20px" | |
| + | | '''L'''<sub>OS</sub> |
| + | | = |
| + | | ''proj''<sub>OS</sub>('''L''') |
| + | | = |
| + | | { (''o'', ''s'') ∈ '''O''' × '''S''' : (''o'', ''s'', ''i'') ∈ '''L''' for some ''i'' ∈ '''I''' } |
| + | |- |
| + | | width="20px" | |
| + | | '''L'''<sub>SO</sub> |
| + | | = |
| + | | ''proj''<sub>SO</sub>('''L''') |
| + | | = |
| + | | { (''s'', ''o'') ∈ '''S''' × '''O''' : (''o'', ''s'', ''i'') ∈ '''L''' for some ''i'' ∈ '''I''' } |
| + | |- |
| + | | width="20px" | |
| + | | '''L'''<sub>IS</sub> |
| + | | = |
| + | | ''proj''<sub>IS</sub>('''L''') |
| + | | = |
| + | | { (''i'', ''s'') ∈ '''I''' × '''S''' : (''o'', ''s'', ''i'') ∈ '''L''' for some ''o'' ∈ '''O''' } |
| + | |- |
| + | | width="20px" | |
| + | | '''L'''<sub>SI</sub> |
| + | | = |
| + | | ''proj''<sub>SI</sub>('''L''') |
| + | | = |
| + | | { (''s'', ''i'') ∈ '''S''' × '''I''' : (''o'', ''s'', ''i'') ∈ '''L''' for some ''o'' ∈ '''O''' } |
| + | |- |
| + | | width="20px" | |
| + | | '''L'''<sub>OI</sub> |
| + | | = |
| + | | ''proj''<sub>OI</sub>('''L''') |
| + | | = |
| + | | { (''o'', ''i'') ∈ '''O''' × '''I''' : (''o'', ''s'', ''i'') ∈ '''L''' for some ''s'' ∈ '''S''' } |
| + | |- |
| + | | width="20px" | |
| + | | '''L'''<sub>IO</sub> |
| + | | = |
| + | | ''proj''<sub>IO</sub>('''L''') |
| + | | = |
| + | | { (''i'', ''o'') ∈ '''I''' × '''O''' : (''o'', ''s'', ''i'') ∈ '''L''' for some ''s'' ∈ '''S''' } |
| + | |} |
| + | <br> |
| + | |
| + | ====Method 1 : Subtitles as Captions==== |
| + | |
| + | {| align="center" style="width:90%" |
| + | | |
| + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |+ ''proj''<sub>OS</sub>('''L'''<sub>A</sub>) |
| + | |- style="background:paleturquoise" |
| + | ! style="width:50%" | Object |
| + | ! style="width:50%" | Sign |
| + | |- |
| + | | '''A''' || '''"A"''' |
| + | |- |
| + | | '''A''' || '''"i"''' |
| + | |- |
| + | | '''B''' || '''"B"''' |
| + | |- |
| + | | '''B''' || '''"u"''' |
| + | |} |
| + | | |
| + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |+ ''proj''<sub>OS</sub>('''L'''<sub>B</sub>) |
| + | |- style="background:paleturquoise" |
| + | ! style="width:50%" | Object |
| + | ! style="width:50%" | Sign |
| + | |- |
| + | | '''A''' || '''"A"''' |
| + | |- |
| + | | '''A''' || '''"u"''' |
| + | |- |
| + | | '''B''' || '''"B"''' |
| + | |- |
| + | | '''B''' || '''"i"''' |
| + | |} |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" style="width:90%" |
| + | | |
| + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |+ ''proj''<sub>SI</sub>('''L'''<sub>A</sub>) |
| + | |- style="background:paleturquoise" |
| + | ! style="width:50%" | Sign |
| + | ! style="width:50%" | Interpretant |
| + | |- |
| + | | '''"A"''' || '''"A"''' |
| + | |- |
| + | | '''"A"''' || '''"i"''' |
| + | |- |
| + | | '''"i"''' || '''"A"''' |
| + | |- |
| + | | '''"i"''' || '''"i"''' |
| + | |- |
| + | | '''"B"''' || '''"B"''' |
| + | |- |
| + | | '''"B"''' || '''"u"''' |
| + | |- |
| + | | '''"u"''' || '''"B"''' |
| + | |- |
| + | | '''"u"''' || '''"u"''' |
| + | |} |
| + | | |
| + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |+ ''proj''<sub>SI</sub>('''L'''<sub>B</sub>) |
| + | |- style="background:paleturquoise" |
| + | ! style="width:50%" | Sign |
| + | ! style="width:50%" | Interpretant |
| + | |- |
| + | | '''"A"''' || '''"A"''' |
| + | |- |
| + | | '''"A"''' || '''"u"''' |
| + | |- |
| + | | '''"u"''' || '''"A"''' |
| + | |- |
| + | | '''"u"''' || '''"u"''' |
| + | |- |
| + | | '''"B"''' || '''"B"''' |
| + | |- |
| + | | '''"B"''' || '''"i"''' |
| + | |- |
| + | | '''"i"''' || '''"B"''' |
| + | |- |
| + | | '''"i"''' || '''"i"''' |
| + | |} |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" style="width:90%" |
| + | | |
| + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |+ ''proj''<sub>OI</sub>('''L'''<sub>A</sub>) |
| + | |- style="background:paleturquoise" |
| + | ! style="width:50%" | Object |
| + | ! style="width:50%" | Interpretant |
| + | |- |
| + | | '''A''' || '''"A"''' |
| + | |- |
| + | | '''A''' || '''"i"''' |
| + | |- |
| + | | '''B''' || '''"B"''' |
| + | |- |
| + | | '''B''' || '''"u"''' |
| + | |} |
| + | | |
| + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |+ ''proj''<sub>OI</sub>('''L'''<sub>B</sub>) |
| + | |- style="background:paleturquoise" |
| + | ! style="width:50%" | Object |
| + | ! style="width:50%" | Interpretant |
| + | |- |
| + | | '''A''' || '''"A"''' |
| + | |- |
| + | | '''A''' || '''"u"''' |
| + | |- |
| + | | '''B''' || '''"B"''' |
| + | |- |
| + | | '''B''' || '''"i"''' |
| + | |} |
| + | |} |
| + | <br> |
| + | |
| + | ====Method 2 : Subtitles as Top Rows==== |
| + | |
| + | {| align="center" style="width:90%" |
| + | | align="center" style="width:45%" | ''proj''<sub>OS</sub>('''L'''<sub>A</sub>) |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |- style="background:paleturquoise" |
| + | ! style="width:50%" | Object |
| + | ! style="width:50%" | Sign |
| + | |- |
| + | | '''A''' || '''"A"''' |
| + | |- |
| + | | '''A''' || '''"i"''' |
| + | |- |
| + | | '''B''' || '''"B"''' |
| + | |- |
| + | | '''B''' || '''"u"''' |
| + | |} |
| + | | align="center" style="width:45%" | ''proj''<sub>OS</sub>('''L'''<sub>B</sub>) |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |- style="background:paleturquoise" |
| + | ! style="width:50%" | Object |
| + | ! style="width:50%" | Sign |
| + | |- |
| + | | '''A''' || '''"A"''' |
| + | |- |
| + | | '''A''' || '''"u"''' |
| + | |- |
| + | | '''B''' || '''"B"''' |
| + | |- |
| + | | '''B''' || '''"i"''' |
| + | |} |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" style="width:90%" |
| + | | align="center" style="width:45%" | ''proj''<sub>SI</sub>('''L'''<sub>A</sub>) |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |- style="background:paleturquoise" |
| + | ! style="width:50%" | Sign |
| + | ! style="width:50%" | Interpretant |
| + | |- |
| + | | '''"A"''' || '''"A"''' |
| + | |- |
| + | | '''"A"''' || '''"i"''' |
| + | |- |
| + | | '''"i"''' || '''"A"''' |
| + | |- |
| + | | '''"i"''' || '''"i"''' |
| + | |- |
| + | | '''"B"''' || '''"B"''' |
| + | |- |
| + | | '''"B"''' || '''"u"''' |
| + | |- |
| + | | '''"u"''' || '''"B"''' |
| + | |- |
| + | | '''"u"''' || '''"u"''' |
| + | |} |
| + | | align="center" style="width:45%" | ''proj''<sub>SI</sub>('''L'''<sub>B</sub>) |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |- style="background:paleturquoise" |
| + | ! style="width:50%" | Sign |
| + | ! style="width:50%" | Interpretant |
| + | |- |
| + | | '''"A"''' || '''"A"''' |
| + | |- |
| + | | '''"A"''' || '''"u"''' |
| + | |- |
| + | | '''"u"''' || '''"A"''' |
| + | |- |
| + | | '''"u"''' || '''"u"''' |
| + | |- |
| + | | '''"B"''' || '''"B"''' |
| + | |- |
| + | | '''"B"''' || '''"i"''' |
| + | |- |
| + | | '''"i"''' || '''"B"''' |
| + | |- |
| + | | '''"i"''' || '''"i"''' |
| + | |} |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" style="width:90%" |
| + | | align="center" style="width:45%" | ''proj''<sub>OI</sub>('''L'''<sub>A</sub>) |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |- style="background:paleturquoise" |
| + | ! style="width:50%" | Object |
| + | ! style="width:50%" | Interpretant |
| + | |- |
| + | | '''A''' || '''"A"''' |
| + | |- |
| + | | '''A''' || '''"i"''' |
| + | |- |
| + | | '''B''' || '''"B"''' |
| + | |- |
| + | | '''B''' || '''"u"''' |
| + | |} |
| + | | align="center" style="width:45%" | ''proj''<sub>OI</sub>('''L'''<sub>B</sub>) |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |- style="background:paleturquoise" |
| + | ! style="width:50%" | Object |
| + | ! style="width:50%" | Interpretant |
| + | |- |
| + | | '''A''' || '''"A"''' |
| + | |- |
| + | | '''A''' || '''"u"''' |
| + | |- |
| + | | '''B''' || '''"B"''' |
| + | |- |
| + | | '''B''' || '''"i"''' |
| + | |} |
| + | |} |
| + | <br> |
| + | |
| + | ===Relation Reduction=== |
| + | |
| + | ====Method 1 : Subtitles as Captions==== |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |
| + | |+ '''L'''<sub>0</sub> = {(''x'', ''y'', ''z'') ∈ '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 0} |
| + | |- style="background:paleturquoise" |
| + | ! X !! Y !! Z |
| + | |- |
| + | | '''0''' || '''0''' || '''0''' |
| + | |- |
| + | | '''0''' || '''1''' || '''1''' |
| + | |- |
| + | | '''1''' || '''0''' || '''1''' |
| + | |- |
| + | | '''1''' || '''1''' || '''0''' |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |
| + | |+ '''L'''<sub>1</sub> = {(''x'', ''y'', ''z'') ∈ '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 1} |
| + | |- style="background:paleturquoise" |
| + | ! X !! Y !! Z |
| + | |- |
| + | | '''0''' || '''0''' || '''1''' |
| + | |- |
| + | | '''0''' || '''1''' || '''0''' |
| + | |- |
| + | | '''1''' || '''0''' || '''0''' |
| + | |- |
| + | | '''1''' || '''1''' || '''1''' |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" style="width:90%" |
| + | | |
| + | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |+ proj<sub>''XY''</sub>('''L'''<sub>0</sub>) |
| + | |- style="background:paleturquoise" |
| + | ! X !! Y |
| + | |- |
| + | | '''0''' || '''0''' |
| + | |- |
| + | | '''0''' || '''1''' |
| + | |- |
| + | | '''1''' || '''0''' |
| + | |- |
| + | | '''1''' || '''1''' |
| + | |} |
| + | | |
| + | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |+ proj<sub>''XZ''</sub>('''L'''<sub>0</sub>) |
| + | |- style="background:paleturquoise" |
| + | ! X !! Z |
| + | |- |
| + | | '''0''' || '''0''' |
| + | |- |
| + | | '''0''' || '''1''' |
| + | |- |
| + | | '''1''' || '''1''' |
| + | |- |
| + | | '''1''' || '''0''' |
| + | |} |
| + | | |
| + | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |+ proj<sub>''YZ''</sub>('''L'''<sub>0</sub>) |
| + | |- style="background:paleturquoise" |
| + | ! Y !! Z |
| + | |- |
| + | | '''0''' || '''0''' |
| + | |- |
| + | | '''1''' || '''1''' |
| + | |- |
| + | | '''0''' || '''1''' |
| + | |- |
| + | | '''1''' || '''0''' |
| + | |} |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" style="width:90%" |
| + | | |
| + | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |+ proj<sub>''XY''</sub>('''L'''<sub>1</sub>) |
| + | |- style="background:paleturquoise" |
| + | ! X !! Y |
| + | |- |
| + | | '''0''' || '''0''' |
| + | |- |
| + | | '''0''' || '''1''' |
| + | |- |
| + | | '''1''' || '''0''' |
| + | |- |
| + | | '''1''' || '''1''' |
| + | |} |
| + | | |
| + | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |+ proj<sub>''XZ''</sub>('''L'''<sub>1</sub>) |
| + | |- style="background:paleturquoise" |
| + | ! X !! Z |
| + | |- |
| + | | '''0''' || '''1''' |
| + | |- |
| + | | '''0''' || '''0''' |
| + | |- |
| + | | '''1''' || '''0''' |
| + | |- |
| + | | '''1''' || '''1''' |
| + | |} |
| + | | |
| + | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |+ proj<sub>''YZ''</sub>('''L'''<sub>1</sub>) |
| + | |- style="background:paleturquoise" |
| + | ! Y !! Z |
| + | |- |
| + | | '''0''' || '''1''' |
| + | |- |
| + | | '''1''' || '''0''' |
| + | |- |
| + | | '''0''' || '''0''' |
| + | |- |
| + | | '''1''' || '''1''' |
| + | |} |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" cellpadding="4" style="text-align:center; width:90%" |
| + | | proj<sub>''XY''</sub>('''L'''<sub>0</sub>) = proj<sub>''XY''</sub>('''L'''<sub>1</sub>) |
| + | | proj<sub>''XZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''XZ''</sub>('''L'''<sub>1</sub>) |
| + | | proj<sub>''YZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''YZ''</sub>('''L'''<sub>1</sub>) |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |
| + | |+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A |
| + | |- style="background:paleturquoise" |
| + | ! style="width:20%" | Object |
| + | ! style="width:20%" | Sign |
| + | ! style="width:20%" | Interpretant |
| + | |- |
| + | | '''A''' || '''"A"''' || '''"A"''' |
| + | |- |
| + | | '''A''' || '''"A"''' || '''"i"''' |
| + | |- |
| + | | '''A''' || '''"i"''' || '''"A"''' |
| + | |- |
| + | | '''A''' || '''"i"''' || '''"i"''' |
| + | |- |
| + | | '''B''' || '''"B"''' || '''"B"''' |
| + | |- |
| + | | '''B''' || '''"B"''' || '''"u"''' |
| + | |- |
| + | | '''B''' || '''"u"''' || '''"B"''' |
| + | |- |
| + | | '''B''' || '''"u"''' || '''"u"''' |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |
| + | |+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B |
| + | |- style="background:paleturquoise" |
| + | ! style="width:20%" | Object |
| + | ! style="width:20%" | Sign |
| + | ! style="width:20%" | Interpretant |
| + | |- |
| + | | '''A''' || '''"A"''' || '''"A"''' |
| + | |- |
| + | | '''A''' || '''"A"''' || '''"u"''' |
| + | |- |
| + | | '''A''' || '''"u"''' || '''"A"''' |
| + | |- |
| + | | '''A''' || '''"u"''' || '''"u"''' |
| + | |- |
| + | | '''B''' || '''"B"''' || '''"B"''' |
| + | |- |
| + | | '''B''' || '''"B"''' || '''"i"''' |
| + | |- |
| + | | '''B''' || '''"i"''' || '''"B"''' |
| + | |- |
| + | | '''B''' || '''"i"''' || '''"i"''' |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" style="width:90%" |
| + | | |
| + | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |+ proj<sub>''XY''</sub>('''L'''<sub>A</sub>) |
| + | |- style="background:paleturquoise" |
| + | ! style="width:50%" | Object |
| + | ! style="width:50%" | Sign |
| + | |- |
| + | | '''A''' || '''"A"''' |
| + | |- |
| + | | '''A''' || '''"i"''' |
| + | |- |
| + | | '''B''' || '''"B"''' |
| + | |- |
| + | | '''B''' || '''"u"''' |
| + | |} |
| + | | |
| + | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |+ proj<sub>''XZ''</sub>('''L'''<sub>A</sub>) |
| + | |- style="background:paleturquoise" |
| + | ! style="width:50%" | Object |
| + | ! style="width:50%" | Interpretant |
| + | |- |
| + | | '''A''' || '''"A"''' |
| + | |- |
| + | | '''A''' || '''"i"''' |
| + | |- |
| + | | '''B''' || '''"B"''' |
| + | |- |
| + | | '''B''' || '''"u"''' |
| + | |} |
| + | | |
| + | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |+ proj<sub>''YZ''</sub>('''L'''<sub>A</sub>) |
| + | |- style="background:paleturquoise" |
| + | ! style="width:50%" | Sign |
| + | ! style="width:50%" | Interpretant |
| + | |- |
| + | | '''"A"''' || '''"A"''' |
| + | |- |
| + | | '''"A"''' || '''"i"''' |
| + | |- |
| + | | '''"i"''' || '''"A"''' |
| + | |- |
| + | | '''"i"''' || '''"i"''' |
| + | |- |
| + | | '''"B"''' || '''"B"''' |
| + | |- |
| + | | '''"B"''' || '''"u"''' |
| + | |- |
| + | | '''"u"''' || '''"B"''' |
| + | |- |
| + | | '''"u"''' || '''"u"''' |
| + | |} |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" style="width:90%" |
| + | | |
| + | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |+ proj<sub>''XY''</sub>('''L'''<sub>B</sub>) |
| + | |- style="background:paleturquoise" |
| + | ! style="width:50%" | Object |
| + | ! style="width:50%" | Sign |
| + | |- |
| + | | '''A''' || '''"A"''' |
| + | |- |
| + | | '''A''' || '''"u"''' |
| + | |- |
| + | | '''B''' || '''"B"''' |
| + | |- |
| + | | '''B''' || '''"i"''' |
| + | |} |
| + | | |
| + | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |+ proj<sub>''XZ''</sub>('''L'''<sub>B</sub>) |
| + | |- style="background:paleturquoise" |
| + | ! style="width:50%" | Object |
| + | ! style="width:50%" | Interpretant |
| + | |- |
| + | | '''A''' || '''"A"''' |
| + | |- |
| + | | '''A''' || '''"u"''' |
| + | |- |
| + | | '''B''' || '''"B"''' |
| + | |- |
| + | | '''B''' || '''"i"''' |
| + | |} |
| + | | |
| + | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |+ proj<sub>''YZ''</sub>('''L'''<sub>B</sub>) |
| + | |- style="background:paleturquoise" |
| + | ! style="width:50%" | Sign |
| + | ! style="width:50%" | Interpretant |
| + | |- |
| + | | '''"A"''' || '''"A"''' |
| + | |- |
| + | | '''"A"''' || '''"u"''' |
| + | |- |
| + | | '''"u"''' || '''"A"''' |
| + | |- |
| + | | '''"u"''' || '''"u"''' |
| + | |- |
| + | | '''"B"''' || '''"B"''' |
| + | |- |
| + | | '''"B"''' || '''"i"''' |
| + | |- |
| + | | '''"i"''' || '''"B"''' |
| + | |- |
| + | | '''"i"''' || '''"i"''' |
| + | |} |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" cellpadding="4" style="text-align:center; width:90%" |
| + | | proj<sub>''XY''</sub>('''L'''<sub>A</sub>) ≠ proj<sub>''XY''</sub>('''L'''<sub>B</sub>) |
| + | | proj<sub>''XZ''</sub>('''L'''<sub>A</sub>) ≠ proj<sub>''XZ''</sub>('''L'''<sub>B</sub>) |
| + | | proj<sub>''YZ''</sub>('''L'''<sub>A</sub>) ≠ proj<sub>''YZ''</sub>('''L'''<sub>B</sub>) |
| + | |} |
| + | <br> |
| + | |
| + | ====Method 2 : Subtitles as Top Rows==== |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |
| + | |+ '''L'''<sub>0</sub> = {(''x'', ''y'', ''z'') ∈ '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 0} |
| + | |- style="background:paleturquoise" |
| + | ! X !! Y !! Z |
| + | |- |
| + | | '''0''' || '''0''' || '''0''' |
| + | |- |
| + | | '''0''' || '''1''' || '''1''' |
| + | |- |
| + | | '''1''' || '''0''' || '''1''' |
| + | |- |
| + | | '''1''' || '''1''' || '''0''' |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |
| + | |+ '''L'''<sub>1</sub> = {(''x'', ''y'', ''z'') ∈ '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 1} |
| + | |- style="background:paleturquoise" |
| + | ! X !! Y !! Z |
| + | |- |
| + | | '''0''' || '''0''' || '''1''' |
| + | |- |
| + | | '''0''' || '''1''' || '''0''' |
| + | |- |
| + | | '''1''' || '''0''' || '''0''' |
| + | |- |
| + | | '''1''' || '''1''' || '''1''' |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" style="width:90%" |
| + | | align="center" | proj<sub>''XY''</sub>('''L'''<sub>0</sub>) |
| + | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |- style="background:paleturquoise" |
| + | ! X !! Y |
| + | |- |
| + | | '''0''' || '''0''' |
| + | |- |
| + | | '''0''' || '''1''' |
| + | |- |
| + | | '''1''' || '''0''' |
| + | |- |
| + | | '''1''' || '''1''' |
| + | |} |
| + | | align="center" | proj<sub>''XZ''</sub>('''L'''<sub>0</sub>) |
| + | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |- style="background:paleturquoise" |
| + | ! X !! Z |
| + | |- |
| + | | '''0''' || '''0''' |
| + | |- |
| + | | '''0''' || '''1''' |
| + | |- |
| + | | '''1''' || '''1''' |
| + | |- |
| + | | '''1''' || '''0''' |
| + | |} |
| + | | align="center" | proj<sub>''YZ''</sub>('''L'''<sub>0</sub>) |
| + | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |- style="background:paleturquoise" |
| + | ! Y !! Z |
| + | |- |
| + | | '''0''' || '''0''' |
| + | |- |
| + | | '''1''' || '''1''' |
| + | |- |
| + | | '''0''' || '''1''' |
| + | |- |
| + | | '''1''' || '''0''' |
| + | |} |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" style="width:90%" |
| + | | align="center" | proj<sub>''XY''</sub>('''L'''<sub>1</sub>) |
| + | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |- style="background:paleturquoise" |
| + | ! X !! Y |
| + | |- |
| + | | '''0''' || '''0''' |
| + | |- |
| + | | '''0''' || '''1''' |
| + | |- |
| + | | '''1''' || '''0''' |
| + | |- |
| + | | '''1''' || '''1''' |
| + | |} |
| + | | align="center" | proj<sub>''XZ''</sub>('''L'''<sub>1</sub>) |
| + | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |- style="background:paleturquoise" |
| + | ! X !! Z |
| + | |- |
| + | | '''0''' || '''1''' |
| + | |- |
| + | | '''0''' || '''0''' |
| + | |- |
| + | | '''1''' || '''0''' |
| + | |- |
| + | | '''1''' || '''1''' |
| + | |} |
| + | | align="center" | proj<sub>''YZ''</sub>('''L'''<sub>1</sub>) |
| + | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |- style="background:paleturquoise" |
| + | ! Y !! Z |
| + | |- |
| + | | '''0''' || '''1''' |
| + | |- |
| + | | '''1''' || '''0''' |
| + | |- |
| + | | '''0''' || '''0''' |
| + | |- |
| + | | '''1''' || '''1''' |
| + | |} |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" cellpadding="4" style="text-align:center; width:90%" |
| + | | proj<sub>''XY''</sub>('''L'''<sub>0</sub>) = proj<sub>''XY''</sub>('''L'''<sub>1</sub>) |
| + | | proj<sub>''XZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''XZ''</sub>('''L'''<sub>1</sub>) |
| + | | proj<sub>''YZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''YZ''</sub>('''L'''<sub>1</sub>) |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |
| + | |+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A |
| + | |- style="background:paleturquoise" |
| + | ! style="width:20%" | Object |
| + | ! style="width:20%" | Sign |
| + | ! style="width:20%" | Interpretant |
| + | |- |
| + | | '''A''' || '''"A"''' || '''"A"''' |
| + | |- |
| + | | '''A''' || '''"A"''' || '''"i"''' |
| + | |- |
| + | | '''A''' || '''"i"''' || '''"A"''' |
| + | |- |
| + | | '''A''' || '''"i"''' || '''"i"''' |
| + | |- |
| + | | '''B''' || '''"B"''' || '''"B"''' |
| + | |- |
| + | | '''B''' || '''"B"''' || '''"u"''' |
| + | |- |
| + | | '''B''' || '''"u"''' || '''"B"''' |
| + | |- |
| + | | '''B''' || '''"u"''' || '''"u"''' |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |
| + | |+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B |
| + | |- style="background:paleturquoise" |
| + | ! style="width:20%" | Object |
| + | ! style="width:20%" | Sign |
| + | ! style="width:20%" | Interpretant |
| + | |- |
| + | | '''A''' || '''"A"''' || '''"A"''' |
| + | |- |
| + | | '''A''' || '''"A"''' || '''"u"''' |
| + | |- |
| + | | '''A''' || '''"u"''' || '''"A"''' |
| + | |- |
| + | | '''A''' || '''"u"''' || '''"u"''' |
| + | |- |
| + | | '''B''' || '''"B"''' || '''"B"''' |
| + | |- |
| + | | '''B''' || '''"B"''' || '''"i"''' |
| + | |- |
| + | | '''B''' || '''"i"''' || '''"B"''' |
| + | |- |
| + | | '''B''' || '''"i"''' || '''"i"''' |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" style="width:90%" |
| + | | align="center" style="width:30%" | proj<sub>''XY''</sub>('''L'''<sub>A</sub>) |
| + | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |- style="background:paleturquoise" |
| + | ! style="width:50%" | Object |
| + | ! style="width:50%" | Sign |
| + | |- |
| + | | '''A''' || '''"A"''' |
| + | |- |
| + | | '''A''' || '''"i"''' |
| + | |- |
| + | | '''B''' || '''"B"''' |
| + | |- |
| + | | '''B''' || '''"u"''' |
| + | |} |
| + | | align="center" style="width:30%" | proj<sub>''XZ''</sub>('''L'''<sub>A</sub>) |
| + | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |- style="background:paleturquoise" |
| + | ! style="width:50%" | Object |
| + | ! style="width:50%" | Interpretant |
| + | |- |
| + | | '''A''' || '''"A"''' |
| + | |- |
| + | | '''A''' || '''"i"''' |
| + | |- |
| + | | '''B''' || '''"B"''' |
| + | |- |
| + | | '''B''' || '''"u"''' |
| + | |} |
| + | | align="center" style="width:30%" | proj<sub>''YZ''</sub>('''L'''<sub>A</sub>) |
| + | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |- style="background:paleturquoise" |
| + | ! style="width:50%" | Sign |
| + | ! style="width:50%" | Interpretant |
| + | |- |
| + | | '''"A"''' || '''"A"''' |
| + | |- |
| + | | '''"A"''' || '''"i"''' |
| + | |- |
| + | | '''"i"''' || '''"A"''' |
| + | |- |
| + | | '''"i"''' || '''"i"''' |
| + | |- |
| + | | '''"B"''' || '''"B"''' |
| + | |- |
| + | | '''"B"''' || '''"u"''' |
| + | |- |
| + | | '''"u"''' || '''"B"''' |
| + | |- |
| + | | '''"u"''' || '''"u"''' |
| + | |} |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" style="width:90%" |
| + | | align="center" style="width:30%" | proj<sub>''XY''</sub>('''L'''<sub>B</sub>) |
| + | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |- style="background:paleturquoise" |
| + | ! style="width:50%" | Object |
| + | ! style="width:50%" | Sign |
| + | |- |
| + | | '''A''' || '''"A"''' |
| + | |- |
| + | | '''A''' || '''"u"''' |
| + | |- |
| + | | '''B''' || '''"B"''' |
| + | |- |
| + | | '''B''' || '''"i"''' |
| + | |} |
| + | | align="center" style="width:30%" | proj<sub>''XZ''</sub>('''L'''<sub>B</sub>) |
| + | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |- style="background:paleturquoise" |
| + | ! style="width:50%" | Object |
| + | ! style="width:50%" | Interpretant |
| + | |- |
| + | | '''A''' || '''"A"''' |
| + | |- |
| + | | '''A''' || '''"u"''' |
| + | |- |
| + | | '''B''' || '''"B"''' |
| + | |- |
| + | | '''B''' || '''"i"''' |
| + | |} |
| + | | align="center" style="width:30%" | proj<sub>''YZ''</sub>('''L'''<sub>B</sub>) |
| + | {| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |- style="background:paleturquoise" |
| + | ! style="width:50%" | Sign |
| + | ! style="width:50%" | Interpretant |
| + | |- |
| + | | '''"A"''' || '''"A"''' |
| + | |- |
| + | | '''"A"''' || '''"u"''' |
| + | |- |
| + | | '''"u"''' || '''"A"''' |
| + | |- |
| + | | '''"u"''' || '''"u"''' |
| + | |- |
| + | | '''"B"''' || '''"B"''' |
| + | |- |
| + | | '''"B"''' || '''"i"''' |
| + | |- |
| + | | '''"i"''' || '''"B"''' |
| + | |- |
| + | | '''"i"''' || '''"i"''' |
| + | |} |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" cellpadding="4" style="text-align:center; width:90%" |
| + | | proj<sub>''XY''</sub>('''L'''<sub>A</sub>) ≠ proj<sub>''XY''</sub>('''L'''<sub>B</sub>) |
| + | | proj<sub>''XZ''</sub>('''L'''<sub>A</sub>) ≠ proj<sub>''XZ''</sub>('''L'''<sub>B</sub>) |
| + | | proj<sub>''YZ''</sub>('''L'''<sub>A</sub>) ≠ proj<sub>''YZ''</sub>('''L'''<sub>B</sub>) |
| + | |} |
| + | <br> |
| + | |
| + | ===Formatted Text Display=== |
| + | |
| + | : So in a triadic fact, say, the example <br> |
| + | {| align="center" cellspacing="8" style="width:72%" |
| + | | align="center" | ''A'' gives ''B'' to ''C'' |
| + | |} |
| + | : we make no distinction in the ordinary logic of relations between the ''[[subject (grammar)|subject]] [[nominative]]'', the ''[[direct object]]'', and the ''[[indirect object]]''. We say that the proposition has three ''logical subjects''. We regard it as a mere affair of English grammar that there are six ways of expressing this: <br> |
| + | {| align="center" cellspacing="8" style="width:72%" |
| + | | style="width:36%" | ''A'' gives ''B'' to ''C'' |
| + | | style="width:36%" | ''A'' benefits ''C'' with ''B'' |
| + | |- |
| + | | ''B'' enriches ''C'' at expense of ''A'' |
| + | | ''C'' receives ''B'' from ''A'' |
| + | |- |
| + | | ''C'' thanks ''A'' for ''B'' |
| + | | ''B'' leaves ''A'' for ''C'' |
| + | |} |
| + | : These six sentences express one and the same indivisible phenomenon. (C.S. Peirce, "The Categories Defended", MS 308 (1903), EP 2, 170-171). |
| + | |
| + | ==Work Area== |
| + | |
| + | {| border="1" cellspacing="0" cellpadding="0" style="text-align:center" |
| + | |+ Binary Operations |
| + | |- |
| + | ! style="width:2em" | x<sub>0</sub> |
| + | ! style="width:2em" | x<sub>1</sub> |
| + | | style="width:2em" | <sup>2</sup>f<sub>0</sub> |
| + | | style="width:2em" | <sup>2</sup>f<sub>1</sub> |
| + | | style="width:2em" | <sup>2</sup>f<sub>2</sub> |
| + | | style="width:2em" | <sup>2</sup>f<sub>3</sub> |
| + | | style="width:2em" | <sup>2</sup>f<sub>4</sub> |
| + | | style="width:2em" | <sup>2</sup>f<sub>5</sub> |
| + | | style="width:2em" | <sup>2</sup>f<sub>6</sub> |
| + | | style="width:2em" | <sup>2</sup>f<sub>7</sub> |
| + | | style="width:2em" | <sup>2</sup>f<sub>8</sub> |
| + | | style="width:2em" | <sup>2</sup>f<sub>9</sub> |
| + | | style="width:2em" | <sup>2</sup>f<sub>10</sub> |
| + | | style="width:2em" | <sup>2</sup>f<sub>11</sub> |
| + | | style="width:2em" | <sup>2</sup>f<sub>12</sub> |
| + | | style="width:2em" | <sup>2</sup>f<sub>13</sub> |
| + | | style="width:2em" | <sup>2</sup>f<sub>14</sub> |
| + | | style="width:2em" | <sup>2</sup>f<sub>15</sub> |
| + | |- |
| + | | 0 || 0 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 |
| + | |- |
| + | | 1 || 0 || 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 |
| + | |- |
| + | | 0 || 1 || 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 |
| + | |- |
| + | | 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 |
| + | |} |
| + | <br> |
| + | |
| + | ===Draft 1=== |
| + | |
| + | <center><table> |
| + | <caption>TRUTH TABLES FOR THE BOOLEAN OPERATIONS OF ARITY UP TO 2</caption> |
| + | <tr valign="top"> |
| + | <td><table border=5 cellspacing=0> |
| + | <caption>Constants</caption> |
| + | <tr><td></td> |
| + | <td><sup>0</sup>f<sub>0</sub></td> <td><sup>0</sup>f<sub>1</sub></td> |
| + | </tr> <tr><td></td> |
| + | <td align=center>0</td> <td align=center>1</td> |
| + | </tr></table></td><td> </td> |
| + | <td><table border=5 cellspacing=0><caption>Unary Operations</caption><tr> |
| + | <td>x<sub>0</sub></td> <td></td> |
| + | <td><sup>1</sup>f<sub>0 </sub></td> <td><sup>1</sup>f<sub>1 </sub></td> |
| + | <td><sup>1</sup>f<sub>2 </sub></td> <td><sup>1</sup>f<sub>3 </sub></td> |
| + | </tr><tr> <td align=center>0</td> <td></td> |
| + | <td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td> |
| + | </tr> <tr> <td align=center>1</td> <td></td> |
| + | <td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td> |
| + | </tr></table></td><td> </td> |
| + | <td><table border=5 cellspacing=0><caption>Binary Operations</caption><tr> |
| + | <td>x<sub>0</sub></td> <td>x<sub>1</sub></td> |
| + | <td></td> |
| + | <td><sup>2</sup>f<sub>0</sub></td> <td><sup>2</sup>f<sub>1 </sub></td> |
| + | <td><sup> 2</sup>f<sub>2 </sub></td> <td><sup>2</sup>f<sub>3 </sub></td> |
| + | <td><sup>2</sup>f<sub>4 </sub></td> <td><sup>2</sup>f<sub>5 </sub></td> |
| + | <td><sup>2</sup>f<sub>6 </sub></td> <td><sup>2</sup>f<sub>7 </sub></td> |
| + | <td><sup>2</sup>f<sub>8 </sub></td> <td><sup>2</sup>f<sub>9 </sub></td> |
| + | <td><sup>2</sup>f<sub>10</sub></td> <td><sup>2</sup>f<sub>11</sub></td> |
| + | <td><sup>2</sup>f<sub>12</sub></td> <td><sup>2</sup>f<sub>13</sub></td> |
| + | <td><sup>2</sup>f<sub>14</sub></td> <td><sup>2</sup>f<sub>15</sub></td> |
| + | </tr><tr> <td align=center>0</td> <td align=center>0</td> <td></td> |
| + | <td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td> |
| + | <td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td> |
| + | <td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td> |
| + | <td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td> |
| + | </tr> <tr> <td align=center>1</td> <td align=center>0</td> <td></td> |
| + | <td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td> |
| + | <td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td> |
| + | <td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td> |
| + | <td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td> |
| + | </tr> <tr> <td align=center>0</td> <td align=center>1</td> <td></td> |
| + | <td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td> |
| + | <td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td> |
| + | <td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td> |
| + | <td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td> |
| + | </tr> <tr> <td align=center>1</td> <td align=center>1</td> <td></td> |
| + | <td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td> |
| + | <td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td> |
| + | <td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td> |
| + | <td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td> |
| + | </tr> </table></td> |
| </table></center> | | </table></center> |
| + | |
| + | ===Draft 2=== |
| + | |
| + | <center><table> |
| + | <caption>TRUTH TABLES FOR THE BOOLEAN OPERATIONS OF ARITY UP TO 2</caption> |
| + | <tr valign="top"> |
| + | <td><table border=5 cellspacing=0> |
| + | <caption>Constants</caption> |
| + | <tr><td></td> |
| + | <td><sup>0</sup>f<sub>0</sub></td> <td><sup>0</sup>f<sub>1</sub></td> |
| + | </tr> <tr><td></td> |
| + | <td align=center>0</td> <td align=center>1</td> |
| + | </tr></table></td><td> </td> |
| + | <td><table border=5 cellspacing=0><caption>Unary Operations</caption><tr> |
| + | <td>x<sub>0</sub></td> <td></td> |
| + | <td><sup>1</sup>f<sub>0 </sub></td> <td><sup>1</sup>f<sub>1 </sub></td> |
| + | <td><sup>1</sup>f<sub>2 </sub></td> <td><sup>1</sup>f<sub>3 </sub></td> |
| + | </tr><tr> <td align=center>0</td> <td></td> |
| + | <td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td> |
| + | </tr> <tr> <td align=center>1</td> <td></td> |
| + | <td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td> |
| + | </tr></table></td><td> </td> |
| + | <td><table border=5 cellspacing=0><caption>Binary Operations</caption><tr> |
| + | <td>x<sub>0</sub></td> <td>x<sub>1</sub></td> |
| + | <td></td> |
| + | <td><sup>2</sup>f<sub>0</sub></td> <td><sup>2</sup>f<sub>1 </sub></td> |
| + | <td><sup> 2</sup>f<sub>2 </sub></td> <td><sup>2</sup>f<sub>3 </sub></td> |
| + | <td><sup>2</sup>f<sub>4 </sub></td> <td><sup>2</sup>f<sub>5 </sub></td> |
| + | <td><sup>2</sup>f<sub>6 </sub></td> <td><sup>2</sup>f<sub>7 </sub></td> |
| + | <td><sup>2</sup>f<sub>8 </sub></td> <td><sup>2</sup>f<sub>9 </sub></td> |
| + | <td><sup>2</sup>f<sub>10</sub></td> <td><sup>2</sup>f<sub>11</sub></td> |
| + | <td><sup>2</sup>f<sub>12</sub></td> <td><sup>2</sup>f<sub>13</sub></td> |
| + | <td><sup>2</sup>f<sub>14</sub></td> <td><sup>2</sup>f<sub>15</sub></td> |
| + | </tr><tr> <td align=center>0</td> <td align=center>0</td> <td></td> |
| + | <td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td> |
| + | <td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td> |
| + | <td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td> |
| + | <td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td> |
| + | </tr> <tr> <td align=center>1</td> <td align=center>0</td> <td></td> |
| + | <td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td> |
| + | <td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td> |
| + | <td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td> |
| + | <td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td> |
| + | </tr> <tr> <td align=center>0</td> <td align=center>1</td> <td></td> |
| + | <td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td> |
| + | <td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td> |
| + | <td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td> |
| + | <td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td> |
| + | </tr> <tr> <td align=center>1</td> <td align=center>1</td> <td></td> |
| + | <td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td> |
| + | <td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td> |
| + | <td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td> |
| + | <td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td> |
| + | </tr> </table></td> |
| + | </table></center> |
| + | |
| + | ==Inquiry and Analogy== |
| + | |
| + | ===Test Patterns=== |
| + | |
| + | {| align="center" |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | |}<br> |
| + | |
| + | {| align="center" |
| + | | style="background:white; color:black" | 1 |
| + | | style="background:black; color:white" | 0 |
| + | | style="background:white; color:black" | 1 |
| + | | style="background:black; color:white" | 0 |
| + | | style="background:white; color:black" | 1 |
| + | | style="background:black; color:white" | 0 |
| + | | style="background:white; color:black" | 1 |
| + | | style="background:black; color:white" | 0 |
| + | |- |
| + | | style="background:black; color:white" | 0 |
| + | | style="background:white; color:black" | 1 |
| + | | style="background:black; color:white" | 0 |
| + | | style="background:white; color:black" | 1 |
| + | | style="background:black; color:white" | 0 |
| + | | style="background:white; color:black" | 1 |
| + | | style="background:black; color:white" | 0 |
| + | | style="background:white; color:black" | 1 |
| + | |}<br> |
| + | |
| + | {| align="center" |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | |}<br> |
| + | |
| + | ===Table 10=== |
| + | |
| + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" |
| + | |+ '''Table 10. Higher Order Propositions (''n'' = 1)''' |
| + | |- style="background:ghostwhite" |
| + | | align="right" | <math>x</math>: |
| + | | 1 0 |
| + | | <math>f</math> |
| + | | <math>m_0</math> |
| + | | <math>m_1</math> |
| + | | <math>m_2</math> |
| + | | <math>m_3</math> |
| + | | <math>m_4</math> |
| + | | <math>m_5</math> |
| + | | <math>m_6</math> |
| + | | <math>m_7</math> |
| + | | <math>m_8</math> |
| + | | <math>m_9</math> |
| + | | <math>m_{10}</math> |
| + | | <math>m_{11}</math> |
| + | | <math>m_{12}</math> |
| + | | <math>m_{13}</math> |
| + | | <math>m_{14}</math> |
| + | | <math>m_{15}</math> |
| + | |- |
| + | | <math>f_0</math> |
| + | | 0 0 |
| + | | <math>0\!</math> |
| + | | 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 |
| + | |- |
| + | | <math>f_1</math> |
| + | | 0 1 |
| + | | <math>(x)\!</math> |
| + | | 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 |
| + | |- |
| + | | <math>f_2</math> |
| + | | 1 0 |
| + | | <math>x\!</math> |
| + | | 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 |
| + | |- |
| + | | <math>f_3</math> |
| + | | 1 1 |
| + | | <math>1\!</math> |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 |
| + | |}<br> |
| + | |
| + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:white; color:black; font-weight:bold; text-align:center; width:96%" |
| + | |+ '''Table 10. Higher Order Propositions (''n'' = 1)''' |
| + | |- style="background:ghostwhite" |
| + | | align="right" | <math>x:</math> |
| + | | 1 0 |
| + | | <math>f\!</math> |
| + | | <math>m_0</math> |
| + | | <math>m_1</math> |
| + | | <math>m_2</math> |
| + | | <math>m_3</math> |
| + | | <math>m_4</math> |
| + | | <math>m_5</math> |
| + | | <math>m_6</math> |
| + | | <math>m_7</math> |
| + | | <math>m_8</math> |
| + | | <math>m_9</math> |
| + | | <math>m_{10}</math> |
| + | | <math>m_{11}</math> |
| + | | <math>m_{12}</math> |
| + | | <math>m_{13}</math> |
| + | | <math>m_{14}</math> |
| + | | <math>m_{15}</math> |
| + | |- |
| + | | <math>f_0</math> |
| + | | 0 0 |
| + | | <math>0\!</math> |
| + | | 0 || style="background:black; color:white" | 1 |
| + | | 0 || style="background:black; color:white" | 1 |
| + | | 0 || style="background:black; color:white" | 1 |
| + | | 0 || style="background:black; color:white" | 1 |
| + | | 0 || style="background:black; color:white" | 1 |
| + | | 0 || style="background:black; color:white" | 1 |
| + | | 0 || style="background:black; color:white" | 1 |
| + | | 0 || style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_1</math> |
| + | | 0 1 |
| + | | <math>(x)\!</math> |
| + | | 0 || 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | 0 || 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | 0 || 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | 0 || 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_2</math> |
| + | | 1 0 |
| + | | <math>x\!</math> |
| + | | 0 || 0 || 0 || 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | 0 || 0 || 0 || 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_3</math> |
| + | | 1 1 |
| + | | <math>1\!</math> |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | |}<br> |
| + | |
| + | ===Table 11=== |
| + | |
| + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" |
| + | |+ '''Table 11. Interpretive Categories for Higher Order Propositions (''n'' = 1)''' |
| + | |- style="background:ghostwhite" |
| + | | Measure |
| + | | Happening |
| + | | Exactness |
| + | | Existence |
| + | | Linearity |
| + | | Uniformity |
| + | | Information |
| + | |- |
| + | | <math>m_0\!</math> |
| + | | Nothing happens |
| + | | |
| + | | |
| + | | |
| + | | |
| + | | |
| + | |- |
| + | | <math>m_1\!</math> |
| + | | |
| + | | Just false |
| + | | Nothing exists |
| + | | |
| + | | |
| + | | |
| + | |- |
| + | | <math>m_2\!</math> |
| + | | |
| + | | Just not <math>x\!</math> |
| + | | |
| + | | |
| + | | |
| + | | |
| + | |- |
| + | | <math>m_3\!</math> |
| + | | |
| + | | |
| + | | Nothing is <math>x\!</math> |
| + | | |
| + | | |
| + | | |
| + | |- |
| + | | <math>m_4\!</math> |
| + | | |
| + | | Just <math>x\!</math> |
| + | | |
| + | | |
| + | | |
| + | | |
| + | |- |
| + | | <math>m_5\!</math> |
| + | | |
| + | | |
| + | | Everything is <math>x\!</math> |
| + | | <math>f\!</math> is linear |
| + | | |
| + | | |
| + | |- |
| + | | <math>m_6\!</math> |
| + | | |
| + | | |
| + | | |
| + | | |
| + | | <math>f\!</math> is not uniform |
| + | | <math>f\!</math> is informed |
| + | |- |
| + | | <math>m_7\!</math> |
| + | | |
| + | | Not just true |
| + | | |
| + | | |
| + | | |
| + | | |
| + | |- |
| + | | <math>m_8\!</math> |
| + | | |
| + | | Just true |
| + | | |
| + | | |
| + | | |
| + | | |
| + | |- |
| + | | <math>m_9\!</math> |
| + | | |
| + | | |
| + | | |
| + | | |
| + | | <math>f\!</math> is uniform |
| + | | <math>f\!</math> is not informed |
| + | |- |
| + | | <math>m_{10}\!</math> |
| + | | |
| + | | |
| + | | Something is not <math>x\!</math> |
| + | | <math>f\!</math> is not linear |
| + | | |
| + | | |
| + | |- |
| + | | <math>m_{11}\!</math> |
| + | | |
| + | | Not just <math>x\!</math> |
| + | | |
| + | | |
| + | | |
| + | | |
| + | |- |
| + | | <math>m_{12}\!</math> |
| + | | |
| + | | |
| + | | Something is <math>x\!</math> |
| + | | |
| + | | |
| + | | |
| + | |- |
| + | | <math>m_{13}\!</math> |
| + | | |
| + | | Not just not <math>x\!</math> |
| + | | |
| + | | |
| + | | |
| + | | |
| + | |- |
| + | | <math>m_{14}\!</math> |
| + | | |
| + | | Not just false |
| + | | Something exists |
| + | | |
| + | | |
| + | | |
| + | |- |
| + | | <math>m_{15}\!</math> |
| + | | Anything happens |
| + | | |
| + | | |
| + | | |
| + | | |
| + | | |
| + | |}<br> |
| + | |
| + | ===Table 12=== |
| + | |
| + | {| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" |
| + | |+ '''Table 12. Higher Order Propositions (''n'' = 2)''' |
| + | |- style="background:ghostwhite" |
| + | | align="right" | <math>x:</math><br><math>y:</math> |
| + | | 1100<br>1010 |
| + | | <math>f\!</math> |
| + | | <math>m_0</math> |
| + | | <math>m_1</math> |
| + | | <math>m_2</math> |
| + | | <math>m_3</math> |
| + | | <math>m_4</math> |
| + | | <math>m_5</math> |
| + | | <math>m_6</math> |
| + | | <math>m_7</math> |
| + | | <math>m_8</math> |
| + | | <math>m_9</math> |
| + | | <math>m_{10}</math> |
| + | | <math>m_{11}</math> |
| + | | <math>m_{12}</math> |
| + | | <math>m_{13}</math> |
| + | | <math>m_{14}</math> |
| + | | <math>m_{15}</math> |
| + | | <math>m_{16}</math> |
| + | | <math>m_{17}</math> |
| + | | <math>m_{18}</math> |
| + | | <math>m_{19}</math> |
| + | | <math>m_{20}</math> |
| + | | <math>m_{21}</math> |
| + | | <math>m_{22}</math> |
| + | | <math>m_{23}</math> |
| + | |- |
| + | | <math>f_0</math> || 0000 || <math>(~)</math> |
| + | | 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 |
| + | | 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 |
| + | | 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 |
| + | |- |
| + | | <math>f_1</math> || 0001 || <math>(x)(y)\!</math> |
| + | | || || 1 || 1 || 0 || 0 || 1 || 1 |
| + | | 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 |
| + | | 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 |
| + | |- |
| + | | <math>f_2</math> || 0010 || <math>(x) y\!</math> |
| + | | || || || || 1 || 1 || 1 || 1 |
| + | | 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 |
| + | | 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 |
| + | |- |
| + | | <math>f_3</math> || 0011 || <math>(x)\!</math> |
| + | | || || || || || || || |
| + | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | |- |
| + | | <math>f_4</math> || 0100 || <math>x (y)\!</math> |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 |
| + | |- |
| + | | <math>f_5</math> || 0101 || <math>(y)\!</math> |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | |- |
| + | | <math>f_6</math> || 0110 || <math>(x, y)\!</math> |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | |- |
| + | | <math>f_7</math> || 0111 || <math>(x y)\!</math> |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | |- |
| + | | <math>f_8</math> || 1000 || <math>x y\!</math> |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | |- |
| + | | <math>f_9</math> || 1001 || <math>((x, y))\!</math> |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | |- |
| + | | <math>f_{10}</math> || 1010 || <math>y\!</math> |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | |- |
| + | | <math>f_{11}</math> || 1011 || <math>(x (y))\!</math> |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | |- |
| + | | <math>f_{12}</math> || 1100 || <math>x\!</math> |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | |- |
| + | | <math>f_{13}</math> || 1101 || <math>((x) y)\!</math> |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | |- |
| + | | <math>f_{14}</math> || 1110 || <math>((x)(y))\!</math> |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | |- |
| + | | <math>f_{15}</math> || 1111 || <math>((~))\!</math> |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | |}<br> |
| + | |
| + | {| align="center" border="1" cellpadding="0" cellspacing="0" style="background:white; color:black; font-weight:bold; text-align:center; width:96%" |
| + | |+ '''Table 12. Higher Order Propositions (''n'' = 2)''' |
| + | |- style="background:ghostwhite" |
| + | | align="right" | <math>u:</math><br><math>v:</math> |
| + | | 1100<br>1010 |
| + | | <math>f\!</math> |
| + | | <math>m_0</math> |
| + | | <math>m_1</math> |
| + | | <math>m_2</math> |
| + | | <math>m_3</math> |
| + | | <math>m_4</math> |
| + | | <math>m_5</math> |
| + | | <math>m_6</math> |
| + | | <math>m_7</math> |
| + | | <math>m_8</math> |
| + | | <math>m_9</math> |
| + | | <math>m_{10}</math> |
| + | | <math>m_{11}</math> |
| + | | <math>m_{12}</math> |
| + | | <math>m_{13}</math> |
| + | | <math>m_{14}</math> |
| + | | <math>m_{15}</math> |
| + | | <math>m_{16}</math> |
| + | | <math>m_{17}</math> |
| + | | <math>m_{18}</math> |
| + | | <math>m_{19}</math> |
| + | | <math>m_{20}</math> |
| + | | <math>m_{21}</math> |
| + | | <math>m_{22}</math> |
| + | | <math>m_{23}</math> |
| + | |- |
| + | | <math>f_0</math> |
| + | | 0000 |
| + | | <math>(~)</math> |
| + | | 0 || style="background:black; color:white" | 1 |
| + | | 0 || style="background:black; color:white" | 1 |
| + | | 0 || style="background:black; color:white" | 1 |
| + | | 0 || style="background:black; color:white" | 1 |
| + | | 0 || style="background:black; color:white" | 1 |
| + | | 0 || style="background:black; color:white" | 1 |
| + | | 0 || style="background:black; color:white" | 1 |
| + | | 0 || style="background:black; color:white" | 1 |
| + | | 0 || style="background:black; color:white" | 1 |
| + | | 0 || style="background:black; color:white" | 1 |
| + | | 0 || style="background:black; color:white" | 1 |
| + | | 0 || style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_1</math> |
| + | | 0001 |
| + | | <math>(u)(v)\!</math> |
| + | | 0 || 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | 0 || 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | 0 || 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | 0 || 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | 0 || 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | 0 || 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_2</math> |
| + | | 0010 |
| + | | <math>(u) v\!</math> |
| + | | 0 || 0 || 0 || 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | 0 || 0 || 0 || 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | 0 || 0 || 0 || 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_3</math> |
| + | | 0011 |
| + | | <math>(u)\!</math> |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | |- |
| + | | <math>f_4</math> |
| + | | 0100 |
| + | | <math>u (v)\!</math> |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_5</math> |
| + | | 0101 |
| + | | <math>(v)\!</math> |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | |- |
| + | | <math>f_6</math> |
| + | | 0110 |
| + | | <math>(u, v)\!</math> |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | |- |
| + | | <math>f_7</math> |
| + | | 0111 |
| + | | <math>(u v)\!</math> |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | |- |
| + | | <math>f_8</math> |
| + | | 1000 |
| + | | <math>u v\!</math> |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | |- |
| + | | <math>f_9</math> |
| + | | 1001 |
| + | | <math>((u, v))\!</math> |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | |- |
| + | | <math>f_{10}</math> |
| + | | 1010 |
| + | | <math>v\!</math> |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | |- |
| + | | <math>f_{11}</math> |
| + | | 1011 |
| + | | <math>(u (v))\!</math> |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | |- |
| + | | <math>f_{12}</math> |
| + | | 1100 |
| + | | <math>u\!</math> |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | |- |
| + | | <math>f_{13}</math> |
| + | | 1101 |
| + | | <math>((u) v)\!</math> |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | |- |
| + | | <math>f_{14}</math> |
| + | | 1110 |
| + | | <math>((u)(v))\!</math> |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | |- |
| + | | <math>f_{15}</math> |
| + | | 1111 |
| + | | <math>((~))\!</math> |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | |}<br> |
| + | |
| + | ===Table 13=== |
| + | |
| + | {| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" |
| + | |+ '''Table 13. Qualifiers of Implication Ordering: <math>\alpha_i f = \Upsilon (f_i, f) = \Upsilon (f_i \Rightarrow f)</math>''' |
| + | |- style="background:ghostwhite" |
| + | | align="right" | <math>u:</math><br><math>v:</math> |
| + | | 1100<br>1010 |
| + | | <math>f\!</math> |
| + | | <math>\alpha_0</math> |
| + | | <math>\alpha_1</math> |
| + | | <math>\alpha_2</math> |
| + | | <math>\alpha_3</math> |
| + | | <math>\alpha_4</math> |
| + | | <math>\alpha_5</math> |
| + | | <math>\alpha_6</math> |
| + | | <math>\alpha_7</math> |
| + | | <math>\alpha_8</math> |
| + | | <math>\alpha_9</math> |
| + | | <math>\alpha_{10}</math> |
| + | | <math>\alpha_{11}</math> |
| + | | <math>\alpha_{12}</math> |
| + | | <math>\alpha_{13}</math> |
| + | | <math>\alpha_{14}</math> |
| + | | <math>\alpha_{15}</math> |
| + | |- |
| + | | <math>f_0</math> |
| + | | 0000 |
| + | | <math>(~)</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_1</math> |
| + | | 0001 |
| + | | <math>(u)(v)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_2</math> |
| + | | 0010 |
| + | | <math>(u) v\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_3</math> |
| + | | 0011 |
| + | | <math>(u)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_4</math> |
| + | | 0100 |
| + | | <math>u (v)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_5</math> |
| + | | 0101 |
| + | | <math>(v)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_6</math> |
| + | | 0110 |
| + | | <math>(u, v)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_7</math> |
| + | | 0111 |
| + | | <math>(u v)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_8</math> |
| + | | 1000 |
| + | | <math>u v\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_9</math> |
| + | | 1001 |
| + | | <math>((u, v))\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_{10}</math> |
| + | | 1010 |
| + | | <math>v\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_{11}</math> |
| + | | 1011 |
| + | | <math>(u (v))\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_{12}</math> |
| + | | 1100 |
| + | | <math>u\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_{13}</math> |
| + | | 1101 |
| + | | <math>((u) v)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_{14}</math> |
| + | | 1110 |
| + | | <math>((u)(v))\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_{15}</math> |
| + | | 1111 |
| + | | <math>((~))</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | |}<br> |
| + | |
| + | ===Table 14=== |
| + | |
| + | {| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" |
| + | |+ '''Table 14. Qualifiers of Implication Ordering: <math>\beta_i f = \Upsilon (f, f_i) = \Upsilon (f \Rightarrow f_i)</math>''' |
| + | |- style="background:ghostwhite" |
| + | | align="right" | <math>u:</math><br><math>v:</math> |
| + | | 1100<br>1010 |
| + | | <math>f\!</math> |
| + | | <math>\beta_0</math> |
| + | | <math>\beta_1</math> |
| + | | <math>\beta_2</math> |
| + | | <math>\beta_3</math> |
| + | | <math>\beta_4</math> |
| + | | <math>\beta_5</math> |
| + | | <math>\beta_6</math> |
| + | | <math>\beta_7</math> |
| + | | <math>\beta_8</math> |
| + | | <math>\beta_9</math> |
| + | | <math>\beta_{10}</math> |
| + | | <math>\beta_{11}</math> |
| + | | <math>\beta_{12}</math> |
| + | | <math>\beta_{13}</math> |
| + | | <math>\beta_{14}</math> |
| + | | <math>\beta_{15}</math> |
| + | |- |
| + | | <math>f_0</math> |
| + | | 0000 |
| + | | <math>(~)</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_1</math> |
| + | | 0001 |
| + | | <math>(u)(v)\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_2</math> |
| + | | 0010 |
| + | | <math>(u) v\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_3</math> |
| + | | 0011 |
| + | | <math>(u)\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_4</math> |
| + | | 0100 |
| + | | <math>u (v)\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_5</math> |
| + | | 0101 |
| + | | <math>(v)\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_6</math> |
| + | | 0110 |
| + | | <math>(u, v)\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_7</math> |
| + | | 0111 |
| + | | <math>(u v)\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_8</math> |
| + | | 1000 |
| + | | <math>u v\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_9</math> |
| + | | 1001 |
| + | | <math>((u, v))\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_{10}</math> |
| + | | 1010 |
| + | | <math>v\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_{11}</math> |
| + | | 1011 |
| + | | <math>(u (v))\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_{12}</math> |
| + | | 1100 |
| + | | <math>u\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_{13}</math> |
| + | | 1101 |
| + | | <math>((u) v)\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_{14}</math> |
| + | | 1110 |
| + | | <math>((u)(v))\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_{15}</math> |
| + | | 1111 |
| + | | <math>((~))\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | |}<br> |
| + | |
| + | ===Figure 15=== |
| + | |
| + | ===Table 16=== |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" |
| + | |+ '''Table 16. Syllogistic Premisses as Higher Order Indicator Functions''' |
| + | | |
| + | <math>\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}</math> |
| + | |}<br> |
| + | |
| + | ===Table 17=== |
| + | |
| + | {| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" |
| + | |+ '''Table 17. Simple Qualifiers of Propositions (Version 1)''' |
| + | |- style="background:ghostwhite" |
| + | | align="right" | <math>u:</math><br><math>v:</math> |
| + | | 1100<br>1010 |
| + | | <math>f\!</math> |
| + | | <math>(\ell_{11})</math><br><math>\text{No } u </math><br><math>\text{is } v </math> |
| + | | <math>(\ell_{10})</math><br><math>\text{No } u </math><br><math>\text{is }(v)</math> |
| + | | <math>(\ell_{01})</math><br><math>\text{No }(u)</math><br><math>\text{is } v </math> |
| + | | <math>(\ell_{00})</math><br><math>\text{No }(u)</math><br><math>\text{is }(v)</math> |
| + | | <math> \ell_{00} </math><br><math>\text{Some }(u)</math><br><math>\text{is }(v)</math> |
| + | | <math> \ell_{01} </math><br><math>\text{Some }(u)</math><br><math>\text{is } v </math> |
| + | | <math> \ell_{10} </math><br><math>\text{Some } u </math><br><math>\text{is }(v)</math> |
| + | | <math> \ell_{11} </math><br><math>\text{Some } u </math><br><math>\text{is } v </math> |
| + | |- |
| + | | <math>f_0</math> |
| + | | 0000 |
| + | | <math>(~)</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_1</math> |
| + | | 0001 |
| + | | <math>(u)(v)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_2</math> |
| + | | 0010 |
| + | | <math>(u) v\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_3</math> |
| + | | 0011 |
| + | | <math>(u)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_4</math> |
| + | | 0100 |
| + | | <math>u (v)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_5</math> |
| + | | 0101 |
| + | | <math>(v)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_6</math> |
| + | | 0110 |
| + | | <math>(u, v)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_7</math> |
| + | | 0111 |
| + | | <math>(u v)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_8</math> |
| + | | 1000 |
| + | | <math>u v\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_9</math> |
| + | | 1001 |
| + | | <math>((u, v))\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_{10}</math> |
| + | | 1010 |
| + | | <math>v\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_{11}</math> |
| + | | 1011 |
| + | | <math>(u (v))\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_{12}</math> |
| + | | 1100 |
| + | | <math>u\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_{13}</math> |
| + | | 1101 |
| + | | <math>((u) v)\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_{14}</math> |
| + | | 1110 |
| + | | <math>((u)(v))\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_{15}</math> |
| + | | 1111 |
| + | | <math>((~))</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | |}<br> |
| + | |
| + | ===Table 18=== |
| + | |
| + | {| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" |
| + | |+ '''Table 18. Simple Qualifiers of Propositions (Version 2)''' |
| + | |- style="background:ghostwhite" |
| + | | align="right" | <math>u:</math><br><math>v:</math> |
| + | | 1100<br>1010 |
| + | | <math>f\!</math> |
| + | | <math>(\ell_{11})</math><br><math>\text{No } u </math><br><math>\text{is } v </math> |
| + | | <math>(\ell_{10})</math><br><math>\text{No } u </math><br><math>\text{is }(v)</math> |
| + | | <math>(\ell_{01})</math><br><math>\text{No }(u)</math><br><math>\text{is } v </math> |
| + | | <math>(\ell_{00})</math><br><math>\text{No }(u)</math><br><math>\text{is }(v)</math> |
| + | | <math> \ell_{00} </math><br><math>\text{Some }(u)</math><br><math>\text{is }(v)</math> |
| + | | <math> \ell_{01} </math><br><math>\text{Some }(u)</math><br><math>\text{is } v </math> |
| + | | <math> \ell_{10} </math><br><math>\text{Some } u </math><br><math>\text{is }(v)</math> |
| + | | <math> \ell_{11} </math><br><math>\text{Some } u </math><br><math>\text{is } v </math> |
| + | |- |
| + | | <math>f_0</math> |
| + | | 0000 |
| + | | <math>(~)</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_1</math> |
| + | | 0001 |
| + | | <math>(u)(v)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_2</math> |
| + | | 0010 |
| + | | <math>(u) v\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_4</math> |
| + | | 0100 |
| + | | <math>u (v)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_8</math> |
| + | | 1000 |
| + | | <math>u v\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_3</math> |
| + | | 0011 |
| + | | <math>(u)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_{12}</math> |
| + | | 1100 |
| + | | <math>u\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_6</math> |
| + | | 0110 |
| + | | <math>(u, v)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_9</math> |
| + | | 1001 |
| + | | <math>((u, v))\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_5</math> |
| + | | 0101 |
| + | | <math>(v)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_{10}</math> |
| + | | 1010 |
| + | | <math>v\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_7</math> |
| + | | 0111 |
| + | | <math>(u v)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_{11}</math> |
| + | | 1011 |
| + | | <math>(u (v))\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_{13}</math> |
| + | | 1101 |
| + | | <math>((u) v)\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_{14}</math> |
| + | | 1110 |
| + | | <math>((u)(v))\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_{15}</math> |
| + | | 1111 |
| + | | <math>((~))</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | |}<br> |
| + | |
| + | ===Table 19=== |
| + | |
| + | {| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" |
| + | |+ '''Table 19. Relation of Quantifiers to Higher Order Propositions''' |
| + | |- style="background:ghostwhite" |
| + | | <math>\text{Mnemonic}</math> |
| + | | <math>\text{Category}</math> |
| + | | <math>\text{Classical Form}</math> |
| + | | <math>\text{Alternate Form}</math> |
| + | | <math>\text{Symmetric Form}</math> |
| + | | <math>\text{Operator}</math> |
| + | |- |
| + | | <math>\text{E}\!</math><br><math>\text{Exclusive}</math> |
| + | | <math>\text{Universal}</math><br><math>\text{Negative}</math> |
| + | | <math>\text{All}\ u\ \text{is}\ (v)</math> |
| + | | |
| + | | <math>\text{No}\ u\ \text{is}\ v </math> |
| + | | <math>(\ell_{11})</math> |
| + | |- |
| + | | <math>\text{A}\!</math><br><math>\text{Absolute}</math> |
| + | | <math>\text{Universal}</math><br><math>\text{Affirmative}</math> |
| + | | <math>\text{All}\ u\ \text{is}\ v </math> |
| + | | |
| + | | <math>\text{No}\ u\ \text{is}\ (v)</math> |
| + | | <math>(\ell_{10})</math> |
| + | |- |
| + | | |
| + | | |
| + | | <math>\text{All}\ v\ \text{is}\ u </math> |
| + | | <math>\text{No}\ v\ \text{is}\ (u)</math> |
| + | | <math>\text{No}\ (u)\ \text{is}\ v </math> |
| + | | <math>(\ell_{01})</math> |
| + | |- |
| + | | |
| + | | |
| + | | <math>\text{All}\ (v)\ \text{is}\ u </math> |
| + | | <math>\text{No}\ (v)\ \text{is}\ (u)</math> |
| + | | <math>\text{No}\ (u)\ \text{is}\ (v)</math> |
| + | | <math>(\ell_{00})</math> |
| + | |- |
| + | | |
| + | | |
| + | | <math>\text{Some}\ (u)\ \text{is}\ (v)</math> |
| + | | |
| + | | <math>\text{Some}\ (u)\ \text{is}\ (v)</math> |
| + | | <math>\ell_{00}\!</math> |
| + | |- |
| + | | |
| + | | |
| + | | <math>\text{Some}\ (u)\ \text{is}\ v</math> |
| + | | |
| + | | <math>\text{Some}\ (u)\ \text{is}\ v</math> |
| + | | <math>\ell_{01}\!</math> |
| + | |- |
| + | | <math>\text{O}\!</math><br><math>\text{Obtrusive}</math> |
| + | | <math>\text{Particular}</math><br><math>\text{Negative}</math> |
| + | | <math>\text{Some}\ u\ \text{is}\ (v)</math> |
| + | | |
| + | | <math>\text{Some}\ u\ \text{is}\ (v)</math> |
| + | | <math>\ell_{10}\!</math> |
| + | |- |
| + | | <math>\text{I}\!</math><br><math>\text{Indefinite}</math> |
| + | | <math>\text{Particular}</math><br><math>\text{Affirmative}</math> |
| + | | <math>\text{Some}\ u\ \text{is}\ v</math> |
| + | | |
| + | | <math>\text{Some}\ u\ \text{is}\ v</math> |
| + | | <math>\ell_{11}\!</math> |
| + | |}<br> |