Line 330: |
Line 330: |
| | @ | (C_1, ..., C_k) | is not true. | is true. | | | | @ | (C_1, ..., C_k) | is not true. | is true. | |
| | | | | | | | | | | | | |
− | o-----------------o-----------------o-----------------o-----------------o
| |
− | </pre>
| |
− | |}
| |
− |
| |
− | {| align="center" cellpadding="6" width="90%"
| |
− | | align="center" |
| |
− | <pre>
| |
− | o-----------------o-----------------o-----------------o-----------------o
| |
− | | Graph | String | Entitative | Existential |
| |
− | o-----------------o-----------------o-----------------o-----------------o
| |
− | | | | | |
| |
− | | @ | " " | untrue. | true. |
| |
− | o-----------------o-----------------o-----------------o-----------------o
| |
− | | | | | |
| |
− | | o | | | |
| |
− | | | | | | |
| |
− | | @ | ( ) | true. | untrue. |
| |
− | o-----------------o-----------------o-----------------o-----------------o
| |
− | | | | | |
| |
− | | r | | | |
| |
− | | @ | r | r. | r. |
| |
− | o-----------------o-----------------o-----------------o-----------------o
| |
− | | | | | |
| |
− | | r | | | |
| |
− | | o | | | |
| |
− | | | | | | |
| |
− | | @ | (r) | not r. | not r. |
| |
− | o-----------------o-----------------o-----------------o-----------------o
| |
− | | | | | |
| |
− | | r s t | | | |
| |
− | | @ | r s t | r or s or t. | r and s and t. |
| |
− | o-----------------o-----------------o-----------------o-----------------o
| |
− | | | | | |
| |
− | | r s t | | | |
| |
− | | o o o | | | |
| |
− | | \|/ | | | |
| |
− | | o | | | |
| |
− | | | | | | |
| |
− | | @ | ((r)(s)(t)) | r and s and t. | r or s or t. |
| |
− | o-----------------o-----------------o-----------------o-----------------o
| |
− | | | | | r implies s. |
| |
− | | | | | |
| |
− | | o r | | | if r then s. |
| |
− | | | | | | |
| |
− | | @ s | (r) s | not r, or s | no r sans s. |
| |
− | o-----------------o-----------------o-----------------o-----------------o
| |
− | | | | | r implies s. |
| |
− | | r s | | | |
| |
− | | o---o | | | if r then s. |
| |
− | | | | | | |
| |
− | | @ | (r (s)) | | no r sans s. |
| |
− | o-----------------o-----------------o-----------------o-----------------o
| |
− | | | | | |
| |
− | | r s | | | |
| |
− | | o---o | | |r exclusive-or s.|
| |
− | | \ / | | | |
| |
− | | @ | (r , s) | |r not equal to s.|
| |
− | o-----------------o-----------------o-----------------o-----------------o
| |
− | | | | | |
| |
− | | r s | | | |
| |
− | | o---o | | | |
| |
− | | \ / | | | |
| |
− | | o | | |r if & only if s.|
| |
− | | | | | | |
| |
− | | @ | ((r , s)) | |r equates with s.|
| |
− | o-----------------o-----------------o-----------------o-----------------o
| |
− | | | | | |
| |
− | | r s t | | | |
| |
− | | o--o--o | | | |
| |
− | | \ / | | | |
| |
− | | \ / | | | just one false |
| |
− | | @ | (r , s , t) | | out of r, s, t. |
| |
− | o-----------------o-----------------o-----------------o-----------------o
| |
− | | | | | |
| |
− | | r s t | | | |
| |
− | | o o o | | | |
| |
− | | | | | | | | |
| |
− | | o--o--o | | | |
| |
− | | \ / | | | |
| |
− | | \ / | | | just one true |
| |
− | | @ | ((r),(s),(t)) | | among r, s, t. |
| |
− | o-----------------o-----------------o-----------------o-----------------o
| |
− | | | | | genus t over |
| |
− | | r s | | | species r, s. |
| |
− | | o o | | | |
| |
− | | t | | | | | partition t |
| |
− | | o--o--o | | | among r & s. |
| |
− | | \ / | | | |
| |
− | | \ / | | | whole pie t: |
| |
− | | @ | ( t ,(r),(s)) | | slices r, s. |
| |
| o-----------------o-----------------o-----------------o-----------------o | | o-----------------o-----------------o-----------------o-----------------o |
| </pre> | | </pre> |
Line 428: |
Line 338: |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="6" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%" | + | {| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%" |
| |+ <math>\text{Table A.}~~\text{Existential Interpretation}</math> | | |+ <math>\text{Table A.}~~\text{Existential Interpretation}</math> |
| |- style="background:#f0f0ff" | | |- style="background:#f0f0ff" |
Line 435: |
Line 345: |
| | <math>\text{Interpretation}\!</math> | | | <math>\text{Interpretation}\!</math> |
| |- | | |- |
− | | | + | | height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]] |
− | <pre>
| |
− | o-------------------o
| |
− | | | | |
− | | @ |
| |
− | | |
| |
− | o-------------------o
| |
− | </pre>
| |
| | <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math> | | | <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math> |
| | <math>\operatorname{true}.</math> | | | <math>\operatorname{true}.</math> |
| |- | | |- |
− | | | + | | height="100px" | [[Image:Cactus Spike Big Fat.jpg|20px]] |
− | <pre>
| |
− | o-------------------o
| |
− | | | | |
− | | o |
| |
− | | | |
| |
− | | @ |
| |
− | | |
| |
− | o-------------------o
| |
− | </pre>
| |
| | <math>\texttt{(~)}</math> | | | <math>\texttt{(~)}</math> |
| | <math>\operatorname{false}.</math> | | | <math>\operatorname{false}.</math> |
| |- | | |- |
− | | | + | | height="100px" | [[Image:Cactus A Big.jpg|20px]] |
− | <pre>
| |
− | o-------------------o
| |
− | | | | |
− | | a |
| |
− | | @ |
| |
− | | |
| |
− | o-------------------o
| |
− | </pre>
| |
| | <math>a\!</math> | | | <math>a\!</math> |
| | <math>a.\!</math> | | | <math>a.\!</math> |
| |- | | |- |
− | | | + | | height="120px" | [[Image:Cactus (A) Big.jpg|20px]] |
− | <pre>
| |
− | o-------------------o
| |
− | | | | |
− | | a |
| |
− | | o |
| |
− | | | |
| |
− | | @ |
| |
− | | |
| |
− | o-------------------o
| |
− | </pre>
| |
| | <math>\texttt{(} a \texttt{)}</math> | | | <math>\texttt{(} a \texttt{)}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
| \tilde{a} | | \tilde{a} |
− | \\[6pt] | + | \\[2pt] |
| a^\prime | | a^\prime |
− | \\[6pt] | + | \\[2pt] |
| \lnot a | | \lnot a |
− | \\[6pt] | + | \\[2pt] |
| \operatorname{not}~ a. | | \operatorname{not}~ a. |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
− | | | + | | height="100px" | [[Image:Cactus ABC Big.jpg|50px]] |
− | <pre>
| |
− | o-------------------o
| |
− | | | | |
− | | a b c |
| |
− | | @ |
| |
− | | |
| |
− | o-------------------o
| |
− | </pre>
| |
| | <math>a~b~c</math> | | | <math>a~b~c</math> |
| | | | | |
Line 511: |
Line 379: |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
− | | | + | | height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]] |
− | <pre>
| |
− | o-------------------o
| |
− | | | | |
− | | a b c |
| |
− | | o o o |
| |
− | | \|/ |
| |
− | | o |
| |
− | | | |
| |
− | | @ |
| |
− | | |
| |
− | o-------------------o
| |
− | </pre>
| |
| | <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math> | | | <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math> |
| | | | | |
Line 532: |
Line 388: |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
− | | | + | | height="120px" | [[Image:Cactus (A(B)) Big.jpg|60px]] |
− | <pre>
| |
− | o-------------------o
| |
− | | | | |
− | | a b |
| |
− | | o---o |
| |
− | | | |
| |
− | | @ |
| |
− | | |
| |
− | o-------------------o
| |
− | </pre>
| |
| | <math>\texttt{(} a \texttt{(} b \texttt{))}</math> | | | <math>\texttt{(} a \texttt{(} b \texttt{))}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
| a \Rightarrow b | | a \Rightarrow b |
− | \\[6pt] | + | \\[2pt] |
| a ~\operatorname{implies}~ b. | | a ~\operatorname{implies}~ b. |
− | \\[6pt] | + | \\[2pt] |
| \operatorname{if}~ a ~\operatorname{then}~ b. | | \operatorname{if}~ a ~\operatorname{then}~ b. |
− | \\[6pt] | + | \\[2pt] |
| \operatorname{not}~ a ~\operatorname{without}~ b. | | \operatorname{not}~ a ~\operatorname{without}~ b. |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
− | | | + | | height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]] |
− | <pre>
| |
− | o-------------------o
| |
− | | | | |
− | | a b |
| |
− | | o---o |
| |
− | | \ / |
| |
− | | @ |
| |
− | | |
| |
− | o-------------------o
| |
− | </pre>
| |
| | <math>\texttt{(} a, b \texttt{)}</math> | | | <math>\texttt{(} a, b \texttt{)}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
| a + b | | a + b |
− | \\[6pt] | + | \\[2pt] |
| a \neq b | | a \neq b |
− | \\[6pt] | + | \\[2pt] |
| a ~\operatorname{exclusive-or}~ b. | | a ~\operatorname{exclusive-or}~ b. |
− | \\[6pt] | + | \\[2pt] |
| a ~\operatorname{not~equal~to}~ b. | | a ~\operatorname{not~equal~to}~ b. |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
− | | | + | | height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]] |
− | <pre>
| |
− | o-------------------o
| |
− | | | | |
− | | a b |
| |
− | | o---o |
| |
− | | \ / |
| |
− | | o |
| |
− | | | |
| |
− | | @ |
| |
− | | |
| |
− | o-------------------o
| |
− | </pre>
| |
| | <math>\texttt{((} a, b \texttt{))}</math> | | | <math>\texttt{((} a, b \texttt{))}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
| a = b | | a = b |
− | \\[6pt] | + | \\[2pt] |
| a \iff b | | a \iff b |
− | \\[6pt] | + | \\[2pt] |
| a ~\operatorname{equals}~ b. | | a ~\operatorname{equals}~ b. |
− | \\[6pt] | + | \\[2pt] |
| a ~\operatorname{if~and~only~if}~ b. | | a ~\operatorname{if~and~only~if}~ b. |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
− | | | + | | height="120px" | [[Image:Cactus (A,B,C) Big.jpg|65px]] |
− | <pre>
| |
− | o-------------------o
| |
− | | | | |
− | | a b c |
| |
− | | o--o--o |
| |
− | | \ / |
| |
− | | \ / |
| |
− | | @ |
| |
− | | |
| |
− | o-------------------o
| |
− | </pre>
| |
| | <math>\texttt{(} a, b, c \texttt{)}</math> | | | <math>\texttt{(} a, b, c \texttt{)}</math> |
| | | | | |
Line 625: |
Line 438: |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
− | | | + | | height="160px" | [[Image:Cactus ((A),(B),(C)) Big.jpg|65px]] |
− | <pre>
| |
− | o-------------------o
| |
− | | | | |
− | | a b c |
| |
− | | o o o |
| |
− | | | | | |
| |
− | | o--o--o |
| |
− | | \ / |
| |
− | | \ / |
| |
− | | @ |
| |
− | | |
| |
− | o-------------------o
| |
− | </pre>
| |
| | <math>\texttt{((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math> | | | <math>\texttt{((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math> |
| | | | | |
Line 649: |
Line 449: |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
− | | | + | | height="160px" | [[Image:Cactus (A,(B),(C)) Big.jpg|65px]] |
− | <pre>
| |
− | o-------------------o
| |
− | | | | |
− | | b c |
| |
− | | o o |
| |
− | | a | | |
| |
− | | o--o--o |
| |
− | | \ / |
| |
− | | \ / |
| |
− | | @ |
| |
− | | |
| |
− | o-------------------o
| |
− | </pre>
| |
| | <math>\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math> | | | <math>\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math> |
| | | | | |
Line 676: |
Line 463: |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="6" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%" | + | {| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%" |
| |+ <math>\text{Table B.}~~\text{Entitative Interpretation}</math> | | |+ <math>\text{Table B.}~~\text{Entitative Interpretation}</math> |
| |- style="background:#f0f0ff" | | |- style="background:#f0f0ff" |
Line 683: |
Line 470: |
| | <math>\text{Interpretation}\!</math> | | | <math>\text{Interpretation}\!</math> |
| |- | | |- |
− | | | + | | height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]] |
− | <pre>
| |
− | o-------------------o
| |
− | | | | |
− | | @ |
| |
− | | |
| |
− | o-------------------o
| |
− | </pre>
| |
| | <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math> | | | <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math> |
| | <math>\operatorname{false}.</math> | | | <math>\operatorname{false}.</math> |
| |- | | |- |
− | | | + | | height="100px" | [[Image:Cactus Spike Big Fat.jpg|20px]] |
− | <pre>
| + | | <math>\texttt{(~)}</math> |
− | o-------------------o
| |
− | | | | |
− | | o |
| |
− | | | |
| |
− | | @ |
| |
− | | |
| |
− | o-------------------o
| |
− | </pre>
| |
− | | <math>\texttt{(~)}</math> | |
| | <math>\operatorname{true}.</math> | | | <math>\operatorname{true}.</math> |
| |- | | |- |
− | | | + | | height="100px" | [[Image:Cactus A Big.jpg|20px]] |
− | <pre>
| |
− | o-------------------o
| |
− | | | | |
− | | a |
| |
− | | @ |
| |
− | | |
| |
− | o-------------------o
| |
− | </pre>
| |
| | <math>a\!</math> | | | <math>a\!</math> |
| | <math>a.\!</math> | | | <math>a.\!</math> |
| |- | | |- |
− | | | + | | height="120px" | [[Image:Cactus (A) Big.jpg|20px]] |
− | <pre>
| |
− | o-------------------o
| |
− | | | | |
− | | a |
| |
− | | o |
| |
− | | | |
| |
− | | @ |
| |
− | | |
| |
− | o-------------------o
| |
− | </pre>
| |
| | <math>\texttt{(} a \texttt{)}</math> | | | <math>\texttt{(} a \texttt{)}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
| \tilde{a} | | \tilde{a} |
− | \\[6pt] | + | \\[2pt] |
| a^\prime | | a^\prime |
− | \\[6pt] | + | \\[2pt] |
| \lnot a | | \lnot a |
− | \\[6pt] | + | \\[2pt] |
| \operatorname{not}~ a. | | \operatorname{not}~ a. |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
− | | | + | | height="100px" | [[Image:Cactus ABC Big.jpg|50px]] |
− | <pre>
| |
− | o-------------------o
| |
− | | | | |
− | | a b c |
| |
− | | @ |
| |
− | | |
| |
− | o-------------------o
| |
− | </pre>
| |
| | <math>a~b~c</math> | | | <math>a~b~c</math> |
| | | | | |
Line 759: |
Line 504: |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
− | | | + | | height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]] |
− | <pre>
| |
− | o-------------------o
| |
− | | | | |
− | | a b c |
| |
− | | o o o |
| |
− | | \|/ |
| |
− | | o |
| |
− | | | |
| |
− | | @ |
| |
− | | |
| |
− | o-------------------o
| |
− | </pre>
| |
| | <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math> | | | <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math> |
| | | | | |
Line 780: |
Line 513: |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
− | | | + | | height="120px" | [[Image:Cactus (A)B Big.jpg|35px]] |
− | <pre>
| |
− | o-------------------o
| |
− | | | | |
− | | o a |
| |
− | | | |
| |
− | | @ b |
| |
− | | |
| |
− | o-------------------o
| |
− | </pre>
| |
| | <math>\texttt{(} a \texttt{)} b</math> | | | <math>\texttt{(} a \texttt{)} b</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
| a \Rightarrow b | | a \Rightarrow b |
− | \\[6pt] | + | \\[2pt] |
| a ~\operatorname{implies}~ b. | | a ~\operatorname{implies}~ b. |
− | \\[6pt] | + | \\[2pt] |
| \operatorname{if}~ a ~\operatorname{then}~ b. | | \operatorname{if}~ a ~\operatorname{then}~ b. |
− | \\[6pt] | + | \\[2pt] |
| \operatorname{not}~ a, ~\operatorname{or}~ b. | | \operatorname{not}~ a, ~\operatorname{or}~ b. |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
− | | | + | | height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]] |
− | <pre>
| |
− | o-------------------o
| |
− | | | | |
− | | a b |
| |
− | | o---o |
| |
− | | \ / |
| |
− | | @ |
| |
− | | |
| |
− | o-------------------o
| |
− | </pre>
| |
| | <math>\texttt{(} a, b \texttt{)}</math> | | | <math>\texttt{(} a, b \texttt{)}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
| a = b | | a = b |
− | \\[6pt] | + | \\[2pt] |
| a \iff b | | a \iff b |
− | \\[6pt] | + | \\[2pt] |
| a ~\operatorname{equals}~ b. | | a ~\operatorname{equals}~ b. |
− | \\[6pt] | + | \\[2pt] |
| a ~\operatorname{if~and~only~if}~ b. | | a ~\operatorname{if~and~only~if}~ b. |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
− | | | + | | height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]] |
− | <pre>
| |
− | o-------------------o
| |
− | | | | |
− | | a b |
| |
− | | o---o |
| |
− | | \ / |
| |
− | | o |
| |
− | | | |
| |
− | | @ |
| |
− | | |
| |
− | o-------------------o
| |
− | </pre>
| |
| | <math>\texttt{((} a, b \texttt{))}</math> | | | <math>\texttt{((} a, b \texttt{))}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
| a + b | | a + b |
− | \\[6pt] | + | \\[2pt] |
| a \neq b | | a \neq b |
− | \\[6pt] | + | \\[2pt] |
| a ~\operatorname{exclusive-or}~ b. | | a ~\operatorname{exclusive-or}~ b. |
− | \\[6pt] | + | \\[2pt] |
| a ~\operatorname{not~equal~to}~ b. | | a ~\operatorname{not~equal~to}~ b. |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
− | | | + | | height="120px" | [[Image:Cactus (A,B,C) Big.jpg|65px]] |
− | <pre>
| |
− | o-------------------o
| |
− | | | | |
− | | a b c |
| |
− | | o--o--o |
| |
− | | \ / |
| |
− | | \ / |
| |
− | | @ |
| |
− | | |
| |
− | o-------------------o
| |
− | </pre>
| |
| | <math>\texttt{(} a, b, c \texttt{)}</math> | | | <math>\texttt{(} a, b, c \texttt{)}</math> |
| | | | | |
Line 872: |
Line 563: |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
− | | | + | | height="160px" | [[Image:Cactus ((A,B,C)) Big.jpg|65px]] |
− | <pre>
| |
− | o-------------------o
| |
− | | | | |
− | | a b c |
| |
− | | o--o--o |
| |
− | | \ / |
| |
− | | \ / |
| |
− | | o |
| |
− | | | |
| |
− | | @ |
| |
− | | |
| |
− | o-------------------o
| |
− | </pre>
| |
| | <math>\texttt{((} a, b, c \texttt{))}</math> | | | <math>\texttt{((} a, b, c \texttt{))}</math> |
| | | | | |
Line 896: |
Line 574: |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
− | | | + | | height="200px" | [[Image:Cactus (((A),B,C)) Big.jpg|65px]] |
− | <pre>
| |
− | o-------------------o
| |
− | | | | |
− | | a |
| |
− | | o |
| |
− | | | b c |
| |
− | | o--o--o |
| |
− | | \ / |
| |
− | | \ / |
| |
− | | o |
| |
− | | | |
| |
− | | @ |
| |
− | | |
| |
− | o-------------------o
| |
− | </pre>
| |
| | <math>\texttt{(((} a \texttt{)}, b, c \texttt{))}</math> | | | <math>\texttt{(((} a \texttt{)}, b, c \texttt{))}</math> |
| | | | | |
Line 923: |
Line 586: |
| |} | | |} |
| | | |
− | ==Differential Logic==
| + | <br> |
| | | |
− | ===Ascii Tables=== | + | {| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%" |
− | | + | |+ <math>\text{Table C.}~~\text{Dualing Interpretations}</math> |
− | <pre> | + | |- style="background:#f0f0ff" |
− | Table A1. Propositional Forms On Two Variables | + | | <math>\text{Graph}\!</math> |
− | o---------o---------o---------o----------o------------------o----------o | + | | <math>\text{String}\!</math> |
− | | L_1 | L_2 | L_3 | L_4 | L_5 | L_6 | | + | | <math>\text{Existential}\!</math> |
− | | | | | | | | | + | | <math>\text{Entitative}\!</math> |
− | | Decimal | Binary | Vector | Cactus | English | Ordinary | | + | |- |
− | o---------o---------o---------o----------o------------------o----------o | + | | height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]] |
− | | | x : 1 1 0 0 | | | | | + | | <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math> |
− | | | y : 1 0 1 0 | | | | | + | | <math>\operatorname{true}.</math> |
− | o---------o---------o---------o----------o------------------o----------o | + | | <math>\operatorname{false}.</math> |
− | | | | | | | | | + | |- |
− | | f_0 | f_0000 | 0 0 0 0 | () | false | 0 | | + | | height="100px" | [[Image:Cactus Spike Big Fat.jpg|20px]] |
− | | | | | | | | | + | | <math>\texttt{(~)}</math> |
− | | f_1 | f_0001 | 0 0 0 1 | (x)(y) | neither x nor y | ~x & ~y | | + | | <math>\operatorname{false}.</math> |
− | | | | | | | | | + | | <math>\operatorname{true}.</math> |
− | | f_2 | f_0010 | 0 0 1 0 | (x) y | y and not x | ~x & y | | + | |- |
− | | | | | | | | | + | | height="100px" | [[Image:Cactus A Big.jpg|20px]] |
− | | f_3 | f_0011 | 0 0 1 1 | (x) | not x | ~x | | + | | <math>a\!</math> |
− | | | | | | | | | + | | <math>a.\!</math> |
− | | f_4 | f_0100 | 0 1 0 0 | x (y) | x and not y | x & ~y | | + | | <math>a.\!</math> |
− | | | | | | | | | + | |- |
− | | f_5 | f_0101 | 0 1 0 1 | (y) | not y | ~y | | + | | height="120px" | [[Image:Cactus (A) Big.jpg|20px]] |
− | | | | | | | | | + | | <math>\texttt{(} a \texttt{)}</math> |
− | | f_6 | f_0110 | 0 1 1 0 | (x, y) | x not equal to y | x + y | | + | | <math>\lnot a</math> |
− | | | | | | | | | + | | <math>\lnot a</math> |
− | | f_7 | f_0111 | 0 1 1 1 | (x y) | not both x and y | ~x v ~y | | + | |- |
− | | | | | | | | | + | | height="100px" | [[Image:Cactus ABC Big.jpg|50px]] |
− | | f_8 | f_1000 | 1 0 0 0 | x y | x and y | x & y | | + | | <math>a~b~c</math> |
− | | | | | | | | | + | | <math>a \land b \land c</math> |
− | | f_9 | f_1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y | | + | | <math>a \lor b \lor c</math> |
− | | | | | | | | | + | |- |
− | | f_10 | f_1010 | 1 0 1 0 | y | y | y | | + | | height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]] |
− | | | | | | | | | + | | <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math> |
− | | f_11 | f_1011 | 1 0 1 1 | (x (y)) | not x without y | x => y | | + | | <math>a \lor b \lor c</math> |
− | | | | | | | | | + | | <math>a \land b \land c</math> |
− | | f_12 | f_1100 | 1 1 0 0 | x | x | x | | + | |- |
− | | | | | | | | | + | | height="120px" | [[Image:Cactus (A(B)) Big.jpg|60px]] |
− | | f_13 | f_1101 | 1 1 0 1 | ((x) y) | not y without x | x <= y | | + | | <math>\texttt{(} a \texttt{(} b \texttt{))}</math> |
− | | | | | | | | | + | | <math>a \Rightarrow b</math> |
− | | f_14 | f_1110 | 1 1 1 0 | ((x)(y)) | x or y | x v y | | + | | |
− | | | | | | | | | + | |- |
− | | f_15 | f_1111 | 1 1 1 1 | (()) | true | 1 | | + | | height="120px" | [[Image:Cactus (A)B Big.jpg|35px]] |
− | | | | | | | | | + | | <math>\texttt{(} a \texttt{)} b</math> |
− | o---------o---------o---------o----------o------------------o----------o | + | | |
− | </pre> | + | | <math>a \Rightarrow b</math> |
− | | + | |- |
− | <pre> | + | | height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]] |
− | Table A2. Propositional Forms On Two Variables | + | | <math>\texttt{(} a, b \texttt{)}</math> |
− | o---------o---------o---------o----------o------------------o----------o | + | | <math>a \neq b</math> |
− | | L_1 | L_2 | L_3 | L_4 | L_5 | L_6 | | + | | <math>a = b\!</math> |
− | | | | | | | | | + | |- |
− | | Decimal | Binary | Vector | Cactus | English | Ordinary | | + | | height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]] |
− | o---------o---------o---------o----------o------------------o----------o | + | | <math>\texttt{((} a, b \texttt{))}</math> |
− | | | x : 1 1 0 0 | | | | | + | | <math>a = b\!</math> |
− | | | y : 1 0 1 0 | | | | | + | | <math>a \neq b\!</math> |
− | o---------o---------o---------o----------o------------------o----------o | + | |- |
− | | | | | | | | | + | | height="120px" | [[Image:Cactus (A,B,C) Big.jpg|65px]] |
− | | f_0 | f_0000 | 0 0 0 0 | () | false | 0 | | + | | <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> |
| + | Table A1. Propositional Forms On Two Variables |
| + | o---------o---------o---------o----------o------------------o----------o |
| + | | L_1 | L_2 | L_3 | L_4 | L_5 | L_6 | |
| + | | | | | | | | |
| + | | Decimal | Binary | Vector | Cactus | English | Ordinary | |
| + | o---------o---------o---------o----------o------------------o----------o |
| + | | | x : 1 1 0 0 | | | | |
| + | | | y : 1 0 1 0 | | | | |
| + | o---------o---------o---------o----------o------------------o----------o |
| + | | | | | | | | |
| + | | f_0 | f_0000 | 0 0 0 0 | () | false | 0 | |
| + | | | | | | | | |
| + | | f_1 | f_0001 | 0 0 0 1 | (x)(y) | neither x nor y | ~x & ~y | |
| + | | | | | | | | |
| + | | f_2 | f_0010 | 0 0 1 0 | (x) y | y and not x | ~x & y | |
| + | | | | | | | | |
| + | | f_3 | f_0011 | 0 0 1 1 | (x) | not x | ~x | |
| + | | | | | | | | |
| + | | f_4 | f_0100 | 0 1 0 0 | x (y) | x and not y | x & ~y | |
| + | | | | | | | | |
| + | | f_5 | f_0101 | 0 1 0 1 | (y) | not y | ~y | |
| + | | | | | | | | |
| + | | f_6 | f_0110 | 0 1 1 0 | (x, y) | x not equal to y | x + y | |
| + | | | | | | | | |
| + | | f_7 | f_0111 | 0 1 1 1 | (x y) | not both x and y | ~x v ~y | |
| + | | | | | | | | |
| + | | f_8 | f_1000 | 1 0 0 0 | x y | x and y | x & y | |
| + | | | | | | | | |
| + | | f_9 | f_1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y | |
| + | | | | | | | | |
| + | | f_10 | f_1010 | 1 0 1 0 | y | y | y | |
| + | | | | | | | | |
| + | | f_11 | f_1011 | 1 0 1 1 | (x (y)) | not x without y | x => y | |
| + | | | | | | | | |
| + | | f_12 | f_1100 | 1 1 0 0 | x | x | x | |
| + | | | | | | | | |
| + | | f_13 | f_1101 | 1 1 0 1 | ((x) y) | not y without x | x <= y | |
| + | | | | | | | | |
| + | | f_14 | f_1110 | 1 1 1 0 | ((x)(y)) | x or y | x v y | |
| + | | | | | | | | |
| + | | f_15 | f_1111 | 1 1 1 1 | (()) | true | 1 | |
| + | | | | | | | | |
| + | o---------o---------o---------o----------o------------------o----------o |
| + | </pre> |
| + | |
| + | <pre> |
| + | Table A2. Propositional Forms On Two Variables |
| + | o---------o---------o---------o----------o------------------o----------o |
| + | | L_1 | L_2 | L_3 | L_4 | L_5 | L_6 | |
| + | | | | | | | | |
| + | | Decimal | Binary | Vector | Cactus | English | Ordinary | |
| + | o---------o---------o---------o----------o------------------o----------o |
| + | | | x : 1 1 0 0 | | | | |
| + | | | y : 1 0 1 0 | | | | |
| + | o---------o---------o---------o----------o------------------o----------o |
| + | | | | | | | | |
| + | | f_0 | f_0000 | 0 0 0 0 | () | false | 0 | |
| | | | | | | | | | | | | | | | | |
| o---------o---------o---------o----------o------------------o----------o | | o---------o---------o---------o----------o------------------o----------o |
Line 2,340: |
Line 2,162: |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%" | + | {| 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> | | |+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math> |
| |- style="background:#f0f0ff" | | |- style="background:#f0f0ff" |
Line 2,597: |
Line 2,419: |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%" | + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
| |+ <math>\text{Table A2.}~~\text{Propositional Forms on Two Variables}</math> | | |+ <math>\text{Table A2.}~~\text{Propositional Forms on Two Variables}</math> |
| |- style="background:#f0f0ff" | | |- style="background:#f0f0ff" |
Line 2,883: |
Line 2,705: |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%" | + | {| 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> | | |+ <math>\text{Table A3.}~~\operatorname{E}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}p, \operatorname{d}q \}</math> |
| |- style="background:#f0f0ff" | | |- style="background:#f0f0ff" |
Line 3,157: |
Line 2,979: |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%" | + | {| 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> | | |+ <math>\text{Table A4.}~~\operatorname{D}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}p, \operatorname{d}q \}</math> |
| |- style="background:#f0f0ff" | | |- style="background:#f0f0ff" |
Line 3,421: |
Line 3,243: |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%" | + | {| 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> | | |+ <math>\text{Table A5.}~~\operatorname{E}f ~\text{Expanded Over Ordinary Features}~ \{ p, q \}</math> |
| |- style="background:#f0f0ff" | | |- style="background:#f0f0ff" |
Line 3,681: |
Line 3,503: |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%" | + | {| 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}~ \{ p, q \}</math> | | |+ <math>\text{Table A6.}~~\operatorname{D}f ~\text{Expanded Over Ordinary Features}~ \{ p, q \}</math> |
| |- style="background:#f0f0ff" | | |- style="background:#f0f0ff" |
Line 3,945: |
Line 3,767: |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%" | + | {| 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> | | |+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math> |
| |- style="background:#f0f0ff" | | |- style="background:#f0f0ff" |
Line 4,096: |
Line 3,918: |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%" | + | {| 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> | | |+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math> |
| |- style="background:#f0f0ff" | | |- style="background:#f0f0ff" |
Line 4,353: |
Line 4,175: |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%" | + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
| |+ <math>\text{Table A2.}~~\text{Propositional Forms on Two Variables}</math> | | |+ <math>\text{Table A2.}~~\text{Propositional Forms on Two Variables}</math> |
| |- style="background:#f0f0ff" | | |- style="background:#f0f0ff" |
Line 4,639: |
Line 4,461: |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%" | + | {| 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}x, \operatorname{d}y \}</math> | | |+ <math>\text{Table A3.}~~\operatorname{E}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}x, \operatorname{d}y \}</math> |
| |- style="background:#f0f0ff" | | |- style="background:#f0f0ff" |
Line 4,913: |
Line 4,735: |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%" | + | {| 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> | | |+ <math>\text{Table A4.}~~\operatorname{D}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}x, \operatorname{d}y \}</math> |
| |- style="background:#f0f0ff" | | |- style="background:#f0f0ff" |
Line 5,177: |
Line 4,999: |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%" | + | {| 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}~ \{ x, y \}</math> | | |+ <math>\text{Table A5.}~~\operatorname{E}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math> |
| |- style="background:#f0f0ff" | | |- style="background:#f0f0ff" |
Line 5,437: |
Line 5,259: |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%" | + | {| 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> | | |+ <math>\text{Table A6.}~~\operatorname{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math> |
| |- style="background:#f0f0ff" | | |- style="background:#f0f0ff" |
Line 5,703: |
Line 5,525: |
| {| 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%" | | {| 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" | | |- style="height:50px" |
− | | width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot</math> | + | | width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math> |
| | width="22%" style="border-bottom:1px solid black" | | | | width="22%" style="border-bottom:1px solid black" | |
| <math>\operatorname{T}_{00}</math> | | <math>\operatorname{T}_{00}</math> |
Line 5,742: |
Line 5,564: |
| {| 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%" | | {| 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" | | |- style="height:50px" |
− | | width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot</math> | + | | width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math> |
| | width="22%" style="border-bottom:1px solid black" | | | | width="22%" style="border-bottom:1px solid black" | |
| <math>\operatorname{e}</math> | | <math>\operatorname{e}</math> |
Line 5,783: |
Line 5,605: |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%" | + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
| |+ <math>\text{Permutation Substitutions in}~ \operatorname{Sym} \{ \mathrm{A}, \mathrm{B}, \mathrm{C} \}</math> | | |+ <math>\text{Permutation Substitutions in}~ \operatorname{Sym} \{ \mathrm{A}, \mathrm{B}, \mathrm{C} \}</math> |
| |- style="background:#f0f0ff" | | |- style="background:#f0f0ff" |
Line 5,845: |
Line 5,667: |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%" | + | {| 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> | | |+ <math>\text{Matrix Representations of Permutations in}~ \operatorname{Sym}(3)</math> |
| |- style="background:#f0f0ff" | | |- style="background:#f0f0ff" |
Line 6,509: |
Line 6,331: |
| \end{tabular}\end{quote} | | \end{tabular}\end{quote} |
| </pre> | | </pre> |
| + | |
| + | ==Group Operation Tables== |
| + | |
| + | <br> |
| + | |
| + | {| 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 32.1}~~\text{Scheme of a Group Operation Table}</math> |
| + | |- style="height:50px" |
| + | | style="border-bottom:1px solid black; border-right:1px solid black" | <math>*\!</math> |
| + | | style="border-bottom:1px solid black" | <math>x_0\!</math> |
| + | | style="border-bottom:1px solid black" | <math>\cdots\!</math> |
| + | | style="border-bottom:1px solid black" | <math>x_j\!</math> |
| + | | style="border-bottom:1px solid black" | <math>\cdots\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>x_0\!</math> |
| + | | <math>x_0 * x_0\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>x_0 * x_j\!</math> |
| + | | <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> |
| + | |
| + | {| 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> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | {| 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" 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 33.1}~~\text{Multiplication Operation of the Group}~V_4</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{e}</math> |
| + | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{f}</math> |
| + | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{g}</math> |
| + | | width="20%" 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> |
| + | |
| + | {| 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> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | {| 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> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | {| 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> |
| + | |
| + | {| 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 34.2}~~\text{Regular Representation of the Group}~Z_4(\cdot)</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{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> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | {| 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 35.1}~~\text{Additive Presentation of the Group}~Z_4(+)</math> |
| + | |- style="height:50px" |
| + | | width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>+\!</math> |
| + | | 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> |
| + | |
| + | {| 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> |
| + | |- 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{0}</math> |
| + | | width="4%" | <math>\{\!</math> |
| + | | 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> |
| + | |
| + | ==Higher Order Propositions== |
| + | |
| + | <br> |
| + | |
| + | <table align="center" cellpadding="4" cellspacing="0" style="text-align:center; width:90%"> |
| + | |
| + | <caption><font size="+2"><math>\text{Table 1.} ~~ \text{Higher Order Propositions} ~ (n = 1)</math></font></caption> |
| + | |
| + | <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> |
| + | |
| + | <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> |
| + | |
| + | <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> |
| + | |
| + | </table> |
| + | |
| + | <br> |
| + | |
| + | <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> |
| + | |
| + | <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> |
| + | |
| + | <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> |
| + | |
| + | <tr> |
| + | <td style="border-right:2px solid black"><math>m_{4}</math></td> |
| + | <td> </td> |
| + | <td>Just <math>x</math></td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-right:2px solid black"><math>m_{5}</math></td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td>Everything is <math>x</math></td> |
| + | <td><math>f</math> is linear</td> |
| + | <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> |
| + | |
| + | <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> |
| + | |
| + | <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> |
| + | |
| + | </table> |
| + | |
| + | <br> |
| + | |
| + | <table align="center" cellpadding="1" cellspacing="0" style="background:white; color:black; text-align:center; width:90%"> |
| + | |
| + | <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> |
| + | |
| + | <br> |
| + | |
| + | <table align="center" cellpadding="1" cellspacing="0" style="text-align:center; width:90%"> |
| + | |
| + | <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> |
| + | |
| + | <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>\alpha_{15}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\alpha_{14}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\alpha_{13}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\alpha_{12}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\alpha_{11}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\alpha_{10}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\alpha_{9}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\alpha_{8}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\alpha_{7}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\alpha_{6}</math></td> |
| + | <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> |
| + | |
| + | <table align="center" cellpadding="1" cellspacing="0" style="text-align:center; width:90%"> |
| + | |
| + | <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> |
| + | |
| + | <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>\beta_{0}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\beta_{1}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\beta_{2}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\beta_{3}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\beta_{4}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\beta_{5}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\beta_{6}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\beta_{7}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\beta_{8}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\beta_{9}</math></td> |
| + | <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> |
| + | |
| + | <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%"> |
| + | |
| + | <caption><font size="+2"><math>\text{Table 8.} ~~ \text{Simple Qualifiers of Propositions (Version 1)}</math></font></caption> |
| + | |
| + | <tr> |
| + | <td width="4%" style="border-bottom:1px solid black" align="right"> |
| + | <math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td> |
| + | <td width="6%" style="border-bottom:1px solid black"> |
| + | <math>\begin{matrix}1100\\1010\end{matrix}</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black; border-right:1px solid black"> |
| + | <math>f</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black"> |
| + | <math>\begin{smallmatrix} |
| + | \texttt{(} \ell_{11} \texttt{)} |
| + | \\ |
| + | \mathrm{No} ~ u |
| + | \\ |
| + | \mathrm{is} ~ v |
| + | \end{smallmatrix}</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black"> |
| + | <math>\begin{smallmatrix} |
| + | \texttt{(} \ell_{10} \texttt{)} |
| + | \\ |
| + | \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><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> |
| + | |
| + | <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_{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> |
| + | |
| + | <tr> |
| + | <td><math>f_{10}</math></td> |
| + | <td><math>1010</math></td> |
| + | <td style="border-right:1px solid black"><math>v</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></tr> |
| + | |
| + | <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> |
| + | |
| + | <br> |
| + | |
| + | <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%"> |
| + | |
| + | <caption><font size="+2"><math>\text{Table 9.} ~~ \text{Simple Qualifiers of Propositions (Version 2)}</math></font></caption> |
| + | |
| + | <tr> |
| + | <td width="4%" style="border-bottom:1px solid black" align="right"> |
| + | <math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td> |
| + | <td width="6%" style="border-bottom:1px solid black"> |
| + | <math>\begin{matrix}1100\\1010\end{matrix}</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black; border-right:1px solid black"> |
| + | <math>f</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black"> |
| + | <math>\begin{smallmatrix} |
| + | \texttt{(} \ell_{11} \texttt{)} |
| + | \\ |
| + | \mathrm{No} ~ u |
| + | \\ |
| + | \mathrm{is} ~ v |
| + | \end{smallmatrix}</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black"> |
| + | <math>\begin{smallmatrix} |
| + | \texttt{(} \ell_{10} \texttt{)} |
| + | \\ |
| + | \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> |
| + | |
| + | <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 style="border-bottom:1px solid black"><math>f_{10}</math></td> |
| + | <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> |
| + | <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: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></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_{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_{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> |
| + | |
| + | <br> |
| + | |
| + | <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%"> |
| + | |
| + | <caption><font size="+2"><math>\text{Table 10.} ~~ \text{Relation of Quantifiers to Higher Order Propositions}</math></font></caption> |
| + | |
| + | <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== | | ==Inquiry Driven Systems== |
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