Difference between revisions of "User:Jon Awbrey/TEST"
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<br> | <br> | ||
| − | <table align="center" cellpadding="1" cellspacing="0" style="text-align:center; width:90%"> | + | <table align="center" cellpadding="1" 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> | <caption><font size="+2"><math>\text{Table 8.} ~~ \text{Simple Qualifiers of Propositions (Version 1)}</math></font></caption> | ||
<tr> | <tr> | ||
| − | <td width="4%" style="border-bottom: | + | <td width="4%" style="border-bottom:1px solid black" align="right"> |
<math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td> | <math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td> | ||
| − | <td width="6%" style="border-bottom: | + | <td width="6%" style="border-bottom:1px solid black"> |
<math>\begin{matrix}1100\\1010\end{matrix}</math></td> | <math>\begin{matrix}1100\\1010\end{matrix}</math></td> | ||
| − | <td width="10%" style="border-bottom: | + | <td width="10%" style="border-bottom:1px solid black; border-right:1px solid black"> |
<math>f</math></td> | <math>f</math></td> | ||
| − | <td width="10%" style="border-bottom: | + | <td width="10%" style="border-bottom:1px solid black"> |
<math>\texttt{(} \ell_{11} \texttt{)}</math><br> | <math>\texttt{(} \ell_{11} \texttt{)}</math><br> | ||
<math>No ~ u</math><br> | <math>No ~ u</math><br> | ||
<math>is ~ v</math></td> | <math>is ~ v</math></td> | ||
| − | <td width="10%" style="border-bottom: | + | <td width="10%" style="border-bottom:1px solid black"> |
<math>\texttt{(} \ell_{10} \texttt{)}</math><br> | <math>\texttt{(} \ell_{10} \texttt{)}</math><br> | ||
<math>No ~ u</math><br> | <math>No ~ u</math><br> | ||
<math>is ~ \texttt{(} v \texttt{)}</math></td> | <math>is ~ \texttt{(} v \texttt{)}</math></td> | ||
| − | <td width="10%" style="border-bottom: | + | <td width="10%" style="border-bottom:1px solid black"> |
<math>\texttt{(} \ell_{01} \texttt{)}</math><br> | <math>\texttt{(} \ell_{01} \texttt{)}</math><br> | ||
<math>No ~ \texttt{(} u \texttt{)}</math><br> | <math>No ~ \texttt{(} u \texttt{)}</math><br> | ||
<math>is ~ v</math></td> | <math>is ~ v</math></td> | ||
| − | <td width="10%" style="border-bottom: | + | <td width="10%" style="border-bottom:1px solid black"> |
<math>\texttt{(} \ell_{00} \texttt{)}</math><br> | <math>\texttt{(} \ell_{00} \texttt{)}</math><br> | ||
<math>No ~ \texttt{(} u \texttt{)}</math><br> | <math>No ~ \texttt{(} u \texttt{)}</math><br> | ||
<math>is ~ \texttt{(} v \texttt{)}</math></td> | <math>is ~ \texttt{(} v \texttt{)}</math></td> | ||
| − | <td width="10%" style="border-bottom: | + | <td width="10%" style="border-bottom:1px solid black"> |
<math>\ell_{00}</math><br> | <math>\ell_{00}</math><br> | ||
<math>Some ~ \texttt{(} u \texttt{)}</math><br> | <math>Some ~ \texttt{(} u \texttt{)}</math><br> | ||
<math>is ~ \texttt{(} v \texttt{)}</math></td> | <math>is ~ \texttt{(} v \texttt{)}</math></td> | ||
| − | <td width="10%" style="border-bottom: | + | <td width="10%" style="border-bottom:1px solid black"> |
<math>\ell_{01}</math><br> | <math>\ell_{01}</math><br> | ||
<math>Some ~ \texttt{(} u \texttt{)}</math><br> | <math>Some ~ \texttt{(} u \texttt{)}</math><br> | ||
<math>is ~ v</math></td> | <math>is ~ v</math></td> | ||
| − | <td width="10%" style="border-bottom: | + | <td width="10%" style="border-bottom:1px solid black"> |
<math>\ell_{10}</math><br> | <math>\ell_{10}</math><br> | ||
<math>Some ~ u</math><br> | <math>Some ~ u</math><br> | ||
<math>is ~ \texttt{(} v \texttt{)}</math></td> | <math>is ~ \texttt{(} v \texttt{)}</math></td> | ||
| − | <td width="10%" style="border-bottom: | + | <td width="10%" style="border-bottom:1px solid black"> |
<math>\ell_{11}</math><br> | <math>\ell_{11}</math><br> | ||
<math>Some ~ u</math><br> | <math>Some ~ u</math><br> | ||
| Line 50: | Line 50: | ||
<td><math>f_{0}</math></td> | <td><math>f_{0}</math></td> | ||
<td><math>0000</math></td> | <td><math>0000</math></td> | ||
| − | <td style="border-right: | + | <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> | ||
| Line 63: | Line 63: | ||
<td><math>f_{1}</math></td> | <td><math>f_{1}</math></td> | ||
<td><math>0001</math></td> | <td><math>0001</math></td> | ||
| − | <td style="border-right: | + | <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:black; color:white">1</td> | ||
| Line 76: | Line 76: | ||
<td><math>f_{2}</math></td> | <td><math>f_{2}</math></td> | ||
<td><math>0010</math></td> | <td><math>0010</math></td> | ||
| − | <td style="border-right: | + | <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:black; color:white">1</td> | <td style="background:black; color:white">1</td> | ||
| Line 89: | Line 89: | ||
<td><math>f_{3}</math></td> | <td><math>f_{3}</math></td> | ||
<td><math>0011</math></td> | <td><math>0011</math></td> | ||
| − | <td style="border-right: | + | <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:black; color:white">1</td> | <td style="background:black; color:white">1</td> | ||
| Line 102: | Line 102: | ||
<td><math>f_{4}</math></td> | <td><math>f_{4}</math></td> | ||
<td><math>0100</math></td> | <td><math>0100</math></td> | ||
| − | <td style="border-right: | + | <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:black; color:white">1</td> | ||
<td style="background:white; color:black">0</td> | <td style="background:white; color:black">0</td> | ||
| Line 115: | Line 115: | ||
<td><math>f_{5}</math></td> | <td><math>f_{5}</math></td> | ||
<td><math>0101</math></td> | <td><math>0101</math></td> | ||
| − | <td style="border-right: | + | <td style="border-right:1px solid black"><math>\texttt{(} v \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> | ||
| Line 128: | Line 128: | ||
<td><math>f_{6}</math></td> | <td><math>f_{6}</math></td> | ||
<td><math>0110</math></td> | <td><math>0110</math></td> | ||
| − | <td style="border-right: | + | <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:white; color:black">0</td> | <td style="background:white; color:black">0</td> | ||
| Line 141: | Line 141: | ||
<td><math>f_{7}</math></td> | <td><math>f_{7}</math></td> | ||
<td><math>0111</math></td> | <td><math>0111</math></td> | ||
| − | <td style="border-right: | + | <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:black; color:white">1</td> | ||
<td style="background:white; color:black">0</td> | <td style="background:white; color:black">0</td> | ||
| Line 154: | Line 154: | ||
<td><math>f_{8}</math></td> | <td><math>f_{8}</math></td> | ||
<td><math>1000</math></td> | <td><math>1000</math></td> | ||
| − | <td style="border-right: | + | <td style="border-right:1px 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:black; color:white">1</td> | <td style="background:black; color:white">1</td> | ||
| Line 167: | Line 167: | ||
<td><math>f_{9}</math></td> | <td><math>f_{9}</math></td> | ||
<td><math>1001</math></td> | <td><math>1001</math></td> | ||
| − | <td style="border-right: | + | <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:black; color:white">1</td> | ||
| Line 180: | Line 180: | ||
<td><math>f_{10}</math></td> | <td><math>f_{10}</math></td> | ||
<td><math>1010</math></td> | <td><math>1010</math></td> | ||
| − | <td style="border-right: | + | <td style="border-right:1px 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:black; color:white">1</td> | <td style="background:black; color:white">1</td> | ||
| Line 193: | Line 193: | ||
<td><math>f_{11}</math></td> | <td><math>f_{11}</math></td> | ||
<td><math>1011</math></td> | <td><math>1011</math></td> | ||
| − | <td style="border-right: | + | <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:black; color:white">1</td> | ||
| Line 206: | Line 206: | ||
<td><math>f_{12}</math></td> | <td><math>f_{12}</math></td> | ||
<td><math>1100</math></td> | <td><math>1100</math></td> | ||
| − | <td style="border-right: | + | <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:white; color:black">0</td> | <td style="background:white; color:black">0</td> | ||
| Line 219: | Line 219: | ||
<td><math>f_{13}</math></td> | <td><math>f_{13}</math></td> | ||
<td><math>1101</math></td> | <td><math>1101</math></td> | ||
| − | <td style="border-right: | + | <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:white; color:black">0</td> | ||
| Line 232: | Line 232: | ||
<td><math>f_{14}</math></td> | <td><math>f_{14}</math></td> | ||
<td><math>1110</math></td> | <td><math>1110</math></td> | ||
| − | <td style="border-right: | + | <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:white; color:black">0</td> | ||
| Line 245: | Line 245: | ||
<td><math>f_{15}</math></td> | <td><math>f_{15}</math></td> | ||
<td><math>1111</math></td> | <td><math>1111</math></td> | ||
| − | <td style="border-right: | + | <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> | ||
| Line 259: | Line 259: | ||
<br> | <br> | ||
| − | <table align="center" cellpadding="1" cellspacing="0" style="text-align:center; width:90%"> | + | <table align="center" cellpadding="1" 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> | <caption><font size="+2"><math>\text{Table 9.} ~~ \text{Simple Qualifiers of Propositions (Version 2)}</math></font></caption> | ||
<tr> | <tr> | ||
| − | <td width="4%" style="border-bottom: | + | <td width="4%" style="border-bottom:1px solid black" align="right"> |
<math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td> | <math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td> | ||
| − | <td width="6%" style="border-bottom: | + | <td width="6%" style="border-bottom:1px solid black"> |
<math>\begin{matrix}1100\\1010\end{matrix}</math></td> | <math>\begin{matrix}1100\\1010\end{matrix}</math></td> | ||
| − | <td width="10%" style="border-bottom: | + | <td width="10%" style="border-bottom:1px solid black; border-right:1px solid black"> |
<math>f</math></td> | <math>f</math></td> | ||
| − | <td width="10%" style="border-bottom: | + | <td width="10%" style="border-bottom:1px solid black"> |
<math>\texttt{(} \ell_{11} \texttt{)}</math><br> | <math>\texttt{(} \ell_{11} \texttt{)}</math><br> | ||
<math>No ~ u</math><br> | <math>No ~ u</math><br> | ||
<math>is ~ v</math></td> | <math>is ~ v</math></td> | ||
| − | <td width="10%" style="border-bottom: | + | <td width="10%" style="border-bottom:1px solid black"> |
<math>\texttt{(} \ell_{10} \texttt{)}</math><br> | <math>\texttt{(} \ell_{10} \texttt{)}</math><br> | ||
<math>No ~ u</math><br> | <math>No ~ u</math><br> | ||
<math>is ~ \texttt{(} v \texttt{)}</math></td> | <math>is ~ \texttt{(} v \texttt{)}</math></td> | ||
| − | <td width="10%" style="border-bottom: | + | <td width="10%" style="border-bottom:1px solid black"> |
<math>\texttt{(} \ell_{01} \texttt{)}</math><br> | <math>\texttt{(} \ell_{01} \texttt{)}</math><br> | ||
<math>No ~ \texttt{(} u \texttt{)}</math><br> | <math>No ~ \texttt{(} u \texttt{)}</math><br> | ||
<math>is ~ v</math></td> | <math>is ~ v</math></td> | ||
| − | <td width="10%" style="border-bottom: | + | <td width="10%" style="border-bottom:1px solid black"> |
<math>\texttt{(} \ell_{00} \texttt{)}</math><br> | <math>\texttt{(} \ell_{00} \texttt{)}</math><br> | ||
<math>No ~ \texttt{(} u \texttt{)}</math><br> | <math>No ~ \texttt{(} u \texttt{)}</math><br> | ||
<math>is ~ \texttt{(} v \texttt{)}</math></td> | <math>is ~ \texttt{(} v \texttt{)}</math></td> | ||
| − | <td width="10%" style="border-bottom: | + | <td width="10%" style="border-bottom:1px solid black"> |
<math>\ell_{00}</math><br> | <math>\ell_{00}</math><br> | ||
<math>Some ~ \texttt{(} u \texttt{)}</math><br> | <math>Some ~ \texttt{(} u \texttt{)}</math><br> | ||
<math>is ~ \texttt{(} v \texttt{)}</math></td> | <math>is ~ \texttt{(} v \texttt{)}</math></td> | ||
| − | <td width="10%" style="border-bottom: | + | <td width="10%" style="border-bottom:1px solid black"> |
<math>\ell_{01}</math><br> | <math>\ell_{01}</math><br> | ||
<math>Some ~ \texttt{(} u \texttt{)}</math><br> | <math>Some ~ \texttt{(} u \texttt{)}</math><br> | ||
<math>is ~ v</math></td> | <math>is ~ v</math></td> | ||
| − | <td width="10%" style="border-bottom: | + | <td width="10%" style="border-bottom:1px solid black"> |
<math>\ell_{10}</math><br> | <math>\ell_{10}</math><br> | ||
<math>Some ~ u</math><br> | <math>Some ~ u</math><br> | ||
<math>is ~ \texttt{(} v \texttt{)}</math></td> | <math>is ~ \texttt{(} v \texttt{)}</math></td> | ||
| − | <td width="10%" style="border-bottom: | + | <td width="10%" style="border-bottom:1px solid black"> |
<math>\ell_{11}</math><br> | <math>\ell_{11}</math><br> | ||
<math>Some ~ u</math><br> | <math>Some ~ u</math><br> | ||
| Line 304: | Line 304: | ||
<tr> | <tr> | ||
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black"><math>f_{0}</math></td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black"><math>0000</math></td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; border-right:1px solid black"><math>\texttt{(~)}</math></td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:white; color:black">0</td></tr> |
<tr> | <tr> | ||
<td><math>f_{1}</math></td> | <td><math>f_{1}</math></td> | ||
<td><math>0001</math></td> | <td><math>0001</math></td> | ||
| − | <td style="border-right: | + | <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:black; color:white">1</td> | ||
| Line 332: | Line 332: | ||
<td><math>f_{2}</math></td> | <td><math>f_{2}</math></td> | ||
<td><math>0010</math></td> | <td><math>0010</math></td> | ||
| − | <td style="border-right: | + | <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:black; color:white">1</td> | <td style="background:black; color:white">1</td> | ||
| Line 345: | Line 345: | ||
<td><math>f_{4}</math></td> | <td><math>f_{4}</math></td> | ||
<td><math>0100</math></td> | <td><math>0100</math></td> | ||
| − | <td style="border-right: | + | <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:black; color:white">1</td> | ||
<td style="background:white; color:black">0</td> | <td style="background:white; color:black">0</td> | ||
| Line 356: | Line 356: | ||
<tr> | <tr> | ||
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black"><math>f_{8}</math></td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black"><math>1000</math></td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; border-right:1px solid black"><math>u ~ v</math></td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:black; color:white">1</td></tr> |
<tr> | <tr> | ||
<td><math>f_{3}</math></td> | <td><math>f_{3}</math></td> | ||
<td><math>0011</math></td> | <td><math>0011</math></td> | ||
| − | <td style="border-right: | + | <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:black; color:white">1</td> | <td style="background:black; color:white">1</td> | ||
| Line 382: | Line 382: | ||
<tr> | <tr> | ||
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black"><math>f_{12}</math></td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black"><math>1100</math></td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; border-right:1px solid black"><math>u</math></td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:black; color:white">1</td></tr> |
<tr> | <tr> | ||
<td><math>f_{6}</math></td> | <td><math>f_{6}</math></td> | ||
<td><math>0110</math></td> | <td><math>0110</math></td> | ||
| − | <td style="border-right: | + | <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:white; color:black">0</td> | <td style="background:white; color:black">0</td> | ||
| Line 408: | Line 408: | ||
<tr> | <tr> | ||
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black"><math>f_{9}</math></td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black"><math>1001</math></td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; border-right:1px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:black; color:white">1</td></tr> |
<tr> | <tr> | ||
<td><math>f_{5}</math></td> | <td><math>f_{5}</math></td> | ||
<td><math>0101</math></td> | <td><math>0101</math></td> | ||
| − | <td style="border-right: | + | <td style="border-right:1px solid black"><math>\texttt{(} v \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> | ||
| Line 434: | Line 434: | ||
<tr> | <tr> | ||
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black"><math>f_{10}</math></td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black"><math>1010</math></td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; border-right:1px solid black"><math>v</math></td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:black; color:white">1</td></tr> |
<tr> | <tr> | ||
<td><math>f_{7}</math></td> | <td><math>f_{7}</math></td> | ||
<td><math>0111</math></td> | <td><math>0111</math></td> | ||
| − | <td style="border-right: | + | <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:black; color:white">1</td> | ||
<td style="background:white; color:black">0</td> | <td style="background:white; color:black">0</td> | ||
| Line 462: | Line 462: | ||
<td><math>f_{11}</math></td> | <td><math>f_{11}</math></td> | ||
<td><math>1011</math></td> | <td><math>1011</math></td> | ||
| − | <td style="border-right: | + | <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:black; color:white">1</td> | ||
| Line 475: | Line 475: | ||
<td><math>f_{13}</math></td> | <td><math>f_{13}</math></td> | ||
<td><math>1101</math></td> | <td><math>1101</math></td> | ||
| − | <td style="border-right: | + | <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:white; color:black">0</td> | ||
| Line 486: | Line 486: | ||
<tr> | <tr> | ||
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black"><math>f_{14}</math></td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black"><math>1110</math></td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; border-right:1px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black; background:black; color:white">1</td></tr> |
<tr> | <tr> | ||
<td><math>f_{15}</math></td> | <td><math>f_{15}</math></td> | ||
<td><math>1111</math></td> | <td><math>1111</math></td> | ||
| − | <td style="border-right: | + | <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> | ||
| Line 515: | Line 515: | ||
<br> | <br> | ||
| − | <table align="center" cellpadding="4" cellspacing="0" style="text-align:center; width:90%"> | + | <table align="center" cellpadding="4" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:90%"> |
<caption><font size="+2"><math>\text{Table 10.} ~~ \text{Relation of Quantifiers to Higher Order Propositions}</math></font></caption> | <caption><font size="+2"><math>\text{Table 10.} ~~ \text{Relation of Quantifiers to Higher Order Propositions}</math></font></caption> | ||
<tr> | <tr> | ||
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black"><math>\mathrm{Mnemonic}</math></td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black"><math>\mathrm{Category}</math></td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black"><math>\mathrm{Classical~Form}</math></td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black"><math>\mathrm{Alternate~Form}</math></td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black"><math>\mathrm{Symmetric~Form}</math></td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black"><math>\mathrm{Operator}</math></td></tr> |
<tr> | <tr> | ||
<td><math>\mathrm{E}</math><br><math>\mathrm{Exclusive}</math></td> | <td><math>\mathrm{E}</math><br><math>\mathrm{Exclusive}</math></td> | ||
| − | <td><math>\mathrm{Universal}</math><br><math>\ | + | <td><math>\mathrm{Universal}</math><br><math>\mathrm{Negative}</math></td> |
<td><math>\mathrm{All} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td> | <td><math>\mathrm{All} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td> | ||
<td> </td> | <td> </td> | ||
| Line 552: | Line 552: | ||
<tr> | <tr> | ||
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black"> </td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black"> </td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black"><math>\mathrm{All} ~ \texttt{(} v \texttt{)} ~ \mathrm{is} ~ u</math></td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black"><math>\mathrm{No} ~ \texttt{(} v \texttt{)} ~ \mathrm{is} ~ \texttt{(} u \texttt{)}</math></td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black"><math>\mathrm{No} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td> |
| − | <td style="border-bottom: | + | <td style="border-bottom:1px solid black"><math>\texttt{(} \ell_{00} \texttt{)}</math></td></tr> |
<tr> | <tr> | ||
Revision as of 02:20, 23 November 2009
The following Tables develop these ideas in more detail.
| \(\begin{matrix}u\!:\\v\!:\end{matrix}\) | \(\begin{matrix}1100\\1010\end{matrix}\) | \(f\) |
\(\texttt{(} \ell_{11} \texttt{)}\) |
\(\texttt{(} \ell_{10} \texttt{)}\) |
\(\texttt{(} \ell_{01} \texttt{)}\) |
\(\texttt{(} \ell_{00} \texttt{)}\) |
\(\ell_{00}\) |
\(\ell_{01}\) |
\(\ell_{10}\) |
\(\ell_{11}\) |
| \(f_{0}\) | \(0000\) | \(\texttt{(~)}\) | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
| \(f_{1}\) | \(0001\) | \(\texttt{(} u \texttt{)(} v \texttt{)}\) | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 |
| \(f_{2}\) | \(0010\) | \(\texttt{(} u\texttt{)} ~ v\) | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 |
| \(f_{3}\) | \(0011\) | \(\texttt{(} u \texttt{)}\) | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 |
| \(f_{4}\) | \(0100\) | \(u ~ \texttt{(} v \texttt{)}\) | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 |
| \(f_{5}\) | \(0101\) | \(\texttt{(} v \texttt{)}\) | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |
| \(f_{6}\) | \(0110\) | \(\texttt{(} u \texttt{,} v \texttt{)}\) | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 |
| \(f_{7}\) | \(0111\) | \(\texttt{(} u ~ v \texttt{)}\) | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 |
| \(f_{8}\) | \(1000\) | \(u ~ v\) | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 |
| \(f_{9}\) | \(1001\) | \(\texttt{((} u \texttt{,} v \texttt{))}\) | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 |
| \(f_{10}\) | \(1010\) | \(v\) | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| \(f_{11}\) | \(1011\) | \(\texttt{(} u ~ \texttt{(} v \texttt{))}\) | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 |
| \(f_{12}\) | \(1100\) | \(u\) | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| \(f_{13}\) | \(1101\) | \(\texttt{((} u \texttt{)} ~ v \texttt{)}\) | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 |
| \(f_{14}\) | \(1110\) | \(\texttt{((} u \texttt{)(} v \texttt{))}\) | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 1 |
| \(f_{15}\) | \(1111\) | \(\texttt{((~))}\) | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| \(\begin{matrix}u\!:\\v\!:\end{matrix}\) | \(\begin{matrix}1100\\1010\end{matrix}\) | \(f\) |
\(\texttt{(} \ell_{11} \texttt{)}\) |
\(\texttt{(} \ell_{10} \texttt{)}\) |
\(\texttt{(} \ell_{01} \texttt{)}\) |
\(\texttt{(} \ell_{00} \texttt{)}\) |
\(\ell_{00}\) |
\(\ell_{01}\) |
\(\ell_{10}\) |
\(\ell_{11}\) |
| \(f_{0}\) | \(0000\) | \(\texttt{(~)}\) | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
| \(f_{1}\) | \(0001\) | \(\texttt{(} u \texttt{)(} v \texttt{)}\) | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 |
| \(f_{2}\) | \(0010\) | \(\texttt{(} u\texttt{)} ~ v\) | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 |
| \(f_{4}\) | \(0100\) | \(u ~ \texttt{(} v \texttt{)}\) | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 |
| \(f_{8}\) | \(1000\) | \(u ~ v\) | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 |
| \(f_{3}\) | \(0011\) | \(\texttt{(} u \texttt{)}\) | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 |
| \(f_{12}\) | \(1100\) | \(u\) | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| \(f_{6}\) | \(0110\) | \(\texttt{(} u \texttt{,} v \texttt{)}\) | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 |
| \(f_{9}\) | \(1001\) | \(\texttt{((} u \texttt{,} v \texttt{))}\) | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 |
| \(f_{5}\) | \(0101\) | \(\texttt{(} v \texttt{)}\) | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |
| \(f_{10}\) | \(1010\) | \(v\) | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| \(f_{7}\) | \(0111\) | \(\texttt{(} u ~ v \texttt{)}\) | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 |
| \(f_{11}\) | \(1011\) | \(\texttt{(} u ~ \texttt{(} v \texttt{))}\) | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 |
| \(f_{13}\) | \(1101\) | \(\texttt{((} u \texttt{)} ~ v \texttt{)}\) | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 |
| \(f_{14}\) | \(1110\) | \(\texttt{((} u \texttt{)(} v \texttt{))}\) | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 1 |
| \(f_{15}\) | \(1111\) | \(\texttt{((~))}\) | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| \(\mathrm{Mnemonic}\) | \(\mathrm{Category}\) | \(\mathrm{Classical~Form}\) | \(\mathrm{Alternate~Form}\) | \(\mathrm{Symmetric~Form}\) | \(\mathrm{Operator}\) |
| \(\mathrm{E}\) \(\mathrm{Exclusive}\) |
\(\mathrm{Universal}\) \(\mathrm{Negative}\) |
\(\mathrm{All} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}\) | \(\mathrm{No} ~ u ~ \mathrm{is} ~ v\) | \(\texttt{(} \ell_{11} \texttt{)}\) | |
| \(\mathrm{A}\) \(\mathrm{Absolute}\) |
\(\mathrm{Universal}\) \(\mathrm{Affirmative}\) |
\(\mathrm{All} ~ u ~ \mathrm{is} ~ v\) | \(\mathrm{No} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}\) | \(\texttt{(} \ell_{10} \texttt{)}\) | |
| \(\mathrm{All} ~ v ~ \mathrm{is} ~ u\) | \(\mathrm{No} ~ v ~ \mathrm{is} ~ \texttt{(} u \texttt{)}\) | \(\mathrm{No} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v\) | \(\texttt{(} \ell_{01} \texttt{)}\) | ||
| \(\mathrm{All} ~ \texttt{(} v \texttt{)} ~ \mathrm{is} ~ u\) | \(\mathrm{No} ~ \texttt{(} v \texttt{)} ~ \mathrm{is} ~ \texttt{(} u \texttt{)}\) | \(\mathrm{No} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}\) | \(\texttt{(} \ell_{00} \texttt{)}\) | ||
| \(\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}\) | \(\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}\) | \(\ell_{00}\) | |||
| \(\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v\) | \(\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v\) | \(\ell_{01}\) | |||
| \(\mathrm{O}\) \(\mathrm{Obtrusive}\) |
\(\mathrm{Particular}\) \(\mathrm{Negative}\) |
\(\mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}\) | \(\mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}\) | \(\ell_{10}\) | |
| \(\mathrm{I}\) \(\mathrm{Indefinite}\) |
\(\mathrm{Particular}\) \(\mathrm{Affirmative}\) |
\(\mathrm{Some} ~ u ~ \mathrm{is} ~ v\) | \(\mathrm{Some} ~ u ~ \mathrm{is} ~ v\) | \(\ell_{11}\) |
