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Revision as of 05:00, 24 November 2009
, 05:00, 24 November 2009→Higher Order Propositions: add new table set
==Higher Order Propositions==
==Higher Order Propositions==
<br>
<br>
<td><math>f_{0}</math></td>
<td><math>f_{0}</math></td>
<td><math>0 ~ 0</math></td>
<td><math>0 ~ 0</math></td>
<td style="border-right:2px solid black"><math>\texttt{( )}</math></td>
<td style="border-right:2px solid black"><math>\texttt{(~)}</math></td>
<td style="background:white; color:black"><math>0</math></td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white"><math>1</math></td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black"><math>0</math></td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white"><math>1</math></td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black"><math>0</math></td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white"><math>1</math></td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black"><math>0</math></td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white"><math>1</math></td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black"><math>0</math></td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white"><math>1</math></td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black"><math>0</math></td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white"><math>1</math></td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black"><math>0</math></td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white"><math>1</math></td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black"><math>0</math></td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white"><math>1</math></td></tr>
<td style="background:black; color:white">1</td></tr>
<tr>
<tr>
<td><math>0 ~ 1</math></td>
<td><math>0 ~ 1</math></td>
<td style="border-right:2px solid black"><math>\texttt{(} x \texttt{)}</math></td>
<td style="border-right:2px solid black"><math>\texttt{(} x \texttt{)}</math></td>
<td style="background:white; color:black"><math>0</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black"><math>0</math></td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white"><math>1</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white"><math>1</math></td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black"><math>0</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black"><math>0</math></td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white"><math>1</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white"><math>1</math></td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black"><math>0</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black"><math>0</math></td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white"><math>1</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white"><math>1</math></td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black"><math>0</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black"><math>0</math></td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white"><math>1</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white"><math>1</math></td></tr>
<td style="background:black; color:white">1</td></tr>
<tr>
<tr>
<td><math>1 ~ 0</math></td>
<td><math>1 ~ 0</math></td>
<td style="border-right:2px solid black"><math>x</math></td>
<td style="border-right:2px solid black"><math>x</math></td>
<td style="background:white; color:black"><math>0</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black"><math>0</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black"><math>0</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black"><math>0</math></td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white"><math>1</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white"><math>1</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white"><math>1</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white"><math>1</math></td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black"><math>0</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black"><math>0</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black"><math>0</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black"><math>0</math></td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white"><math>1</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white"><math>1</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white"><math>1</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white"><math>1</math></td></tr>
<td style="background:black; color:white">1</td></tr>
<tr>
<tr>
<td><math>f_{3}</math></td>
<td><math>f_{3}</math></td>
<td><math>1 ~ 1</math></td>
<td><math>1 ~ 1</math></td>
<td style="border-right:2px solid black"><math>\texttt{(( ))}</math></td>
<td style="border-right:2px solid black"><math>\texttt{((~))}</math></td>
<td style="background:white; color:black"><math>0</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black"><math>0</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black"><math>0</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black"><math>0</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black"><math>0</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black"><math>0</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black"><math>0</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black"><math>0</math></td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white"><math>1</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white"><math>1</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white"><math>1</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white"><math>1</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white"><math>1</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white"><math>1</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white"><math>1</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white"><math>1</math></td></tr>
<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>
</table>
