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Directory:Jon Awbrey/Papers/Functional Logic : Higher Order Propositions (view source)

Revision as of 03:10, 23 November 2009

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With this interpretation in mind we note the following correspondences between classical quantifications and higher order indicator functions:
 
With this interpretation in mind we note the following correspondences between classical quantifications and higher order indicator functions:
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<br>
    
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
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\end{array}</math>
 
\end{array}</math>
 
|}
 
|}
 +
 +
<br>
 +
 +
The following Tables develop these ideas in more detail.
 +
 +
<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{matrix}
 +
\texttt{(} \ell_{11} \texttt{)}
 +
\\
 +
\mathrm{No} ~ u
 +
\\
 +
\mathrm{is} ~ v
 +
\end{matrix}</math></td>
 +
<td width="10%" style="border-bottom:1px solid black">
 +
<math>\begin{matrix}
 +
\texttt{(} \ell_{10} \texttt{)}
 +
\\
 +
\mathrm{No} ~ u
 +
\\
 +
\mathrm{is} ~ \texttt{(} v \texttt{)}
 +
\end{matrix}</math></td>
 +
<td width="10%" style="border-bottom:1px solid black">
 +
<math>\begin{matrix}
 +
\texttt{(} \ell_{01} \texttt{)}
 +
\\
 +
\mathrm{No} ~ \texttt{(} u \texttt{)}
 +
\\
 +
\mathrm{is} ~ v
 +
\end{matrix}</math></td>
 +
<td width="10%" style="border-bottom:1px solid black">
 +
<math>\begin{matrix}
 +
\texttt{(} \ell_{00} \texttt{)}
 +
\\
 +
\mathrm{No} ~ \texttt{(} u \texttt{)}
 +
\\
 +
\mathrm{is} ~ \texttt{(} v \texttt{)}
 +
\end{matrix}</math></td>
 +
<td width="10%" style="border-bottom:1px solid black">
 +
<math>\begin{matrix}
 +
\ell_{00}
 +
\\
 +
\mathrm{Some} ~ \texttt{(} u \texttt{)}
 +
\\
 +
\mathrm{is} ~ \texttt{(} v \texttt{)}
 +
\end{matrix}</math></td>
 +
<td width="10%" style="border-bottom:1px solid black">
 +
<math>\begin{matrix}
 +
\ell_{01}
 +
\\
 +
\mathrm{Some} ~ \texttt{(} u \texttt{)}
 +
\\
 +
\mathrm{is} ~ v
 +
\end{matrix}</math></td>
 +
<td width="10%" style="border-bottom:1px solid black">
 +
<math>\begin{matrix}
 +
\ell_{10}
 +
\\
 +
\mathrm{Some} ~ u
 +
\\
 +
\mathrm{is} ~ \texttt{(} v \texttt{)}
 +
\end{matrix}</math></td>
 +
<td width="10%" style="border-bottom:1px solid black">
 +
<math>\begin{matrix}
 +
\ell_{11}
 +
\\
 +
\mathrm{Some} ~ u
 +
\\
 +
\mathrm{is} ~ v
 +
\end{matrix}</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{matrix}
 +
\texttt{(} \ell_{11} \texttt{)}
 +
\\
 +
\mathrm{No} ~ u
 +
\\
 +
\mathrm{is} ~ v
 +
\end{matrix}</math></td>
 +
<td width="10%" style="border-bottom:1px solid black">
 +
<math>\begin{matrix}
 +
\texttt{(} \ell_{10} \texttt{)}
 +
\\
 +
\mathrm{No} ~ u
 +
\\
 +
\mathrm{is} ~ \texttt{(} v \texttt{)}
 +
\end{matrix}</math></td>
 +
<td width="10%" style="border-bottom:1px solid black">
 +
<math>\begin{matrix}
 +
\texttt{(} \ell_{01} \texttt{)}
 +
\\
 +
\mathrm{No} ~ \texttt{(} u \texttt{)}
 +
\\
 +
\mathrm{is} ~ v
 +
\end{matrix}</math></td>
 +
<td width="10%" style="border-bottom:1px solid black">
 +
<math>\begin{matrix}
 +
\texttt{(} \ell_{00} \texttt{)}
 +
\\
 +
\mathrm{No} ~ \texttt{(} u \texttt{)}
 +
\\
 +
\mathrm{is} ~ \texttt{(} v \texttt{)}
 +
\end{matrix}</math></td>
 +
<td width="10%" style="border-bottom:1px solid black">
 +
<math>\begin{matrix}
 +
\ell_{00}
 +
\\
 +
\mathrm{Some} ~ \texttt{(} u \texttt{)}
 +
\\
 +
\mathrm{is} ~ \texttt{(} v \texttt{)}
 +
\end{matrix}</math></td>
 +
<td width="10%" style="border-bottom:1px solid black">
 +
<math>\begin{matrix}
 +
\ell_{01}
 +
\\
 +
\mathrm{Some} ~ \texttt{(} u \texttt{)}
 +
\\
 +
\mathrm{is} ~ v
 +
\end{matrix}</math></td>
 +
<td width="10%" style="border-bottom:1px solid black">
 +
<math>\begin{matrix}
 +
\ell_{10}
 +
\\
 +
\mathrm{Some} ~ u
 +
\\
 +
\mathrm{is} ~ \texttt{(} v \texttt{)}
 +
\end{matrix}</math></td>
 +
<td width="10%" style="border-bottom:1px solid black">
 +
<math>\begin{matrix}
 +
\ell_{11}
 +
\\
 +
\mathrm{Some} ~ u
 +
\\
 +
\mathrm{is} ~ v
 +
\end{matrix}</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>\mathrm{E}</math><br><math>\mathrm{Exclusive}</math></td>
 +
<td><math>\mathrm{Universal}</math><br><math>\mathrm{Negative}</math></td>
 +
<td><math>\mathrm{All} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
 +
<td>&nbsp;</td>
 +
<td><math>\mathrm{No} ~ u ~ \mathrm{is} ~ v</math></td>
 +
<td><math>\texttt{(} \ell_{11} \texttt{)}</math></td></tr>
 +
 +
<tr>
 +
<td><math>\mathrm{A}</math><br><math>\mathrm{Absolute}</math></td>
 +
<td><math>\mathrm{Universal}</math><br><math>\mathrm{Affirmative}</math></td>
 +
<td><math>\mathrm{All} ~ u ~ \mathrm{is} ~ v</math></td>
 +
<td>&nbsp;</td>
 +
<td><math>\mathrm{No} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
 +
<td><math>\texttt{(} \ell_{10} \texttt{)}</math></td></tr>
 +
 +
<tr>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</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">&nbsp;</td>
 +
<td style="border-bottom:1px solid black">&nbsp;</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>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
 +
<td>&nbsp;</td>
 +
<td><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
 +
<td><math>\ell_{00}</math></td></tr>
 +
 +
<tr>
 +
<td>&nbsp;</td>
 +
<td>&nbsp;</td>
 +
<td><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td>
 +
<td>&nbsp;</td>
 +
<td><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td>
 +
<td><math>\ell_{01}</math></td></tr>
 +
 +
<tr>
 +
<td><math>\mathrm{O}</math><br><math>\mathrm{Obtrusive}</math></td>
 +
<td><math>\mathrm{Particular}</math><br><math>\mathrm{Negative}</math></td>
 +
<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
 +
<td>&nbsp;</td>
 +
<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
 +
<td><math>\ell_{10}</math></td></tr>
 +
 +
<tr>
 +
<td><math>\mathrm{I}</math><br><math>\mathrm{Indefinite}</math></td>
 +
<td><math>\mathrm{Particular}</math><br><math>\mathrm{Affirmative}</math></td>
 +
<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ v</math></td>
 +
<td>&nbsp;</td>
 +
<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ v</math></td>
 +
<td><math>\ell_{11}</math></td></tr>
 +
 +
</table>
 +
 +
<br>
    
==References==
 
==References==
Jon Awbrey
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