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===Extending the Existential Interpretation to Quantificational Logic=== | ===Extending the Existential Interpretation to Quantificational Logic=== | ||
| − | |||
The forms commonly viewed as quantified propositions may be viewed again as propositions about propositions, indeed, there is every reason to regard higher order propositions as the genus of quantification under which the more familiar species appear. | The forms commonly viewed as quantified propositions may be viewed again as propositions about propositions, indeed, there is every reason to regard higher order propositions as the genus of quantification under which the more familiar species appear. | ||
| − | Let us return to the 2-dimensional case <math>X^\circ = | + | Let us return to the 2-dimensional case <math>X^\circ = [u, v]</math>. In order to provide a bridge between propositions and quantifications it serves to define a set of qualifiers <math>\ell_{ij} : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}</math> that have the following characters: |
| − | + | {| align="center" cellpadding="6" | |
| − | + | | | |
| − | + | <math>\begin{array}{*{11}{l}} | |
\ell_{00} f | \ell_{00} f | ||
| − | & = & | + | & = & \ell_{\texttt{(} u \texttt{)(} v \texttt{)}} f |
| − | \ell_{\texttt{(} u \texttt{)(} v \texttt{)}} f | + | & = & \alpha_{1} f |
| − | & = & | + | & = & \Upsilon_{\texttt{(} u \texttt{)(} v \texttt{)}} f |
| − | \alpha_{1} f | + | & = & \Upsilon_{\texttt{(} u \texttt{)(} v \texttt{)} ~ \Rightarrow f} |
| − | & = & | + | & = & f ~ \operatorname{likes} ~ \texttt{(} u \texttt{)(} v \texttt{)} |
| − | \Upsilon_{\texttt{(} u \texttt{)(} v \texttt{)}} f | ||
| − | & = & | ||
| − | \Upsilon_{\texttt{(} u \texttt{)(} v \texttt{)} ~ \Rightarrow f} | ||
| − | & = & | ||
| − | f ~ \operatorname{likes} ~ \texttt{(} u \texttt{)(} v \texttt{)} | ||
\\ | \\ | ||
\ell_{01} f | \ell_{01} f | ||
| − | & = & | + | & = & \ell_{\texttt{(} u \texttt{)} ~ v} f |
| − | \ell_{\texttt{(} u \texttt{)} ~ v} f | + | & = & \alpha_{2} f |
| − | & = & | + | & = & \Upsilon_{\texttt{(} u \texttt{)} ~ v} f |
| − | \alpha_{2} f | + | & = & \Upsilon_{\texttt{(} u \texttt{)} ~ v ~ \Rightarrow f} |
| − | & = & | + | & = & f ~ \operatorname{likes} ~ \texttt{(} u \texttt{)} ~ v |
| − | \Upsilon_{\texttt{(} u \texttt{)} ~ v} f | ||
| − | & = & | ||
| − | \Upsilon_{\texttt{(} u \texttt{)} ~ v ~ \Rightarrow f} | ||
| − | & = & | ||
| − | f ~ \operatorname{likes} ~ \texttt{(} u \texttt{)} ~ v | ||
\\ | \\ | ||
\ell_{10} f | \ell_{10} f | ||
| − | & = & | + | & = & \ell_{u ~ \texttt{(} v \texttt{)}} f |
| − | \ell_{u ~ \texttt{(} v \texttt{)}} f | + | & = & \alpha_{4} f |
| − | & = & | + | & = & \Upsilon_{u ~ \texttt{(} v \texttt{)}} f |
| − | \alpha_{4} f | + | & = & \Upsilon_{u ~ \texttt{(} v \texttt{)} ~ \Rightarrow f} |
| − | & = & | + | & = & f ~ \operatorname{likes} ~ u ~ \texttt{(} v \texttt{)} |
| − | \Upsilon_{u ~ \texttt{(} v \texttt{)}} f | ||
| − | & = & | ||
| − | \Upsilon_{u ~ \texttt{(} v \texttt{)} ~ \Rightarrow f} | ||
| − | & = & | ||
| − | f ~ \operatorname{likes} ~ u ~ \texttt{(} v \texttt{)} | ||
\\ | \\ | ||
\ell_{11} f | \ell_{11} f | ||
| − | & = & | + | & = & \ell_{u ~ v} f |
| − | \ell_{u ~ v} f | + | & = & \alpha_{8} f |
| − | & = & | + | & = & \Upsilon_{u ~ v} f |
| − | \alpha_{8} f | + | & = & \Upsilon_{u ~ v ~ \Rightarrow f} |
| − | & = & | + | & = & f ~ \operatorname{likes} ~ u ~ v |
| − | \Upsilon_{u ~ v} f | + | \end{array}</math> |
| − | & = & | + | |} |
| − | \Upsilon_{u ~ v ~ \Rightarrow f} | ||
| − | & = & | ||
| − | f ~ \operatorname{likes} ~ u ~ v | ||
| − | }</ | ||
| − | |||
| − | Intuitively, the <math>\ell_{ij}<math> operators may be thought of as qualifying propositions according to the elements of the universe of discourse that each proposition positively values. Taken together, these measures provide us with the means to express many useful observations about the propositions in <math>X^\circ = | + | Intuitively, the <math>\ell_{ij}</math> operators may be thought of as qualifying propositions according to the elements of the universe of discourse that each proposition positively values. Taken together, these measures provide us with the means to express many useful observations about the propositions in <math>X^\circ = [u, v],</math> and so they mediate a subtext <math>[\ell_{00}, \ell_{01}, \ell_{10}, \ell_{11}]</math> that takes place within the higher order universe of discourse <math>X^{\circ 2} = [X^\circ] = [[u, v]].</math>. Figure 6 summarizes the action of the <math>\ell_{ij}</math> operators on the <math>f_{i}</math> within <math>X^{\circ 2}.</math> |
| + | <pre> | ||
<div align="center" style="text-align:center"> | <div align="center" style="text-align:center"> | ||
Revision as of 14:16, 22 November 2009
Extending the Existential Interpretation to Quantificational Logic
The forms commonly viewed as quantified propositions may be viewed again as propositions about propositions, indeed, there is every reason to regard higher order propositions as the genus of quantification under which the more familiar species appear.
Let us return to the 2-dimensional case \(X^\circ = [u, v]\). In order to provide a bridge between propositions and quantifications it serves to define a set of qualifiers \(\ell_{ij} : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}\) that have the following characters:
|
\(\begin{array}{*{11}{l}} \ell_{00} f & = & \ell_{\texttt{(} u \texttt{)(} v \texttt{)}} f & = & \alpha_{1} f & = & \Upsilon_{\texttt{(} u \texttt{)(} v \texttt{)}} f & = & \Upsilon_{\texttt{(} u \texttt{)(} v \texttt{)} ~ \Rightarrow f} & = & f ~ \operatorname{likes} ~ \texttt{(} u \texttt{)(} v \texttt{)} \\ \ell_{01} f & = & \ell_{\texttt{(} u \texttt{)} ~ v} f & = & \alpha_{2} f & = & \Upsilon_{\texttt{(} u \texttt{)} ~ v} f & = & \Upsilon_{\texttt{(} u \texttt{)} ~ v ~ \Rightarrow f} & = & f ~ \operatorname{likes} ~ \texttt{(} u \texttt{)} ~ v \\ \ell_{10} f & = & \ell_{u ~ \texttt{(} v \texttt{)}} f & = & \alpha_{4} f & = & \Upsilon_{u ~ \texttt{(} v \texttt{)}} f & = & \Upsilon_{u ~ \texttt{(} v \texttt{)} ~ \Rightarrow f} & = & f ~ \operatorname{likes} ~ u ~ \texttt{(} v \texttt{)} \\ \ell_{11} f & = & \ell_{u ~ v} f & = & \alpha_{8} f & = & \Upsilon_{u ~ v} f & = & \Upsilon_{u ~ v ~ \Rightarrow f} & = & f ~ \operatorname{likes} ~ u ~ v \end{array}\) |
Intuitively, the \(\ell_{ij}\) operators may be thought of as qualifying propositions according to the elements of the universe of discourse that each proposition positively values. Taken together, these measures provide us with the means to express many useful observations about the propositions in \(X^\circ = [u, v],\) and so they mediate a subtext \([\ell_{00}, \ell_{01}, \ell_{10}, \ell_{11}]\) that takes place within the higher order universe of discourse \(X^{\circ 2} = [X^\circ] = [[u, v]].\). Figure 6 summarizes the action of the \(\ell_{ij}\) operators on the \(f_{i}\) within \(X^{\circ 2}.\)
<div align="center" style="text-align:center">
<font size="+2">\(\texttt{Figure 6.} ~~ \texttt{Higher Order Universe of Discourse} ~ \left[ \ell_{00}, \ell_{01}, \ell_{10}, \ell_{11} \right] \subseteq \left[\left[ u, v \right]\right]\)</font>
</div>
Application of Higher Order Propositions to Quantification Theory
Our excursion into the vastening landscape of higher order propositions has finally come round to the stage where we can bring its returns to bear on opening up new perspectives for quantificational logic.
It's hard to tell if it makes any difference from a purely formal point of view, but it serves intuition to devise a slightly different interpretation for the two-valued space that we use as the target of our basic indicator functions. Therefore, let us declare the type of _existential-valued functions_ \(f : \mathbb{B}^k \to \mathbb{E}<math>, where <math>\mathbb{E} = \{ -e, +e \} = \{ \operatorname{empty}, \operatorname{exist} \}<math> is a pair of values that indicate whether or not anything exists in the cells of the underlying universe of discourse. As usual, let's not be too fussy about the coding of these functions, reverting to binary codes whenever the intended interpretation is clear enough.
With these qualifications in mind we note the following correspondences between classical quantifications and higher order indicator functions:
<font size="+1">
<table align="center" cellpadding="10" cellspacing="0" width="80%">
<caption><font size="+2"><math>\texttt{Table 7.} ~~ \texttt{Syllogistic Premisses as Higher Order Indicator Functions}\)</font></caption>
<tr>
<td align="center">\(\operatorname{A}\)</td>
<td>\(Absolute\)</td>
<td>\(Universal Affirmative\)</td>
<td align="center">\(All ~ u ~ is ~ v\)</td>
<td>\(Indicator of u ~ \texttt{(} v \texttt{)} = 0\)</td></tr>
<tr>
<td align="center">\(\operatorname{E}\)</td>
<td>\(Exclusive\)</td>
<td>\(Universal Negative\)</td>
<td align="center">\(All ~ u ~ is ~ \texttt{(} v \texttt{)}\)</td>
<td>\(Indicator of ~ u ~ \cdot ~ v = 0\)</td></tr>
<tr>
<td align="center">\(\operatorname{I}\)</td>
<td>\(Indefinite\)</td>
<td>\(Particular Affirmative\)</td>
<td align="center">\(Some ~ u ~ is ~ v\)</td>
<td>\(Indicator of ~ u ~ \cdot ~ v = 1\)</td></tr>
<tr>
<td align="center">\(\operatorname{O}\)</td>
<td>\(Obtrusive\)</td>
<td>\(Particular Negative\)</td>
<td align="center">\(Some ~ u ~ is ~ \texttt{(} v \texttt{)}\)</td>
<td>\(Indicator of ~ u ~ \texttt{(} v \texttt{)} = 1\)</td></tr>
</table></font>
The following Tables develop these ideas in more detail.
<table align="center" cellpadding="1" cellspacing="0" style="text-align:center; width:90%">
<caption><font size="+2">\(\texttt{Table 8.} ~~ \texttt{Simple Qualifiers of Propositions (Version 1)}\)</font></caption>
<tr>
<td width="4%" style="border-bottom:2px solid black" align="right">
\(u:\)<br>
\(v:\)</td>
<td width="6%" style="border-bottom:2px solid black">
\(1100\)<br>
\(1010\)</td>
<td width="10%" style="border-bottom:2px solid black; border-right:2px solid black">
\(f\)</td>
<td width="10%" style="border-bottom:2px solid black">
\(\texttt{(} \ell_{11} \texttt{)}\)<br>
\(No ~ u\)<br>
\(is ~ v\)</td>
<td width="10%" style="border-bottom:2px solid black">
\(\texttt{(} \ell_{10} \texttt{)}\)<br>
\(No ~ u\)<br>
\(is ~ \texttt{(} v \texttt{)}\)</td>
<td width="10%" style="border-bottom:2px solid black">
\(\texttt{(} \ell_{01} \texttt{)}\)<br>
\(No ~ \texttt{(} u \texttt{)}\)<br>
\(is ~ v\)</td>
<td width="10%" style="border-bottom:2px solid black">
\(\texttt{(} \ell_{00} \texttt{)}\)<br>
\(No ~ \texttt{(} u \texttt{)}\)<br>
\(is ~ \texttt{(} v \texttt{)}\)</td>
<td width="10%" style="border-bottom:2px solid black">
\(\ell_{00}\)<br>
\(Some ~ \texttt{(} u \texttt{)}\)<br>
\(is ~ \texttt{(} v \texttt{)}\)</td>
<td width="10%" style="border-bottom:2px solid black">
\(\ell_{01}\)<br>
\(Some ~ \texttt{(} u \texttt{)}\)<br>
\(is ~ v\)</td>
<td width="10%" style="border-bottom:2px solid black">
\(\ell_{10}\)<br>
\(Some ~ u\)<br>
\(is ~ \texttt{(} v \texttt{)}\)</td>
<td width="10%" style="border-bottom:2px solid black">
\(\ell_{11}\)<br>
\(Some ~ u\)<br>
\(is ~ v\)</td></tr>
<tr>
<td>\(f_{0}\)</td>
<td>\(0000\)</td>
<td style="border-right:2px solid black">\(\texttt{(~)}\)</td>
<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>\(f_{1}\)</td>
<td>\(0001\)</td>
<td style="border-right:2px solid black">\(\texttt{(} u \texttt{)(} v \texttt{)}\)</td>
<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>\(f_{2}\)</td>
<td>\(0010\)</td>
<td style="border-right:2px solid black">\(\texttt{(} u\texttt{)} ~ v\)</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>\(f_{3}\)</td>
<td>\(0011\)</td>
<td style="border-right:2px solid black">\(\texttt{(} u \texttt{)}\)</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>\(f_{4}\)</td>
<td>\(0100\)</td>
<td style="border-right:2px solid black">\(u ~ \texttt{(} v \texttt{)}\)</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>\(f_{5}\)</td>
<td>\(0101\)</td>
<td style="border-right:2px solid black">\(\texttt{(} v \texttt{)}\)</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>\(f_{6}\)</td>
<td>\(0110\)</td>
<td style="border-right:2px solid black">\(\texttt{(} u \texttt{,} v \texttt{)}\)</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>\(f_{7}\)</td>
<td>\(0111\)</td>
<td style="border-right:2px solid black">\(\texttt{(} u ~ v \texttt{)}\)</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>\(f_{8}\)</td>
<td>\(1000\)</td>
<td style="border-right:2px solid black">\(u ~ v\)</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>\(f_{9}\)</td>
<td>\(1001\)</td>
<td style="border-right:2px solid black">\(\texttt{((} u \texttt{,} v \texttt{))}\)</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>\(f_{10}\)</td>
<td>\(1010\)</td>
<td style="border-right:2px solid black">\(v\)</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>\(f_{11}\)</td>
<td>\(1011\)</td>
<td style="border-right:2px solid black">\(\texttt{(} u ~ \texttt{(} v \texttt{))}\)</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>\(f_{12}\)</td>
<td>\(1100\)</td>
<td style="border-right:2px solid black">\(u\)</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>\(f_{13}\)</td>
<td>\(1101\)</td>
<td style="border-right:2px solid black">\(\texttt{((} u \texttt{)} ~ v \texttt{)}\)</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>\(f_{14}\)</td>
<td>\(1110\)</td>
<td style="border-right:2px solid black">\(\texttt{((} u \texttt{)(} v \texttt{))}\)</td>
<td style="background: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>\(f_{15}\)</td>
<td>\(1111\)</td>
<td style="border-right:2px solid black">\(\texttt{((~))}\)</td>
<td style="background:white; color:black">0</td>
<td style="background: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="1" cellspacing="0" style="text-align:center; width:90%">
<caption><font size="+2">\(\texttt{Table 9.} ~~ \texttt{Simple Qualifiers of Propositions (Version 2)}\)</font></caption>
<tr>
<td width="4%" style="border-bottom:2px solid black" align="right">
\(u:\)<br>
\(v:\)</td>
<td width="6%" style="border-bottom:2px solid black">
\(1100\)<br>
\(1010\)</td>
<td width="10%" style="border-bottom:2px solid black; border-right:2px solid black">
\(f\)</td>
<td width="10%" style="border-bottom:2px solid black">
\(\texttt{(} \ell_{11} \texttt{)}\)<br>
\(No ~ u\)<br>
\(is ~ v\)</td>
<td width="10%" style="border-bottom:2px solid black">
\(\texttt{(} \ell_{10} \texttt{)}\)<br>
\(No ~ u\)<br>
\(is ~ \texttt{(} v \texttt{)}\)</td>
<td width="10%" style="border-bottom:2px solid black">
\(\texttt{(} \ell_{01} \texttt{)}\)<br>
\(No ~ \texttt{(} u \texttt{)}\)<br>
\(is ~ v\)</td>
<td width="10%" style="border-bottom:2px solid black">
\(\texttt{(} \ell_{00} \texttt{)}\)<br>
\(No ~ \texttt{(} u \texttt{)}\)<br>
\(is ~ \texttt{(} v \texttt{)}\)</td>
<td width="10%" style="border-bottom:2px solid black">
\(\ell_{00}\)<br>
\(Some ~ \texttt{(} u \texttt{)}\)<br>
\(is ~ \texttt{(} v \texttt{)}\)</td>
<td width="10%" style="border-bottom:2px solid black">
\(\ell_{01}\)<br>
\(Some ~ \texttt{(} u \texttt{)}\)<br>
\(is ~ v\)</td>
<td width="10%" style="border-bottom:2px solid black">
\(\ell_{10}\)<br>
\(Some ~ u\)<br>
\(is ~ \texttt{(} v \texttt{)}\)</td>
<td width="10%" style="border-bottom:2px solid black">
\(\ell_{11}\)<br>
\(Some ~ u\)<br>
\(is ~ v\)</td></tr>
<tr>
<td style="border-bottom:2px solid black">\(f_{0}\)</td>
<td style="border-bottom:2px solid black">\(0000\)</td>
<td style="border-bottom:2px solid black; border-right:2px solid black">\(\texttt{(~)}\)</td>
<td style="border-bottom:2px solid black; background:black; color:white">1</td>
<td style="border-bottom:2px solid black; background:black; color:white">1</td>
<td style="border-bottom:2px solid black; background:black; color:white">1</td>
<td style="border-bottom:2px solid black; background:black; color:white">1</td>
<td style="border-bottom:2px solid black; background:white; color:black">0</td>
<td style="border-bottom:2px solid black; background:white; color:black">0</td>
<td style="border-bottom:2px solid black; background:white; color:black">0</td>
<td style="border-bottom:2px solid black; background:white; color:black">0</td></tr>
<tr>
<td>\(f_{1}\)</td>
<td>\(0001\)</td>
<td style="border-right:2px solid black">\(\texttt{(} u \texttt{)(} v \texttt{)}\)</td>
<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>\(f_{2}\)</td>
<td>\(0010\)</td>
<td style="border-right:2px solid black">\(\texttt{(} u\texttt{)} ~ v\)</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>\(f_{4}\)</td>
<td>\(0100\)</td>
<td style="border-right:2px solid black">\(u ~ \texttt{(} v \texttt{)}\)</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:2px solid black">\(f_{8}\)</td>
<td style="border-bottom:2px solid black">\(1000\)</td>
<td style="border-bottom:2px solid black; border-right:2px solid black">\(u ~ v\)</td>
<td style="border-bottom:2px solid black; background:white; color:black">0</td>
<td style="border-bottom:2px solid black; background:black; color:white">1</td>
<td style="border-bottom:2px solid black; background:black; color:white">1</td>
<td style="border-bottom:2px solid black; background:black; color:white">1</td>
<td style="border-bottom:2px solid black; background:white; color:black">0</td>
<td style="border-bottom:2px solid black; background:white; color:black">0</td>
<td style="border-bottom:2px solid black; background:white; color:black">0</td>
<td style="border-bottom:2px solid black; background:black; color:white">1</td></tr>
<tr>
<td>\(f_{3}\)</td>
<td>\(0011\)</td>
<td style="border-right:2px solid black">\(\texttt{(} u \texttt{)}\)</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:2px solid black">\(f_{12}\)</td>
<td style="border-bottom:2px solid black">\(1100\)</td>
<td style="border-bottom:2px solid black; border-right:2px solid black">\(u\)</td>
<td style="border-bottom:2px solid black; background:white; color:black">0</td>
<td style="border-bottom:2px solid black; background:white; color:black">0</td>
<td style="border-bottom:2px solid black; background:black; color:white">1</td>
<td style="border-bottom:2px solid black; background:black; color:white">1</td>
<td style="border-bottom:2px solid black; background:white; color:black">0</td>
<td style="border-bottom:2px solid black; background:white; color:black">0</td>
<td style="border-bottom:2px solid black; background:black; color:white">1</td>
<td style="border-bottom:2px solid black; background:black; color:white">1</td></tr>
<tr>
<td>\(f_{6}\)</td>
<td>\(0110\)</td>
<td style="border-right:2px solid black">\(\texttt{(} u \texttt{,} v \texttt{)}\)</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:2px solid black">\(f_{9}\)</td>
<td style="border-bottom:2px solid black">\(1001\)</td>
<td style="border-bottom:2px solid black; border-right:2px solid black">\(\texttt{((} u \texttt{,} v \texttt{))}\)</td>
<td style="border-bottom:2px solid black; background:white; color:black">0</td>
<td style="border-bottom:2px solid black; background:black; color:white">1</td>
<td style="border-bottom:2px solid black; background:black; color:white">1</td>
<td style="border-bottom:2px solid black; background:white; color:black">0</td>
<td style="border-bottom:2px solid black; background:black; color:white">1</td>
<td style="border-bottom:2px solid black; background:white; color:black">0</td>
<td style="border-bottom:2px solid black; background:white; color:black">0</td>
<td style="border-bottom:2px solid black; background:black; color:white">1</td></tr>
<tr>
<td>\(f_{5}\)</td>
<td>\(0101\)</td>
<td style="border-right:2px solid black">\(\texttt{(} v \texttt{)}\)</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:2px solid black">\(f_{10}\)</td>
<td style="border-bottom:2px solid black">\(1010\)</td>
<td style="border-bottom:2px solid black; border-right:2px solid black">\(v\)</td>
<td style="border-bottom:2px solid black; background:white; color:black">0</td>
<td style="border-bottom:2px solid black; background:black; color:white">1</td>
<td style="border-bottom:2px solid black; background:white; color:black">0</td>
<td style="border-bottom:2px solid black; background:black; color:white">1</td>
<td style="border-bottom:2px solid black; background:white; color:black">0</td>
<td style="border-bottom:2px solid black; background:black; color:white">1</td>
<td style="border-bottom:2px solid black; background:white; color:black">0</td>
<td style="border-bottom:2px solid black; background:black; color:white">1</td></tr>
<tr>
<td>\(f_{7}\)</td>
<td>\(0111\)</td>
<td style="border-right:2px solid black">\(\texttt{(} u ~ v \texttt{)}\)</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>\(f_{11}\)</td>
<td>\(1011\)</td>
<td style="border-right:2px solid black">\(\texttt{(} u ~ \texttt{(} v \texttt{))}\)</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>\(f_{13}\)</td>
<td>\(1101\)</td>
<td style="border-right:2px solid black">\(\texttt{((} u \texttt{)} ~ v \texttt{)}\)</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:2px solid black">\(f_{14}\)</td>
<td style="border-bottom:2px solid black">\(1110\)</td>
<td style="border-bottom:2px solid black; border-right:2px solid black">\(\texttt{((} u \texttt{)(} v \texttt{))}\)</td>
<td style="border-bottom:2px solid black; background:white; color:black">0</td>
<td style="border-bottom:2px solid black; background:white; color:black">0</td>
<td style="border-bottom:2px solid black; background:white; color:black">0</td>
<td style="border-bottom:2px solid black; background:black; color:white">1</td>
<td style="border-bottom:2px solid black; background:white; color:black">0</td>
<td style="border-bottom:2px solid black; background:black; color:white">1</td>
<td style="border-bottom:2px solid black; background:black; color:white">1</td>
<td style="border-bottom:2px solid black; background:black; color:white">1</td></tr>
<tr>
<td>\(f_{15}\)</td>
<td>\(1111\)</td>
<td style="border-right:2px solid black">\(\texttt{((~))}\)</td>
<td style="background:white; color:black">0</td>
<td style="background: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>
<font size="+1">
<table align="center" cellpadding="4" cellspacing="0" style="text-align:center; width:90%">
<caption><font size="+2">\(\texttt{Table 10.} ~~ \texttt{Relation of Quantifiers to Higher Order Propositions}\)</font></caption>
<tr>
<td style="border-bottom:2px solid black">\(\texttt{Mnemonic}\)</td>
<td style="border-bottom:2px solid black">\(\texttt{Category}\)</td>
<td style="border-bottom:2px solid black">\(\texttt{Classical Form}\)</td>
<td style="border-bottom:2px solid black">\(\texttt{Alternate Form}\)</td>
<td style="border-bottom:2px solid black">\(\texttt{Symmetric Form}\)</td>
<td style="border-bottom:2px solid black">\(\texttt{Operator}\)</td></tr>
<tr>
<td>\(\texttt{E}\)<br>\(\texttt{Exclusive}\)</td>
<td>\(\texttt{Universal}\)<br>\(\texttt{Negative}\)</td>
<td>\(\texttt{All} ~ u ~ \texttt{is} ~ \texttt{(} v \texttt{)}\)</td>
<td> </td>
<td>\(\texttt{No} ~ u ~ \texttt{is} ~ v\)</td>
<td>\(\texttt{(} \ell_{11} \texttt{)}\)</td></tr>
<tr>
<td>\(\texttt{A}\)<br>\(\texttt{Absolute}\)</td>
<td>\(\texttt{Universal}\)<br>\(\texttt{Affirmative}\)</td>
<td>\(\texttt{All} ~ u ~ \texttt{is} ~ v\)</td>
<td> </td>
<td>\(\texttt{No} ~ u ~ \texttt{is} ~ \texttt{(} v \texttt{)}\)</td>
<td>\(\texttt{(} \ell_{10} \texttt{)}\)</td></tr>
<tr>
<td> </td>
<td> </td>
<td>\(\texttt{All} ~ v ~ \texttt{is} ~ u\)</td>
<td>\(\texttt{No} ~ v ~ \texttt{is} ~ \texttt{(} u \texttt{)}\)</td>
<td>\(\texttt{No} ~ \texttt{(} u \texttt{)} ~ \texttt{is} ~ v\)</td>
<td>\(\texttt{(} \ell_{01} \texttt{)}\)</td></tr>
<tr>
<td style="border-bottom:2px solid black"> </td>
<td style="border-bottom:2px solid black"> </td>
<td style="border-bottom:2px solid black">\(\texttt{All} ~ \texttt{(} v \texttt{)} ~ \texttt{is} ~ u\)</td>
<td style="border-bottom:2px solid black">\(\texttt{No} ~ \texttt{(} v \texttt{)} ~ \texttt{is} ~ \texttt{(} u \texttt{)}\)</td>
<td style="border-bottom:2px solid black">\(\texttt{No} ~ \texttt{(} u \texttt{)} ~ \texttt{is} ~ \texttt{(} v \texttt{)}\)</td>
<td style="border-bottom:2px solid black">\(\texttt{(} \ell_{00} \texttt{)}\)</td></tr>
<tr>
<td> </td>
<td> </td>
<td>\(\texttt{Some} ~ \texttt{(} u \texttt{)} ~ \texttt{is} ~ \texttt{(} v \texttt{)}\)</td>
<td> </td>
<td>\(\texttt{Some} ~ \texttt{(} u \texttt{)} ~ \texttt{is} ~ \texttt{(} v \texttt{)}\)</td>
<td>\(\ell_{00}\)</td></tr>
<tr>
<td> </td>
<td> </td>
<td>\(\texttt{Some} ~ \texttt{(} u \texttt{)} ~ \texttt{is} ~ v\)</td>
<td> </td>
<td>\(\texttt{Some} ~ \texttt{(} u \texttt{)} ~ \texttt{is} ~ v\)</td>
<td>\(\ell_{01}\)</td></tr>
<tr>
<td>\(\texttt{O}\)<br>\(\texttt{Obtrusive}\)</td>
<td>\(\texttt{Particular}\)<br>\(\texttt{Negative}\)</td>
<td>\(\texttt{Some} ~ u ~ \texttt{is} ~ \texttt{(} v \texttt{)}\)</td>
<td> </td>
<td>\(\texttt{Some} ~ u ~ \texttt{is} ~ \texttt{(} v \texttt{)}\)</td>
<td>\(\ell_{10}\)</td></tr>
<tr>
<td>\(\texttt{I}\)<br>\(\texttt{Indefinite}\)</td>
<td>\(\texttt{Particular}\)<br>\(\texttt{Affirmative}\)</td>
<td>\(\texttt{Some} ~ u ~ \texttt{is} ~ v\)</td>
<td> </td>
<td>\(\texttt{Some} ~ u ~ \texttt{is} ~ v\)</td>
<td>\(\ell_{11}\)</td></tr>
</table></font>
