User:Jon Awbrey/TEST
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Jump to navigationJump to searchExtending 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 \(X^\circ = \left[ u, v \right]<math>. In order to provide a bridge between propositions and quantifications it serves to define a set of qualifiers <math>\ell_{ij} : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}<math> that have the following characters:
<div markdown="1"><font size="+1">
\)\(\array{
\arrayopts{\colalign{left}}
\ell_{00} f
& = &
\ell_{\texttt{(} u \texttt{)(} v \texttt{)}} f
& = &
\alpha_{1} f
& = &
\Upsilon_{\texttt{(} u \texttt{)(} v \texttt{)}} f
& = &
\Upsilon_{\texttt{(} u \texttt{)(} v \texttt{)} ~ \Rightarrow f}
& = &
f ~ \operatorname{likes} ~ \texttt{(} u \texttt{)(} v \texttt{)}
\\
\ell_{01} f
& = &
\ell_{\texttt{(} u \texttt{)} ~ v} f
& = &
\alpha_{2} f
& = &
\Upsilon_{\texttt{(} u \texttt{)} ~ v} f
& = &
\Upsilon_{\texttt{(} u \texttt{)} ~ v ~ \Rightarrow f}
& = &
f ~ \operatorname{likes} ~ \texttt{(} u \texttt{)} ~ v
\\
\ell_{10} f
& = &
\ell_{u ~ \texttt{(} v \texttt{)}} f
& = &
\alpha_{4} f
& = &
\Upsilon_{u ~ \texttt{(} v \texttt{)}} f
& = &
\Upsilon_{u ~ \texttt{(} v \texttt{)} ~ \Rightarrow f}
& = &
f ~ \operatorname{likes} ~ u ~ \texttt{(} v \texttt{)}
\\
\ell_{11} f
& = &
\ell_{u ~ v} f
& = &
\alpha_{8} f
& = &
\Upsilon_{u ~ v} f
& = &
\Upsilon_{u ~ v ~ \Rightarrow f}
& = &
f ~ \operatorname{likes} ~ u ~ v
}\)\(
</font></div>
Intuitively, the <math>\ell_{ij}<math> operators may be thought of as qualifying propositions according to the elements of the universe of discourse that each proposition positively values. Taken together, these measures provide us with the means to express many useful observations about the propositions in <math>X^\circ = \left[ u, v \right]<math>, and so they mediate a subtext <math>\left[ \ell_{00}, \ell_{01}, \ell_{10}, \ell_{11} \right]<math> that takes place within the higher order universe of discourse <math>X^{\circ 2} = \left[ X^\circ \right] = \left[\left[ u, v \right]\right]<math>. Figure 6 summarizes the action of the <math>\ell_{ij}<math> operators on the <math>f_{i}<math> within <math>X^{\circ 2}<math>.
<div align="center" style="text-align:center">
<font size="+2"><math>\texttt{Figure 6.} ~~ \texttt{Higher Order Universe of Discourse} ~ \left[ \ell_{00}, \ell_{01}, \ell_{10}, \ell_{11} \right] \subseteq \left[\left[ u, v \right]\right]\)</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>
References
- Quine, W.V. (1969/1981), "On the Limits of Decision", Akten des XIV. Internationalen Kongresses für Philosophie, vol. 3 (1969). Reprinted, pp. 156–163 in Quine (ed., 1981), Theories and Things, Harvard University Press, Cambridge, MA.
Related Topics
Appendix : Generalized Umpire Operators
In order to get a handle on the space of higher order propositions and eventually to carry out a functional approach to quantification theory, it serves to construct some specialized tools. Specifically, I define a higher order operator \(\Upsilon,\) called the umpire operator, which takes up to three propositions as arguments and returns a single truth value as the result. Formally, this so-called multigrade property of \(\Upsilon\) can be expressed as a union of function types, in the following manner:
| \(\Upsilon : \bigcup_{\ell = 1, 2, 3} ((\mathbb{B}^k \to \mathbb{B})^\ell \to \mathbb{B}).\) |
In contexts of application the intended sense can be discerned by the number of arguments that actually appear in the argument list. Often, the first and last arguments appear as indices, the one in the middle being treated as the main argument while the other two arguments serve to modify the sense of the operation in question. Thus, we have the following forms:
| \(\Upsilon_p^r q = \Upsilon (p, q, r)\) |
| \(\Upsilon_p^r : (\mathbb{B}^k \to \mathbb{B}) \to \mathbb{B}\) |
The intention of this operator is that we evaluate the proposition \(q\) on each model of the proposition \(p\) and combine the results according to the method indicated by the connective parameter \(r.\) In principle, the index \(r\) might specify any connective on as many as \(2^k\) arguments, but usually we have in mind a much simpler form of combination, most often either collective products or collective sums. By convention, each of the accessory indices \(p, r\) is assigned a default value that is understood to be in force when the corresponding argument place is left blank, specifically, the constant proposition \(1 : \mathbb{B}^k \to \mathbb{B}\) for the lower index \(p,\) and the continued conjunction or continued product operation \(\textstyle\prod\) for the upper index \(r.\) Taking the upper default value gives license to the following readings:
| \(\Upsilon_p (q) = \Upsilon (p, q) = \Upsilon (p, q, \textstyle\prod).\) |
| \(\Upsilon_p = \Upsilon (p, \underline{~~}, \textstyle\prod) : (\mathbb{B}^k \to \mathbb{B}) \to \mathbb{B}.\) |
This means that \(\Upsilon_p (q) = 1\) if and only if \(q\) holds for all models of \(p.\) In propositional terms, this is tantamount to the assertion that \(p \Rightarrow q,\) or that \(\texttt{(} p \texttt{(} q \texttt{))} = 1.\)
Throwing in the lower default value permits the following abbreviations:
| \(\Upsilon q = \Upsilon (q) = \Upsilon_1 (q) = \Upsilon (1, q, \textstyle\prod).\) |
| \(\Upsilon = \Upsilon (1, \underline{~~}, \textstyle\prod)) : (\mathbb{B}^k\ \to \mathbb{B}) \to \mathbb{B}.\) |
This means that \(\Upsilon q = 1\) if and only if \(q\) holds for the whole universe of discourse in question, that is, if and only \(q\) is the constantly true proposition \(1 : \mathbb{B}^k \to \mathbb{B}.\) The ambiguities of this usage are not a problem so long as we distinguish the context of definition from the context of application and restrict all shorthand notations to the latter.
Document History
Note. The above material is excerpted from a project report on Charles Sanders Peirce's conceptions of inquiry and analogy. Online formatting of the original document and continuation of the initial project are currently in progress under the title Functional Logic : Inquiry and Analogy.
| Author: | Jon Awbrey | November 1, 1995 |
| Course: | Engineering 690, Graduate Project | Cont'd from Winter 1995 |
| Supervisors: | F. Mili & M.A. Zohdy | Oakland University |
| Version: Draft 3.25 | Created: 01 Jan 1995 | Relayed: 01 Nov 1995 | Revised: 24 Dec 2001 | Revised: 12 Mar 2004
