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===Application of Higher Order Propositions to Quantification Theory===
===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.
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_ <math>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.
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'' <math>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:
With these qualifications in mind we note the following correspondences between classical quantifications and higher order indicator functions:
{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|+ '''Table 7. Syllogistic Premisses as Higher Order Indicator Functions'''
|
<math>\begin{array}{clcl}
\mathrm{A} &
\mathrm{Universal~Affirmative} &
\mathrm{All}\ u\ \mathrm{is}\ v &
\mathrm{Indicator~of}\ u (v) = 0 \\
\mathrm{E} &
\mathrm{Universal~Negative} &
\mathrm{All}\ u\ \mathrm{is}\ (v) &
\mathrm{Indicator~of}\ u \cdot v = 0 \\
\mathrm{I} &
\mathrm{Particular~Affirmative} &
\mathrm{Some}\ u\ \mathrm{is}\ v &
\mathrm{Indicator~of}\ u \cdot v = 1 \\
\mathrm{O} &
\mathrm{Particular~Negative} &
\mathrm{Some}\ u\ \mathrm{is}\ (v) &
\mathrm{Indicator~of}\ u (v) = 1 \\
\end{array}</math>
|}
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<pre>
<table align="center" cellpadding="10" cellspacing="0" width="80%">
<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}</math></font></caption>
<caption><font size="+2"><math>\text{Table 7.} ~~ \text{Syllogistic Premisses as Higher Order Indicator Functions}</math></font></caption>
<tr>
<tr>
<td align="center"><math>\operatorname{A}</math></td>
<td align="center"><math>\operatorname{A}</math></td>
<td><math>Absolute</math></td>
<td><math>\text{Absolute}</math></td>
<td><math>Universal Affirmative</math></td>
<td><math>\text{Universal Affirmative}</math></td>
<td align="center"><math>All ~ u ~ is ~ v</math></td>
<td align="center"><math>All ~ u ~ is ~ v</math></td>
<td><math>Indicator of u ~ \texttt{(} v \texttt{)} = 0</math></td></tr>
<td><math>Indicator of u ~ \texttt{(} v \texttt{)} = 0</math></td></tr>
</table></font>
</table></font>
<br>
The following Tables develop these ideas in more detail.
The following Tables develop these ideas in more detail.
<br>
<table align="center" cellpadding="1" cellspacing="0" style="text-align:center; width:90%">
<table align="center" cellpadding="1" cellspacing="0" style="text-align:center; width:90%">
<caption><font size="+2"><math>\texttt{Table 8.} ~~ \texttt{Simple Qualifiers of Propositions (Version 1)}</math></font></caption>
<caption><font size="+2"><math>\text{Table 8.} ~~ \text{Simple Qualifiers of Propositions (Version 1)}</math></font></caption>
<tr>
<tr>
<table align="center" cellpadding="1" cellspacing="0" style="text-align:center; width:90%">
<table align="center" cellpadding="1" cellspacing="0" style="text-align:center; width:90%">
<caption><font size="+2"><math>\texttt{Table 9.} ~~ \texttt{Simple Qualifiers of Propositions (Version 2)}</math></font></caption>
<caption><font size="+2"><math>\text{Table 9.} ~~ \text{Simple Qualifiers of Propositions (Version 2)}</math></font></caption>
<tr>
<tr>
<br>
<br>
<table align="center" cellpadding="4" cellspacing="0" style="text-align:center; width:90%">
<table align="center" cellpadding="4" cellspacing="0" style="text-align:center; width:90%">
<caption><font size="+2"><math>\texttt{Table 10.} ~~ \texttt{Relation of Quantifiers to Higher Order Propositions}</math></font></caption>
<caption><font size="+2"><math>\text{Table 10.} ~~ \text{Relation of Quantifiers to Higher Order Propositions}</math></font></caption>
<tr>
<tr>
</table></font>
</table></font>
</pre>
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<pre>
