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<tr>
<tr>
<td style="border-bottom:2px solid black" align="right"><math>x:</math></td>
<td style="border-bottom:2px solid black" align="right"><math>x:\!</math></td>
<td style="border-bottom:2px solid black"><math>1 ~ 0</math></td>
<td style="border-bottom:2px solid black"><math>1 ~ 0</math></td>
<td style="border-bottom:2px solid black; border-right:2px solid black"><math>f</math></td>
<td style="border-bottom:2px solid black; border-right:2px solid black"><math>f\!</math></td>
<td style="border-bottom:2px solid black"><math>m_{0}</math></td>
<td style="border-bottom:2px solid black"><math>m_{0}\!</math></td>
<td style="border-bottom:2px solid black"><math>m_{1}</math></td>
<td style="border-bottom:2px solid black"><math>m_{1}\!</math></td>
<td style="border-bottom:2px solid black"><math>m_{2}</math></td>
<td style="border-bottom:2px solid black"><math>m_{2}\!</math></td>
<td style="border-bottom:2px solid black"><math>m_{3}</math></td>
<td style="border-bottom:2px solid black"><math>m_{3}\!</math></td>
<td style="border-bottom:2px solid black"><math>m_{4}</math></td>
<td style="border-bottom:2px solid black"><math>m_{4}\!</math></td>
<td style="border-bottom:2px solid black"><math>m_{5}</math></td>
<td style="border-bottom:2px solid black"><math>m_{5}\!</math></td>
<td style="border-bottom:2px solid black"><math>m_{6}</math></td>
<td style="border-bottom:2px solid black"><math>m_{6}\!</math></td>
<td style="border-bottom:2px solid black"><math>m_{7}</math></td>
<td style="border-bottom:2px solid black"><math>m_{7}\!</math></td>
<td style="border-bottom:2px solid black"><math>m_{8}</math></td>
<td style="border-bottom:2px solid black"><math>m_{8}\!</math></td>
<td style="border-bottom:2px solid black"><math>m_{9}</math></td>
<td style="border-bottom:2px solid black"><math>m_{9}\!</math></td>
<td style="border-bottom:2px solid black"><math>m_{10}</math></td>
<td style="border-bottom:2px solid black"><math>m_{10}\!</math></td>
<td style="border-bottom:2px solid black"><math>m_{11}</math></td>
<td style="border-bottom:2px solid black"><math>m_{11}\!</math></td>
<td style="border-bottom:2px solid black"><math>m_{12}</math></td>
<td style="border-bottom:2px solid black"><math>m_{12}\!</math></td>
<td style="border-bottom:2px solid black"><math>m_{13}</math></td>
<td style="border-bottom:2px solid black"><math>m_{13}\!</math></td>
<td style="border-bottom:2px solid black"><math>m_{14}</math></td>
<td style="border-bottom:2px solid black"><math>m_{14}\!</math></td>
<td style="border-bottom:2px solid black"><math>m_{15}</math></td></tr>
<td style="border-bottom:2px solid black"><math>m_{15}\!</math></td></tr>
<tr>
<tr>
<td><math>f_{0}</math></td>
<td><math>f_{0}\!</math></td>
<td><math>0 ~ 0</math></td>
<td><math>0 ~ 0</math></td>
<td style="border-right:2px solid black"><math>\texttt{(~)}</math></td>
<td style="border-right:2px solid black"><math>\texttt{(~)}\!</math></td>
<td style="background:white; color:black">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_{1}</math></td>
<td><math>f_{1}\!</math></td>
<td><math>0 ~ 1</math></td>
<td><math>0 ~ 1</math></td>
<td style="border-right:2px solid black"><math>\texttt{(} x \texttt{)}</math></td>
<td style="border-right:2px solid black"><math>\texttt{(} x \texttt{)}\!</math></td>
<td style="background:white; color:black">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_{2}</math></td>
<td><math>f_{2}\!</math></td>
<td><math>1 ~ 0</math></td>
<td><math>1 ~ 0</math></td>
<td style="border-right:2px solid black"><math>x</math></td>
<td style="border-right:2px solid black"><math>x\!</math></td>
<td style="background:white; color:black">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_{3}</math></td>
<td><math>f_{3}\!</math></td>
<td><math>1 ~ 1</math></td>
<td><math>1 ~ 1</math></td>
<td style="border-right:2px solid black"><math>\texttt{((~))}</math></td>
<td style="border-right:2px solid black"><math>\texttt{((~))}\!</math></td>
<td style="background:white; color:black">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>
{| align="center" cellpadding="10" style="text-align:center"
{| align="center" cellpadding="10" style="text-align:center"
| [[Image:Venn Diagram 4 Dimensions UV Cacti 8 Inch.png]]
| [[Image:Venn Diagram 4 Dimensions UV Cacti 8 Inch.jpg]]
|-
|-
| <math>\text{Figure 6.} ~~ \text{Higher Order Universe of Discourse} ~ [\ell_{00}, \ell_{01}, \ell_{10}, \ell_{11}] \subseteq [[u, v]]</math>
| <math>\text{Figure 6.} ~~ \text{Higher Order Universe of Discourse} ~ [\ell_{00}, \ell_{01}, \ell_{10}, \ell_{11}] \subseteq [[u, v]]</math>
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:
<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%"
|}
|}
<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%">
* [[Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory|Functional Logic : Quantification Theory]]
<caption><font size="+2"><math>\text{Table 8.} ~~ \text{Simple Qualifiers of Propositions (Version 1)}</math></font></caption>
==Appendix : Generalized Umpire Operators==
<tr>
<td width="4%" style="border-bottom:1px solid black" align="right">
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 <math>\Upsilon,</math> 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 operator|multigrade]]'' property of <math>\Upsilon</math> can be expressed as a union of function types, in the following manner:
<math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td>
<td width="6%" style="border-bottom:1px solid black">
{| align="center" cellpadding="8" style="text-align:center"
<math>\begin{matrix}1100\\1010\end{matrix}</math></td>
| <math>\Upsilon : \bigcup_{\ell = 1, 2, 3} ((\mathbb{B}^k \to \mathbb{B})^\ell \to \mathbb{B}).</math>
<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">
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:
<math>\begin{smallmatrix}
\texttt{(} \ell_{11} \texttt{)}
{| align="center" cellpadding="8" style="text-align:center"
\\
| <math>\Upsilon_p^r q = \Upsilon (p, q, r)</math>
\mathrm{No} ~ u
|-
\\
| <math>\Upsilon_p^r : (\mathbb{B}^k \to \mathbb{B}) \to \mathbb{B}</math>
\mathrm{is} ~ v
|}
\end{smallmatrix}</math></td>
<td width="10%" style="border-bottom:1px solid black">
The intention of this operator is that we evaluate the proposition <math>q</math> on each model of the proposition <math>p</math> and combine the results according to the method indicated by the connective parameter <math>r.</math> In principle, the index <math>r</math> might specify any connective on as many as <math>2^k</math> 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 <math>p, r</math> is assigned a default value that is understood to be in force when the corresponding argument place is left blank, specifically, the constant proposition <math>1 : \mathbb{B}^k \to \mathbb{B}</math> for the lower index <math>p,</math> and the continued conjunction or continued product operation <math>\textstyle\prod</math> for the upper index <math>r.</math> Taking the upper default value gives license to the following readings:
<math>\begin{smallmatrix}
\texttt{(} \ell_{10} \texttt{)}
{| align="center" cellpadding="8" style="text-align:center"
\\
| <math>\Upsilon_p (q) = \Upsilon (p, q) = \Upsilon (p, q, \textstyle\prod).</math>
\mathrm{No} ~ u
|-
\\
| <math>\Upsilon_p = \Upsilon (p, \underline{~~}, \textstyle\prod) : (\mathbb{B}^k \to \mathbb{B}) \to \mathbb{B}.</math>
\mathrm{is} ~ \texttt{(} v \texttt{)}
|}
\end{smallmatrix}</math></td>
<td width="10%" style="border-bottom:1px solid black">
This means that <math>\Upsilon_p (q) = 1</math> if and only if <math>q</math> holds for all models of <math>p.</math> In propositional terms, this is tantamount to the assertion that <math>p \Rightarrow q,</math> or that <math>\texttt{(} p \texttt{(} q \texttt{))} = 1.</math>
<math>\begin{smallmatrix}
\texttt{(} \ell_{01} \texttt{)}
Throwing in the lower default value permits the following abbreviations:
\\
\mathrm{No} ~ \texttt{(} u \texttt{)}
{| align="center" cellpadding="8" style="text-align:center"
\\
| <math>\Upsilon q = \Upsilon (q) = \Upsilon_1 (q) = \Upsilon (1, q, \textstyle\prod).</math>
\mathrm{is} ~ v
|-
\end{smallmatrix}</math></td>
| <math>\Upsilon = \Upsilon (1, \underline{~~}, \textstyle\prod)) : (\mathbb{B}^k\ \to \mathbb{B}) \to \mathbb{B}.</math>
<td width="10%" style="border-bottom:1px solid black">
|}
<math>\begin{smallmatrix}
\texttt{(} \ell_{00} \texttt{)}
This means that <math>\Upsilon q = 1</math> if and only if <math>q</math> holds for the whole universe of discourse in question, that is, if and only <math>q</math> is the constantly true proposition <math>1 : \mathbb{B}^k \to \mathbb{B}.</math> 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.
\\
\mathrm{No} ~ \texttt{(} u \texttt{)}
\\
\mathrm{is} ~ \texttt{(} v \texttt{)}
\end{smallmatrix}</math></td>
<td width="10%" style="border-bottom:1px solid black">
<math>\begin{smallmatrix}
\ell_{00}
\\
\mathrm{Some} ~ \texttt{(} u \texttt{)}
\\
\mathrm{is} ~ \texttt{(} v \texttt{)}
\end{smallmatrix}</math></td>
<td width="10%" style="border-bottom:1px solid black">
<math>\begin{smallmatrix}
\ell_{01}
\\
\mathrm{Some} ~ \texttt{(} u \texttt{)}
\\
\mathrm{is} ~ v
\end{smallmatrix}</math></td>
<td width="10%" style="border-bottom:1px solid black">
<math>\begin{smallmatrix}
\ell_{10}
\\
\mathrm{Some} ~ u
\\
\mathrm{is} ~ \texttt{(} v \texttt{)}
\end{smallmatrix}</math></td>
<td width="10%" style="border-bottom:1px solid black">
<math>\begin{smallmatrix}
\ell_{11}
\\
\mathrm{Some} ~ u
\\
\mathrm{is} ~ v
\end{smallmatrix}</math></td></tr>
<tr>
<td><math>f_{0}</math></td>
<td><math>0000</math></td>
<td style="border-right:1px solid black"><math>\texttt{(~)}</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td></tr>
<tr>
<td><math>f_{1}</math></td>
<td><math>0001</math></td>
<td style="border-right:1px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td></tr>
<tr>
<td><math>f_{2}</math></td>
<td><math>0010</math></td>
<td style="border-right:1px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td></tr>
<tr>
<td><math>f_{3}</math></td>
<td><math>0011</math></td>
<td style="border-right:1px solid black"><math>\texttt{(} u \texttt{)}</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td></tr>
<tr>
<td><math>f_{4}</math></td>
<td><math>0100</math></td>
<td style="border-right:1px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td></tr>
<tr>
<td><math>f_{5}</math></td>
<td><math>0101</math></td>
<td style="border-right:1px solid black"><math>\texttt{(} v \texttt{)}</math></td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td></tr>
<tr>
<td><math>f_{6}</math></td>
<td><math>0110</math></td>
<td style="border-right:1px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td></tr>
<tr>
<td><math>f_{7}</math></td>
<td><math>0111</math></td>
<td style="border-right:1px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td></tr>
<tr>
<td><math>f_{8}</math></td>
<td><math>1000</math></td>
<td style="border-right:1px solid black"><math>u ~ v</math></td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{9}</math></td>
<td><math>1001</math></td>
<td style="border-right:1px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{10}</math></td>
<td><math>1010</math></td>
<td style="border-right:1px solid black"><math>v</math></td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{11}</math></td>
<td><math>1011</math></td>
<td style="border-right:1px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{12}</math></td>
<td><math>1100</math></td>
<td style="border-right:1px solid black"><math>u</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{13}</math></td>
<td><math>1101</math></td>
<td style="border-right:1px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{14}</math></td>
<td><math>1110</math></td>
<td style="border-right:1px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{15}</math></td>
<td><math>1111</math></td>
<td style="border-right:1px solid black"><math>\texttt{((~))}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
</table>
<br>
<table align="center" cellpadding="4" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:90%">
<caption><font size="+2"><math>\text{Table 9.} ~~ \text{Simple Qualifiers of Propositions (Version 2)}</math></font></caption>
<tr>
<td width="4%" style="border-bottom:1px solid black" align="right">
<math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td>
<td width="6%" style="border-bottom:1px solid black">
<math>\begin{matrix}1100\\1010\end{matrix}</math></td>
<td width="10%" style="border-bottom:1px solid black; border-right:1px solid black">
<math>f</math></td>
<td width="10%" style="border-bottom:1px solid black">
<math>\begin{smallmatrix}
\texttt{(} \ell_{11} \texttt{)}
\\
\mathrm{No} ~ u
\\
\mathrm{is} ~ v
\end{smallmatrix}</math></td>
<td width="10%" style="border-bottom:1px solid black">
<math>\begin{smallmatrix}
\texttt{(} \ell_{10} \texttt{)}
\\
\mathrm{No} ~ u
\\
\mathrm{is} ~ \texttt{(} v \texttt{)}
\end{smallmatrix}</math></td>
<td width="10%" style="border-bottom:1px solid black">
<math>\begin{smallmatrix}
\texttt{(} \ell_{01} \texttt{)}
\\
\mathrm{No} ~ \texttt{(} u \texttt{)}
\\
\mathrm{is} ~ v
\end{smallmatrix}</math></td>
<td width="10%" style="border-bottom:1px solid black">
<math>\begin{smallmatrix}
\texttt{(} \ell_{00} \texttt{)}
\\
\mathrm{No} ~ \texttt{(} u \texttt{)}
\\
\mathrm{is} ~ \texttt{(} v \texttt{)}
\end{smallmatrix}</math></td>
<td width="10%" style="border-bottom:1px solid black">
<math>\begin{smallmatrix}
\ell_{00}
\\
\mathrm{Some} ~ \texttt{(} u \texttt{)}
\\
\mathrm{is} ~ \texttt{(} v \texttt{)}
\end{smallmatrix}</math></td>
<td width="10%" style="border-bottom:1px solid black">
<math>\begin{smallmatrix}
\ell_{01}
\\
\mathrm{Some} ~ \texttt{(} u \texttt{)}
\\
\mathrm{is} ~ v
\end{smallmatrix}</math></td>
<td width="10%" style="border-bottom:1px solid black">
<math>\begin{smallmatrix}
\ell_{10}
\\
\mathrm{Some} ~ u
\\
\mathrm{is} ~ \texttt{(} v \texttt{)}
\end{smallmatrix}</math></td>
<td width="10%" style="border-bottom:1px solid black">
<math>\begin{smallmatrix}
\ell_{11}
\\
\mathrm{Some} ~ u
\\
\mathrm{is} ~ v
\end{smallmatrix}</math></td></tr>
<tr>
<td style="border-bottom:1px solid black"><math>f_{0}</math></td>
<td style="border-bottom:1px solid black"><math>0000</math></td>
<td style="border-bottom:1px solid black; border-right:1px solid black"><math>\texttt{(~)}</math></td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td></tr>
<tr>
<td><math>f_{1}</math></td>
<td><math>0001</math></td>
<td style="border-right:1px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td></tr>
<tr>
<td><math>f_{2}</math></td>
<td><math>0010</math></td>
<td style="border-right:1px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td></tr>
<tr>
<td><math>f_{4}</math></td>
<td><math>0100</math></td>
<td style="border-right:1px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td></tr>
<tr>
<td style="border-bottom:1px solid black"><math>f_{8}</math></td>
<td style="border-bottom:1px solid black"><math>1000</math></td>
<td style="border-bottom:1px solid black; border-right:1px solid black"><math>u ~ v</math></td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td></tr>
<tr>
<td><math>f_{3}</math></td>
<td><math>0011</math></td>
<td style="border-right:1px solid black"><math>\texttt{(} u \texttt{)}</math></td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td></tr>
<tr>
<td style="border-bottom:1px solid black"><math>f_{12}</math></td>
<td style="border-bottom:1px solid black"><math>1100</math></td>
<td style="border-bottom:1px solid black; border-right:1px solid black"><math>u</math></td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td></tr>
<tr>
<td><math>f_{6}</math></td>
<td><math>0110</math></td>
<td style="border-right:1px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td></tr>
<tr>
<td style="border-bottom:1px solid black"><math>f_{9}</math></td>
<td style="border-bottom:1px solid black"><math>1001</math></td>
<td style="border-bottom:1px solid black; border-right:1px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td></tr>
<tr>
<td><math>f_{5}</math></td>
<td><math>0101</math></td>
<td style="border-right:1px solid black"><math>\texttt{(} v \texttt{)}</math></td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td></tr>
<tr>
<td style="border-bottom:1px solid black"><math>f_{10}</math></td>
<td style="border-bottom:1px solid black"><math>1010</math></td>
<td style="border-bottom:1px solid black; border-right:1px solid black"><math>v</math></td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td></tr>
<tr>
<td><math>f_{7}</math></td>
<td><math>0111</math></td>
<td style="border-right:1px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td></tr>
<tr>
<td><math>f_{11}</math></td>
<td><math>1011</math></td>
<td style="border-right:1px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td><math>f_{13}</math></td>
<td><math>1101</math></td>
<td style="border-right:1px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
<tr>
<td style="border-bottom:1px solid black"><math>f_{14}</math></td>
<td style="border-bottom:1px solid black"><math>1110</math></td>
<td style="border-bottom:1px solid black; border-right:1px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:white; color:black">0</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td>
<td style="border-bottom:1px solid black; background:black; color:white">1</td></tr>
<tr>
<td><math>f_{15}</math></td>
<td><math>1111</math></td>
<td style="border-right:1px solid black"><math>\texttt{((~))}</math></td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:white; color:black">0</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td>
<td style="background:black; color:white">1</td></tr>
</table>
<br>
<table align="center" cellpadding="4" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:90%">
<caption><font size="+2"><math>\text{Table 10.} ~~ \text{Relation of Quantifiers to Higher Order Propositions}</math></font></caption>
<tr>
<td style="border-bottom:1px solid black"><math>\mathrm{Mnemonic}</math></td>
<td style="border-bottom:1px solid black"><math>\mathrm{Category}</math></td>
<td style="border-bottom:1px solid black"><math>\mathrm{Classical~Form}</math></td>
<td style="border-bottom:1px solid black"><math>\mathrm{Alternate~Form}</math></td>
<td style="border-bottom:1px solid black"><math>\mathrm{Symmetric~Form}</math></td>
<td style="border-bottom:1px solid black"><math>\mathrm{Operator}</math></td></tr>
<tr>
<td><math>\begin{matrix}
\mathrm{E}
\\
\mathrm{Exclusive}
\end{matrix}</math></td>
<td><math>\begin{matrix}
\mathrm{Universal}
\\
\mathrm{Negative}
\end{matrix}</math></td>
<td><math>\mathrm{All} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
<td> </td>
<td><math>\mathrm{No} ~ u ~ \mathrm{is} ~ v</math></td>
<td><math>\texttt{(} \ell_{11} \texttt{)}</math></td></tr>
<tr>
<td style="border-bottom:1px solid black">
<math>\begin{matrix}
\mathrm{A}
\\
\mathrm{Absolute}
\end{matrix}</math></td>
<td style="border-bottom:1px solid black">
<math>\begin{matrix}
\mathrm{Universal}
\\
\mathrm{Affirmative}
\end{matrix}</math></td>
<td style="border-bottom:1px solid black"><math>\mathrm{All} ~ u ~ \mathrm{is} ~ v</math></td>
<td style="border-bottom:1px solid black"> </td>
<td style="border-bottom:1px solid black"><math>\mathrm{No} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
<td style="border-bottom:1px solid black"><math>\texttt{(} \ell_{10} \texttt{)}</math></td></tr>
<tr>
<td> </td>
<td> </td>
<td><math>\mathrm{All} ~ v ~ \mathrm{is} ~ u</math></td>
<td><math>\mathrm{No} ~ v ~ \mathrm{is} ~ \texttt{(} u \texttt{)}</math></td>
<td><math>\mathrm{No} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td>
<td><math>\texttt{(} \ell_{01} \texttt{)}</math></td></tr>
<tr>
<td style="border-bottom:1px solid black"> </td>
<td style="border-bottom:1px solid black"> </td>
<td style="border-bottom:1px solid black"><math>\mathrm{All} ~ \texttt{(} v \texttt{)} ~ \mathrm{is} ~ u</math></td>
<td style="border-bottom:1px solid black"><math>\mathrm{No} ~ \texttt{(} v \texttt{)} ~ \mathrm{is} ~ \texttt{(} u \texttt{)}</math></td>
<td style="border-bottom:1px solid black"><math>\mathrm{No} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
<td style="border-bottom:1px solid black"><math>\texttt{(} \ell_{00} \texttt{)}</math></td></tr>
<tr>
<td> </td>
<td> </td>
<td><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
<td> </td>
<td><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
<td><math>\ell_{00}</math></td></tr>
<tr>
<td style="border-bottom:1px solid black"> </td>
<td style="border-bottom:1px solid black"> </td>
<td style="border-bottom:1px solid black"><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td>
<td style="border-bottom:1px solid black"> </td>
<td style="border-bottom:1px solid black"><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td>
<td style="border-bottom:1px solid black"><math>\ell_{01}</math></td></tr>
<tr>
<td><math>\begin{matrix}
\mathrm{O}
\\
\mathrm{Obtrusive}
\end{matrix}</math></td>
<td><math>\begin{matrix}
\mathrm{Particular}
\\
\mathrm{Negative}
\end{matrix}</math></td>
<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
<td> </td>
<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
<td><math>\ell_{10}</math></td></tr>
<tr>
<td><math>\begin{matrix}
\mathrm{I}
\\
\mathrm{Indefinite}
\end{matrix}</math></td>
<td><math>\begin{matrix}
\mathrm{Particular}
\\
\mathrm{Affirmative}
\end{matrix}</math></td>
<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ v</math></td>
<td> </td>
<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ v</math></td>
<td><math>\ell_{11}</math></td></tr>
</table>
<br>
==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==
* [[Directory:Jon_Awbrey/Papers/Functional_Logic_:_Inquiry_and_Analogy|Functional Logic : Inquiry and Analogy]]
* [[Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory|Functional Logic : Quantification Theory]]
==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 <math>\Upsilon,</math> 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 operator|multigrade]]'' property of <math>\Upsilon</math> can be expressed as a union of function types, in the following manner:
{| align="center" cellpadding="8" style="text-align:center"
| <math>\Upsilon : \bigcup_{\ell = 1, 2, 3} ((\mathbb{B}^k \to \mathbb{B})^\ell \to \mathbb{B}).</math>
|}
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:
{| align="center" cellpadding="8" style="text-align:center"
| <math>\Upsilon_p^r q = \Upsilon (p, q, r)</math>
|-
| <math>\Upsilon_p^r : (\mathbb{B}^k \to \mathbb{B}) \to \mathbb{B}</math>
|}
The intention of this operator is that we evaluate the proposition <math>q</math> on each model of the proposition <math>p</math> and combine the results according to the method indicated by the connective parameter <math>r.</math> In principle, the index <math>r</math> might specify any connective on as many as <math>2^k</math> 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 <math>p, r</math> is assigned a default value that is understood to be in force when the corresponding argument place is left blank, specifically, the constant proposition <math>1 : \mathbb{B}^k \to \mathbb{B}</math> for the lower index <math>p,</math> and the continued conjunction or continued product operation <math>\textstyle\prod</math> for the upper index <math>r.</math> Taking the upper default value gives license to the following readings:
{| align="center" cellpadding="8" style="text-align:center"
| <math>\Upsilon_p (q) = \Upsilon (p, q) = \Upsilon (p, q, \textstyle\prod).</math>
|-
| <math>\Upsilon_p = \Upsilon (p, \underline{~~}, \textstyle\prod) : (\mathbb{B}^k \to \mathbb{B}) \to \mathbb{B}.</math>
|}
This means that <math>\Upsilon_p (q) = 1</math> if and only if <math>q</math> holds for all models of <math>p.</math> In propositional terms, this is tantamount to the assertion that <math>p \Rightarrow q,</math> or that <math>\texttt{(} p \texttt{(} q \texttt{))} = 1.</math>
Throwing in the lower default value permits the following abbreviations:
{| align="center" cellpadding="8" style="text-align:center"
| <math>\Upsilon q = \Upsilon (q) = \Upsilon_1 (q) = \Upsilon (1, q, \textstyle\prod).</math>
|-
| <math>\Upsilon = \Upsilon (1, \underline{~~}, \textstyle\prod)) : (\mathbb{B}^k\ \to \mathbb{B}) \to \mathbb{B}.</math>
|}
This means that <math>\Upsilon q = 1</math> if and only if <math>q</math> holds for the whole universe of discourse in question, that is, if and only <math>q</math> is the constantly true proposition <math>1 : \mathbb{B}^k \to \mathbb{B}.</math> 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==
==Document History==
| Revised: 12 Mar 2004
| Revised: 12 Mar 2004
</pre>
</pre>
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