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

Revision as of 03:12, 4 July 2013

1,523 bytes added ,  03:12, 4 July 2013
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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>
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<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>
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<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>
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<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>
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{| 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>
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<math>f</math></td>
 
<math>f</math></td>
 
<td width="10%" style="border-bottom:1px solid black">
 
<td width="10%" style="border-bottom:1px solid black">
−
<math>\begin{matrix}
+
<math>\begin{smallmatrix}
 
\texttt{(} \ell_{11} \texttt{)}
 
\texttt{(} \ell_{11} \texttt{)}
 
\\
 
\\
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\\
 
\\
 
\mathrm{is} ~ v
 
\mathrm{is} ~ v
−
\end{matrix}</math></td>
+
\end{smallmatrix}</math></td>
 
<td width="10%" style="border-bottom:1px solid black">
 
<td width="10%" style="border-bottom:1px solid black">
−
<math>\begin{matrix}
+
<math>\begin{smallmatrix}
 
\texttt{(} \ell_{10} \texttt{)}
 
\texttt{(} \ell_{10} \texttt{)}
 
\\
 
\\
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\\
 
\\
 
\mathrm{is} ~ \texttt{(} v \texttt{)}
 
\mathrm{is} ~ \texttt{(} v \texttt{)}
−
\end{matrix}</math></td>
+
\end{smallmatrix}</math></td>
 
<td width="10%" style="border-bottom:1px solid black">
 
<td width="10%" style="border-bottom:1px solid black">
−
<math>\begin{matrix}
+
<math>\begin{smallmatrix}
 
\texttt{(} \ell_{01} \texttt{)}
 
\texttt{(} \ell_{01} \texttt{)}
 
\\
 
\\
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\\
 
\\
 
\mathrm{is} ~ v
 
\mathrm{is} ~ v
−
\end{matrix}</math></td>
+
\end{smallmatrix}</math></td>
 
<td width="10%" style="border-bottom:1px solid black">
 
<td width="10%" style="border-bottom:1px solid black">
−
<math>\begin{matrix}
+
<math>\begin{smallmatrix}
 
\texttt{(} \ell_{00} \texttt{)}
 
\texttt{(} \ell_{00} \texttt{)}
 
\\
 
\\
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\\
 
\\
 
\mathrm{is} ~ \texttt{(} v \texttt{)}
 
\mathrm{is} ~ \texttt{(} v \texttt{)}
−
\end{matrix}</math></td>
+
\end{smallmatrix}</math></td>
 
<td width="10%" style="border-bottom:1px solid black">
 
<td width="10%" style="border-bottom:1px solid black">
−
<math>\begin{matrix}
+
<math>\begin{smallmatrix}
 
\ell_{00}
 
\ell_{00}
 
\\
 
\\
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\\
 
\\
 
\mathrm{is} ~ \texttt{(} v \texttt{)}
 
\mathrm{is} ~ \texttt{(} v \texttt{)}
−
\end{matrix}</math></td>
+
\end{smallmatrix}</math></td>
 
<td width="10%" style="border-bottom:1px solid black">
 
<td width="10%" style="border-bottom:1px solid black">
−
<math>\begin{matrix}
+
<math>\begin{smallmatrix}
 
\ell_{01}
 
\ell_{01}
 
\\
 
\\
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\\
 
\\
 
\mathrm{is} ~ v
 
\mathrm{is} ~ v
−
\end{matrix}</math></td>
+
\end{smallmatrix}</math></td>
 
<td width="10%" style="border-bottom:1px solid black">
 
<td width="10%" style="border-bottom:1px solid black">
−
<math>\begin{matrix}
+
<math>\begin{smallmatrix}
 
\ell_{10}
 
\ell_{10}
 
\\
 
\\
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\\
 
\\
 
\mathrm{is} ~ \texttt{(} v \texttt{)}
 
\mathrm{is} ~ \texttt{(} v \texttt{)}
−
\end{matrix}</math></td>
+
\end{smallmatrix}</math></td>
 
<td width="10%" style="border-bottom:1px solid black">
 
<td width="10%" style="border-bottom:1px solid black">
−
<math>\begin{matrix}
+
<math>\begin{smallmatrix}
 
\ell_{11}
 
\ell_{11}
 
\\
 
\\
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\\
 
\\
 
\mathrm{is} ~ v
 
\mathrm{is} ~ v
−
\end{matrix}</math></td></tr>
+
\end{smallmatrix}</math></td></tr>
    
<tr>
 
<tr>
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<math>f</math></td>
 
<math>f</math></td>
 
<td width="10%" style="border-bottom:1px solid black">
 
<td width="10%" style="border-bottom:1px solid black">
−
<math>\begin{matrix}
+
<math>\begin{smallmatrix}
 
\texttt{(} \ell_{11} \texttt{)}
 
\texttt{(} \ell_{11} \texttt{)}
 
\\
 
\\
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\\
 
\\
 
\mathrm{is} ~ v
 
\mathrm{is} ~ v
−
\end{matrix}</math></td>
+
\end{smallmatrix}</math></td>
 
<td width="10%" style="border-bottom:1px solid black">
 
<td width="10%" style="border-bottom:1px solid black">
−
<math>\begin{matrix}
+
<math>\begin{smallmatrix}
 
\texttt{(} \ell_{10} \texttt{)}
 
\texttt{(} \ell_{10} \texttt{)}
 
\\
 
\\
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\\
 
\\
 
\mathrm{is} ~ \texttt{(} v \texttt{)}
 
\mathrm{is} ~ \texttt{(} v \texttt{)}
−
\end{matrix}</math></td>
+
\end{smallmatrix}</math></td>
 
<td width="10%" style="border-bottom:1px solid black">
 
<td width="10%" style="border-bottom:1px solid black">
−
<math>\begin{matrix}
+
<math>\begin{smallmatrix}
 
\texttt{(} \ell_{01} \texttt{)}
 
\texttt{(} \ell_{01} \texttt{)}
 
\\
 
\\
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\\
 
\\
 
\mathrm{is} ~ v
 
\mathrm{is} ~ v
−
\end{matrix}</math></td>
+
\end{smallmatrix}</math></td>
 
<td width="10%" style="border-bottom:1px solid black">
 
<td width="10%" style="border-bottom:1px solid black">
−
<math>\begin{matrix}
+
<math>\begin{smallmatrix}
 
\texttt{(} \ell_{00} \texttt{)}
 
\texttt{(} \ell_{00} \texttt{)}
 
\\
 
\\
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\\
 
\\
 
\mathrm{is} ~ \texttt{(} v \texttt{)}
 
\mathrm{is} ~ \texttt{(} v \texttt{)}
−
\end{matrix}</math></td>
+
\end{smallmatrix}</math></td>
 
<td width="10%" style="border-bottom:1px solid black">
 
<td width="10%" style="border-bottom:1px solid black">
−
<math>\begin{matrix}
+
<math>\begin{smallmatrix}
 
\ell_{00}
 
\ell_{00}
 
\\
 
\\
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\\
 
\\
 
\mathrm{is} ~ \texttt{(} v \texttt{)}
 
\mathrm{is} ~ \texttt{(} v \texttt{)}
−
\end{matrix}</math></td>
+
\end{smallmatrix}</math></td>
 
<td width="10%" style="border-bottom:1px solid black">
 
<td width="10%" style="border-bottom:1px solid black">
−
<math>\begin{matrix}
+
<math>\begin{smallmatrix}
 
\ell_{01}
 
\ell_{01}
 
\\
 
\\
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\\
 
\\
 
\mathrm{is} ~ v
 
\mathrm{is} ~ v
−
\end{matrix}</math></td>
+
\end{smallmatrix}</math></td>
 
<td width="10%" style="border-bottom:1px solid black">
 
<td width="10%" style="border-bottom:1px solid black">
−
<math>\begin{matrix}
+
<math>\begin{smallmatrix}
 
\ell_{10}
 
\ell_{10}
 
\\
 
\\
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\\
 
\\
 
\mathrm{is} ~ \texttt{(} v \texttt{)}
 
\mathrm{is} ~ \texttt{(} v \texttt{)}
−
\end{matrix}</math></td>
+
\end{smallmatrix}</math></td>
 
<td width="10%" style="border-bottom:1px solid black">
 
<td width="10%" style="border-bottom:1px solid black">
−
<math>\begin{matrix}
+
<math>\begin{smallmatrix}
 
\ell_{11}
 
\ell_{11}
 
\\
 
\\
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\\
 
\\
 
\mathrm{is} ~ v
 
\mathrm{is} ~ v
−
\end{matrix}</math></td></tr>
+
\end{smallmatrix}</math></td></tr>
    
<tr>
 
<tr>
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<tr>
 
<tr>
−
<td><math>\mathrm{E}</math><br><math>\mathrm{Exclusive}</math></td>
+
<td><math>\begin{matrix}
−
<td><math>\mathrm{Universal}</math><br><math>\mathrm{Negative}</math></td>
+
\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><math>\mathrm{All} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
 
<td>&nbsp;</td>
 
<td>&nbsp;</td>
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<tr>
 
<tr>
−
<td><math>\mathrm{A}</math><br><math>\mathrm{Absolute}</math></td>
+
<td style="border-bottom:1px solid black">
−
<td><math>\mathrm{Universal}</math><br><math>\mathrm{Affirmative}</math></td>
+
<math>\begin{matrix}
−
<td><math>\mathrm{All} ~ u ~ \mathrm{is} ~ v</math></td>
+
\mathrm{A}
−
<td>&nbsp;</td>
+
\\
−
<td><math>\mathrm{No} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
+
\mathrm{Absolute}
−
<td><math>\texttt{(} \ell_{10} \texttt{)}</math></td></tr>
+
\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">&nbsp;</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>
 
<tr>
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<tr>
 
<tr>
−
<td>&nbsp;</td>
+
<td style="border-bottom:1px solid black">&nbsp;</td>
−
<td>&nbsp;</td>
+
<td style="border-bottom:1px solid black">&nbsp;</td>
−
<td><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td>
+
<td style="border-bottom:1px solid black"><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td>
−
<td>&nbsp;</td>
+
<td style="border-bottom:1px solid black">&nbsp;</td>
−
<td><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td>
+
<td style="border-bottom:1px solid black"><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td>
−
<td><math>\ell_{01}</math></td></tr>
+
<td style="border-bottom:1px solid black"><math>\ell_{01}</math></td></tr>
    
<tr>
 
<tr>
−
<td><math>\mathrm{O}</math><br><math>\mathrm{Obtrusive}</math></td>
+
<td><math>\begin{matrix}
−
<td><math>\mathrm{Particular}</math><br><math>\mathrm{Negative}</math></td>
+
\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><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
 
<td>&nbsp;</td>
 
<td>&nbsp;</td>
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<tr>
 
<tr>
−
<td><math>\mathrm{I}</math><br><math>\mathrm{Indefinite}</math></td>
+
<td><math>\begin{matrix}
−
<td><math>\mathrm{Particular}</math><br><math>\mathrm{Affirmative}</math></td>
+
\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><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ v</math></td>
 
<td>&nbsp;</td>
 
<td>&nbsp;</td>
Line 2,341: Line 2,375:  
| Revised:  12 Mar 2004
 
| Revised:  12 Mar 2004
 
</pre>
 
</pre>
 +
 +
[[Category:Adaptive Systems]]
 +
[[Category:Artificial Intelligence]]
 +
[[Category:Combinatorics]]
 +
[[Category:Computer Science]]
 +
[[Category:Cybernetics]]
 +
[[Category:Differential Logic]]
 +
[[Category:Discrete Systems]]
 +
[[Category:Dynamical Systems]]
 +
[[Category:Formal Languages]]
 +
[[Category:Formal Sciences]]
 +
[[Category:Formal Systems]]
 +
[[Category:Functional Logic]]
 +
[[Category:Graph Theory]]
 +
[[Category:Group Theory]]
 +
[[Category:Inquiry]]
 +
[[Category:Knowledge Representation]]
 +
[[Category:Linguistics]]
 +
[[Category:Logic]]
 +
[[Category:Logical Graphs]]
 +
[[Category:Mathematics]]
 +
[[Category:Mathematical Systems Theory]]
 +
[[Category:Science]]
 +
[[Category:Semiotics]]
 +
[[Category:Philosophy]]
 +
[[Category:Systems Science]]
 +
[[Category:Visualization]]
Jon Awbrey
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