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Differential Logic and Dynamic Systems • Document History

2026-04-17T15:14:35Z

Jon Awbrey: + Differential Logic and Dynamic Systems • Document History

Differential Logic and Dynamic Systems • References

2026-04-17T14:32:20Z

Jon Awbrey: delete {{anchor ...}}s

Differential Logic and Dynamic Systems • References

2026-04-17T14:15:34Z

Jon Awbrey: + Differential Logic and Dynamic Systems • References

Differential Logic and Dynamic Systems • Appendices

2026-04-12T22:21:01Z

Jon Awbrey: add Differential Logic and Dynamic Systems • Appendices

<b>Author: [[User:Jon Awbrey|Jon Awbrey]]</b>

----
<div align="center">
• [[Differential Logic and Dynamic Systems • Overview|Overview]]
• [[Differential Logic and Dynamic Systems • Part 1|Part 1]]
• [[Differential Logic and Dynamic Systems • Part 2|Part 2]]
• [[Differential Logic and Dynamic Systems • Part 3|Part 3]]
• [[Differential Logic and Dynamic Systems • Part 4|Part 4]]
• [[Differential Logic and Dynamic Systems • Part 5|Part 5]]
• [[Differential Logic and Dynamic Systems • Appendices|Appendices]]
• [[Differential Logic and Dynamic Systems • References|References]]
• [[Differential Logic and Dynamic Systems • Document History|Document History]]
•
</div>
----

==Appendices==

===Appendix 1. Propositional Forms and Differential Expansions===

====Table A1. Propositional Forms on Two Variables====

<br>

{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table A1.} ~~ \text{Propositional Forms on Two Variables}</math>
|- style="background:ghostwhite"
| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math>
| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math>
|- style="background:ghostwhite"
|  
| align="right" | <math>x\colon</math>
| <math>1~1~0~0</math>
|   ||   ||  
|- style="background:ghostwhite"
|  
| align="right" | <math>y\colon</math>
| <math>1~0~1~0</math>
|   ||   ||  
|-
|
<math>\begin{matrix}
f_{0}\\f_{1}\\f_{2}\\f_{3}\\f_{4}\\f_{5}\\f_{6}\\f_{7}
\end{matrix}</math>
|
<math>\begin{matrix}
f_{0000}\\f_{0001}\\f_{0010}\\f_{0011}\\f_{0100}\\f_{0101}\\f_{0110}\\f_{0111}
\end{matrix}</math>
|
<math>\begin{matrix}
0~0~0~0\\0~0~0~1\\0~0~1~0\\0~0~1~1\\0~1~0~0\\0~1~0~1\\0~1~1~0\\0~1~1~1
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} ~~ \texttt{)}
\\
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)} ~~ y ~~
\\
\texttt{(} x \texttt{)} ~ ~ ~ ~
\\
~~ x ~~ \texttt{(} y \texttt{)}
\\
~ ~ ~ ~ \texttt{(} y \texttt{)}
\\
\texttt{(} x \texttt{,} ~~ y \texttt{)}
\\
\texttt{(} x ~ ~ ~ y \texttt{)}
\end{matrix}</math>
|
<math>\begin{matrix}
\text{false}
\\
\text{neither}~ x ~\text{nor}~ y
\\
y ~\text{without}~ x
\\
\text{not}~ x
\\
x ~\text{without}~ y
\\
\text{not}~ y
\\
x ~\text{not equal to}~ y
\\
\text{not both}~ x ~\text{and}~ y
\end{matrix}</math>
|
<math>\begin{matrix}
0
\\
\lnot x \land \lnot y
\\
\lnot x \land y
\\
\lnot x
\\
x \land \lnot y
\\
\lnot y
\\
x \ne y
\\
\lnot x \lor \lnot y
\end{matrix}</math>
|-
|
<math>\begin{matrix}
f_{8}\\f_{9}\\f_{10}\\f_{11}\\f_{12}\\f_{13}\\f_{14}\\f_{15}
\end{matrix}</math>
|
<math>\begin{matrix}
f_{1000}\\f_{1001}\\f_{1010}\\f_{1011}\\f_{1100}\\f_{1101}\\f_{1110}\\f_{1111}
\end{matrix}</math>
|
<math>\begin{matrix}
1~0~0~0\\1~0~0~1\\1~0~1~0\\1~0~1~1\\1~1~0~0\\1~1~0~1\\1~1~1~0\\1~1~1~1
\end{matrix}</math>
|
<math>\begin{matrix}
x ~ ~ ~ y
\\
\texttt{((} x \texttt{,} ~~ y \texttt{))}
\\
~ ~ ~ ~ y
\\
~~ \texttt{(} x ~~ \texttt{(} y \texttt{))}
\\
x ~ ~ ~ ~
\\
\texttt{((} x \texttt{)} ~~ y \texttt{)} ~~
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\\
\texttt{((} ~~ \texttt{))}
\end{matrix}</math>
|
<math>\begin{matrix}
x ~\text{and}~ y
\\
x ~\text{equal to}~ y
\\
y
\\
\text{not}~ x ~\text{without}~ y
\\
x
\\
\text{not}~ y ~\text{without}~ x
\\
x ~\text{or}~ y
\\
\text{true}
\end{matrix}</math>
|
<math>\begin{matrix}
x \land y
\\
x = y
\\
y
\\
x \Rightarrow y
\\
x
\\
x \Leftarrow y
\\
x \lor y
\\
1
\end{matrix}</math>
|}

<br>

====Table A2. Propositional Forms on Two Variables====

<br>

{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table A2.} ~~ \text{Propositional Forms on Two Variables}</math>
|- style="background:ghostwhite"
| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math>
| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math>
|- style="background:ghostwhite"
|  
| align="right" | <math>x\colon</math>
| <math>1~1~0~0</math>
|   ||   ||  
|- style="background:ghostwhite"
|  
| align="right" | <math>y\colon</math>
| <math>1~0~1~0</math>
|   ||   ||  
|-
| <math>f_{0}</math>
| <math>f_{0000}</math>
| <math>0~0~0~0</math>
| <math>\texttt{(} ~~ \texttt{)}</math>
| <math>\text{false}</math>
| <math>0</math>
|-
|
<math>\begin{matrix}
f_{1}\\f_{2}\\f_{4}\\f_{8}
\end{matrix}</math>
|
<math>\begin{matrix}
f_{0001}\\f_{0010}\\f_{0100}\\f_{1000}
\end{matrix}</math>
|
<math>\begin{matrix}
0~0~0~1\\0~0~1~0\\0~1~0~0\\1~0~0~0
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)} ~~ y ~~
\\
~~ x ~~ \texttt{(} y \texttt{)}
\\
x ~ ~ ~ y
\end{matrix}</math>
|
<math>\begin{matrix}
\text{neither}~ x ~\text{nor}~ y
\\
y ~\text{without}~ x
\\
x ~\text{without}~ y
\\
x ~\text{and}~ y
\end{matrix}</math>
|
<math>\begin{matrix}
\lnot x \land \lnot y
\\
\lnot x \land y
\\
x \land \lnot y
\\
x \land y
\end{matrix}</math>
|-
|
<math>\begin{matrix}
f_{3}\\f_{12}
\end{matrix}</math>
|
<math>\begin{matrix}
f_{0011}\\f_{1100}
\end{matrix}</math>
|
<math>\begin{matrix}
0~0~1~1\\1~1~0~0
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
x
\end{matrix}</math>
|
<math>\begin{matrix}
\text{not}~ x
\\
x
\end{matrix}</math>
|
<math>\begin{matrix}
\lnot x
\\
x
\end{matrix}</math>
|-
|
<math>\begin{matrix}
f_{6}\\f_{9}
\end{matrix}</math>
|
<math>\begin{matrix}
f_{0110}\\f_{1001}
\end{matrix}</math>
|
<math>\begin{matrix}
0~1~1~0\\1~0~0~1
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} x \texttt{,} ~~ y \texttt{)}
\\
\texttt{((} x \texttt{,} ~~ y \texttt{))}
\end{matrix}</math>
|
<math>\begin{matrix}
x ~\text{not equal to}~ y
\\
x ~\text{equal to}~ y
\end{matrix}</math>
|
<math>\begin{matrix}
x \ne y
\\
x = y
\end{matrix}</math>
|-
|
<math>\begin{matrix}
f_{5}\\f_{10}
\end{matrix}</math>
|
<math>\begin{matrix}
f_{0101}\\f_{1010}
\end{matrix}</math>
|
<math>\begin{matrix}
0~1~0~1\\1~0~1~0
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
y
\end{matrix}</math>
|
<math>\begin{matrix}
\text{not}~ y
\\
y
\end{matrix}</math>
|
<math>\begin{matrix}
\lnot y
\\
y
\end{matrix}</math>
|-
|
<math>\begin{matrix}
f_{7}\\f_{11}\\f_{13}\\f_{14}
\end{matrix}</math>
|
<math>\begin{matrix}
f_{0111}\\f_{1011}\\f_{1101}\\f_{1110}
\end{matrix}</math>
|
<math>\begin{matrix}
0~1~1~1\\1~0~1~1\\1~1~0~1\\1~1~1~0
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} x ~ ~ ~ y \texttt{)}
\\
~~ \texttt{(} x ~~ \texttt{(} y \texttt{))}
\\
\texttt{((} x \texttt{)} ~~ y \texttt{)} ~~
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\end{matrix}</math>
|
<math>\begin{matrix}
\text{not both}~ x ~\text{and}~ y
\\
\text{not}~ x ~\text{without}~ y
\\
\text{not}~ y ~\text{without}~ x
\\
x ~\text{or}~ y
\end{matrix}</math>
|
<math>\begin{matrix}
\lnot x \lor \lnot y
\\
x \Rightarrow y
\\
x \Leftarrow y
\\
x \lor y
\end{matrix}</math>
|-
| <math>f_{15}</math>
| <math>f_{1111}</math>
| <math>1~1~1~1</math>
| <math>\texttt{((} ~~ \texttt{))}</math>
| <math>\text{true}</math>
| <math>1</math>
|}

<br>

====Table A3. E''f'' Expanded Over Differential Features====

<br>

{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table A3.} ~~ \mathrm{E}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}</math>
|- style="background:ghostwhite"
| style="width:10%; border-bottom:1px solid black" |  
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f</math>
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |
<math>\begin{matrix}\mathrm{T}_{11}f\\\mathrm{E}f|_{\mathrm{d}x ~ \mathrm{d}y}\end{matrix}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}\mathrm{T}_{10}f\\\mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}\end{matrix}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}\mathrm{T}_{01}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}\end{matrix}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}\mathrm{T}_{00}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{0}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{1}\\f_{2}\\f_{4}\\f_{8}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)} ~~ y ~~
\\
~~ x ~~ \texttt{(} y \texttt{)}
\\
x ~ ~ ~ y
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
x ~ ~ ~ y
\\
~~ x ~~ \texttt{(} y \texttt{)}
\\
\texttt{(} x \texttt{)} ~~ y ~~
\\
\texttt{(} x \texttt{)(} y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~~ x ~~ \texttt{(} y \texttt{)}
\\
x ~ ~ ~ y
\\
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)} ~~ y ~~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)} ~~ y ~~
\\
\texttt{(} x \texttt{)(} y \texttt{)}
\\
x ~ ~ ~ y
\\
~~ x ~~ \texttt{(} y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)} ~~ y ~~
\\
~~ x ~~ \texttt{(} y \texttt{)}
\\
x ~ ~ ~ y
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{3}\\f_{12}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
x
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
x
\\
\texttt{(} x \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
x
\\
\texttt{(} x \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
x
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
x
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{6}\\f_{9}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{,} ~~ y \texttt{)}
\\
\texttt{((} x \texttt{,} ~~ y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{(} x \texttt{,} ~~ y \texttt{)}
\\
\texttt{((} x \texttt{,} ~~ y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} x \texttt{,} ~~ y \texttt{))}
\\
\texttt{(} x \texttt{,} ~~ y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} x \texttt{,} ~~ y \texttt{))}
\\
\texttt{(} x \texttt{,} ~~ y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{,} ~~ y \texttt{)}
\\
\texttt{((} x \texttt{,} ~~ y \texttt{))}
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{5}\\f_{10}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
y
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
y
\\
\texttt{(} y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
y
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
y
\\
\texttt{(} y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
y
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{7}\\f_{11}\\f_{13}\\f_{14}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)}
\\
\texttt{(} ~~ x ~~ \texttt{(} y \texttt{))}
\\
\texttt{((} x \texttt{)} ~~ y ~~ \texttt{)}
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{((} x \texttt{)(} y \texttt{))}
\\
\texttt{((} x \texttt{)} ~~ y ~~ \texttt{)}
\\
\texttt{(} ~~ x ~~ \texttt{(} y \texttt{))}
\\
\texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} x \texttt{)} ~~ y ~~ \texttt{)}
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\\
\texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)}
\\
\texttt{(} ~~ x ~~ \texttt{(} y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ x ~~ \texttt{(} y \texttt{))}
\\
\texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)}
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\\
\texttt{((} x \texttt{)} ~~ y ~~ \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)}
\\
\texttt{(} ~~ x ~~ \texttt{(} y \texttt{))}
\\
\texttt{((} x \texttt{)} ~~ y ~~ \texttt{)}
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{15}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>1</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
|- style="background:ghostwhite"
| style="border-top:1px solid black" colspan="2" | <math>\text{Fixed Point Total}</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>4</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>4</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>4</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>16</math>
|}

<br>

====Table A4. D''f'' Expanded Over Differential Features====

<br>

{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table A4.} ~~ \mathrm{D}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}</math>
|- style="background:ghostwhite"
| style="width:10%; border-bottom:1px solid black" |  
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f</math>
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |
<math>\mathrm{D}f|_{\mathrm{d}x ~ \mathrm{d}y}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{D}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}</math>
|-
| style="border-top:1px solid black" | <math>f_{0}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{1}\\f_{2}\\f_{4}\\f_{8}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)} ~~ y ~~
\\
~~ x ~~ \texttt{(} y \texttt{)}
\\
x ~ ~ ~ y
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{((} x \texttt{,} ~~ y \texttt{))}
\\
\texttt{(} x \texttt{,} ~~ y \texttt{)}
\\
\texttt{(} x \texttt{,} ~~ y \texttt{)}
\\
\texttt{((} x \texttt{,} ~~ y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
y
\\
\texttt{(} y \texttt{)}
\\
y
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
\texttt{(} x \texttt{)}
\\
x
\\
x
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
x
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}1\\1\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}1\\1\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{,} ~~ y \texttt{)}
\\
\texttt{((} x \texttt{,} ~~ y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}0\\0\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}1\\1\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}1\\1\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
y
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}1\\1\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}1\\1\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)}
\\
\texttt{(} ~~ x ~~ \texttt{(} y \texttt{))}
\\
\texttt{((} x \texttt{)} ~~ y ~~ \texttt{)}
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{((} x \texttt{,} ~~ y \texttt{))}
\\
\texttt{(} x \texttt{,} ~~ y \texttt{)}
\\
\texttt{(} x \texttt{,} ~~ y \texttt{)}
\\
\texttt{((} x \texttt{,} ~~ y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
y
\\
\texttt{(} y \texttt{)}
\\
y
\\
\texttt{(} y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
x
\\
x
\\
\texttt{(} x \texttt{)}
\\
\texttt{(} x \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{15}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
|}

<br>

====Table A5. E''f'' Expanded Over Ordinary Features====

<br>

{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table A5.} ~~ \mathrm{E}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math>
|- style="background:ghostwhite"
| style="width:10%; border-bottom:1px solid black" |  
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f</math>
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |
<math>\mathrm{E}f|_{xy}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{E}f|_{x \texttt{(} y \texttt{)}}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{E}f|_{\texttt{(} x \texttt{)} y}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{E}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math>
|-
| style="border-top:1px solid black" | <math>f_{0}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{1}\\f_{2}\\f_{4}\\f_{8}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)} ~~ y ~~
\\
~~ x ~~ \texttt{(} y \texttt{)}
\\
x ~ ~ ~ y
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}x ~ ~ ~ \mathrm{d}y
\\
~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)}
\\
\texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~
\\
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)}
\\
\mathrm{d}x ~ ~ ~ \mathrm{d}y
\\
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}
\\
\texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~
\\
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}
\\
\mathrm{d}x ~ ~ ~ \mathrm{d}y
\\
~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}
\\
\texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~
\\
~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)}
\\
\mathrm{d}x ~ ~ ~ \mathrm{d}y
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{3}\\f_{12}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
x
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}x
\\
\texttt{(} \mathrm{d}x \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\mathrm{d}x
\\
\texttt{(} \mathrm{d}x \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{)}
\\
\mathrm{d}x
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{)}
\\
\mathrm{d}x
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{6}\\f_{9}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{,} ~~ y \texttt{)}
\\
\texttt{((} x \texttt{,} ~~ y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)}
\\
\texttt{((} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{))}
\\
\texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{))}
\\
\texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)}
\\
\texttt{((} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{))}
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{5}\\f_{10}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
y
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}y
\\
\texttt{(} \mathrm{d}y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}y \texttt{)}
\\
\mathrm{d}y
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\mathrm{d}y
\\
\texttt{(} \mathrm{d}y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}y \texttt{)}
\\
\mathrm{d}y
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{7}\\f_{11}\\f_{13}\\f_{14}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)}
\\
\texttt{(} ~~ x ~~ \texttt{(} y \texttt{))}
\\
\texttt{((} x \texttt{)} ~~ y ~~ \texttt{)}
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\\
\texttt{((} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~ \texttt{)}
\\
\texttt{(} ~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{))}
\\
\texttt{(} ~~ \mathrm{d}x ~ ~ ~ \mathrm{d}y ~~ \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~ \texttt{)}
\\
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\\
\texttt{(} ~~ \mathrm{d}x ~ ~ ~ \mathrm{d}y ~~ \texttt{)}
\\
\texttt{(} ~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{))}
\\
\texttt{(} ~~ \mathrm{d}x ~ ~ ~ \mathrm{d}y ~~ \texttt{)}
\\
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\\
\texttt{((} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~ \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ \mathrm{d}x ~ ~ ~ \mathrm{d}y ~~ \texttt{)}
\\
\texttt{(} ~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{))}
\\
\texttt{((} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~ \texttt{)}
\\
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{15}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>1</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
|}

<br>

====Table A6. D''f'' Expanded Over Ordinary Features====

<br>

{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table A6.} ~~ \mathrm{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math>
|- style="background:ghostwhite"
| style="width:10%; border-bottom:1px solid black" |  
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>f</math>
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |
<math>\mathrm{D}f|_{xy}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{D}f|_{x \texttt{(} y \texttt{)}}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{D}f|_{\texttt{(} x \texttt{)} y}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{D}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math>
|-
| style="border-top:1px solid black" | <math>f_{0}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)} ~~ y ~~
\\
~~ x ~~ \texttt{(} y \texttt{)}
\\
x ~ ~ ~ y
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}x ~ ~ ~ \mathrm{d}y
\\
~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)}
\\
\texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~
\\
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)}
\\
\mathrm{d}x ~ ~ ~ \mathrm{d}y
\\
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\\
\texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~
\\
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\\
\mathrm{d}x ~ ~ ~ \mathrm{d}y
\\
~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\\
\texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~
\\
~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)}
\\
\mathrm{d}x ~ ~ ~ \mathrm{d}y
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
x
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}x
\\
\mathrm{d}x
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\mathrm{d}x
\\
\mathrm{d}x
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\mathrm{d}x
\\
\mathrm{d}x
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\mathrm{d}x
\\
\mathrm{d}x
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{,} ~~ y \texttt{)}
\\
\texttt{((} x \texttt{,} ~~ y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)}
\\
\texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)}
\\
\texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)}
\\
\texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)}
\\
\texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)}
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
y
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}y
\\
\mathrm{d}y
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\mathrm{d}y
\\
\mathrm{d}y
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\mathrm{d}y
\\
\mathrm{d}y
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\mathrm{d}y
\\
\mathrm{d}y
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)}
\\
\texttt{(} ~~ x ~~ \texttt{(} y \texttt{))}
\\
\texttt{((} x \texttt{)} ~~ y ~~ \texttt{)}
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\\
\texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~
\\
~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)}
\\
\mathrm{d}x ~ ~ ~ \mathrm{d}y
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~
\\
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\\
\mathrm{d}x ~ ~ ~ \mathrm{d}y
\\
~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)}
\\
\mathrm{d}x ~ ~ ~ \mathrm{d}y
\\
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\\
\texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\mathrm{d}x ~ ~ ~ \mathrm{d}y
\\
~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)}
\\
\texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~
\\
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{15}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
|}

<br>

===Appendix 2. Differential Forms===

The actions of the difference operator <math>\mathrm{D}</math> and the tangent operator <math>\mathrm{d}</math> on the 16 bivariate propositions are shown in Tables A7 and A8.

Table A7 expands the differential forms that result over a ''logical basis'':

{| align="center" cellpadding="6" style="text-align:center"
|
<math>\{~ \texttt{(}\mathrm{d}x\texttt{)(}\mathrm{d}y\texttt{)}, ~\mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}, ~\texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.</math>
|}

This set consists of the singular propositions in the first order differential variables, indicating mutually exclusive and exhaustive ''cells'' of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the cell-basis, point-basis, or singular differential basis. In this setting it is frequently convenient to use the following abbreviations:

{| align="center" cellpadding="6" style="text-align:center"
|
<math>\partial x ~=~ \mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}</math>     and     <math>\partial y ~=~ \texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y.</math>
|}

Table A8 expands the differential forms that result over an ''algebraic basis'':

{| align="center" cellpadding="6" style="text-align:center"
| <math>\{~ 1, ~\mathrm{d}x, ~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.</math>
|}

This set consists of the ''positive propositions'' in the first order differential variables, indicating overlapping positive regions of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the ''positive differential basis''.

====Table A7. Differential Forms Expanded on a Logical Basis====

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\text{Table A7.} ~~ \text{Differential Forms Expanded on a Logical Basis}</math>
|- style="background:ghostwhite; height:40px"
|  
| style="border-right:none" | <math>f</math>
| style="border-left:4px double black" | <math>\mathrm{D}f</math>
| <math>\mathrm{d}f</math>
|-
| <math>f_{0}</math>
| style="border-right:none" | <math>\texttt{(} ~~ \texttt{)}</math>
| style="border-left:4px double black" | <math>0</math>
| <math>0</math>
|-
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math>
| style="border-right:none" |
<math>\begin{matrix}
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)} ~~ y ~~
\\
~~ x ~~ \texttt{(} y \texttt{)}
\\
x ~ ~ ~ y
\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
& + &
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
& + &
\texttt{((} x \texttt{,} ~~ y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y
\\
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
& + &
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
& + &
\texttt{(} x \texttt{,} ~~ y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y
\\
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
& + &
x & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
& + &
\texttt{(} x \texttt{,} ~~ y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y
\\
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
& + &
x & \texttt{(} \mathrm{d}x) ~ \mathrm{d}y
& + &
\texttt{((} x \texttt{,} ~~ y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} y \texttt{)} ~\partial x
& + &
\texttt{(} x \texttt{)} ~\partial y
\\
~~ y ~~ ~\partial x
& + &
\texttt{(} x \texttt{)} ~\partial y
\\
\texttt{(} y \texttt{)} ~\partial x
& + &
~~ x ~~ ~\partial y
\\
~~ y ~~ ~\partial x
& + &
~~ x ~~ ~\partial y
\end{matrix}</math>
|-
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
| style="border-right:none" |
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
x
\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \mathrm{d}x ~ \mathrm{d}y
\\
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \mathrm{d}x ~ \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\partial x
\\
\partial x
\end{matrix}</math>
|-
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
| style="border-right:none" |
<math>\begin{matrix}
\texttt{(} x \texttt{,} ~~ y \texttt{)}
\\
\texttt{((} x \texttt{,} ~~ y \texttt{))}
\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
\\
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\partial x & + & \partial y
\\
\partial x & + & \partial y
\end{matrix}</math>
|-
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
| style="border-right:none" |
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
y
\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \mathrm{d}x ~ \mathrm{d}y
\\
\texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \mathrm{d}x ~ \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\partial y
\\
\partial y
\end{matrix}</math>
|-
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
| style="border-right:none" |
<math>\begin{matrix}
\texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)}
\\
\texttt{(} ~~ x ~~ \texttt{(} y \texttt{))}
\\
\texttt{((} x \texttt{)} ~~ y ~~ \texttt{)}
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
& + &
x & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
& + &
\texttt{((} x \texttt{,} ~~ y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y
\\
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
& + &
x & \texttt{(} \mathrm{d}x) ~ \mathrm{d}y
& + &
\texttt{(} x \texttt{,} ~~ y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y
\\
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
& + &
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
& + &
\texttt{(} x \texttt{,} ~~ y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y
\\
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
& + &
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
& + &
\texttt{((} x \texttt{,} ~~ y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
~~ y ~~ ~\partial x
& + &
~~ x ~~ ~\partial y
\\
\texttt{(} y \texttt{)} ~\partial x
& + &
~~ x ~~ ~\partial y
\\
~~ y ~~ ~\partial x
& + &
\texttt{(} x \texttt{)} ~\partial y
\\
\texttt{(} y \texttt{)} ~\partial x
& + &
\texttt{(} x \texttt{)} ~\partial y
\end{matrix}</math>
|-
| <math>f_{15}</math>
| style="border-right:none" | <math>\texttt{((} ~~ \texttt{))}</math>
| style="border-left:4px double black" | <math>0</math>
| <math>0</math>
|}

<br>

====Table A8. Differential Forms Expanded on an Algebraic Basis====

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\text{Table A8.} ~~ \text{Differential Forms Expanded on an Algebraic Basis}</math>
|- style="background:ghostwhite; height:40px"
|  
| style="border-right:none" | <math>f</math>
| style="border-left:4px double black" | <math>\mathrm{D}f</math>
| <math>\mathrm{d}f</math>
|-
| <math>f_{0}</math>
| style="border-right:none" | <math>\texttt{(} ~~ \texttt{)}</math>
| style="border-left:4px double black" | <math>0</math>
| <math>0</math>
|-
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math>
| style="border-right:none" |
<math>\begin{matrix}
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)} ~~ y ~~
\\
~~ x ~~ \texttt{(} y \texttt{)}
\\
x ~ ~ ~ y
\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)} ~\mathrm{d}x & + & \texttt{(} x \texttt{)} ~\mathrm{d}y & + & \mathrm{d}x ~\mathrm{d}y
\\
~~ y ~~ ~\mathrm{d}x & + & \texttt{(} x \texttt{)} ~\mathrm{d}y & + & \mathrm{d}x ~\mathrm{d}y
\\
\texttt{(} y \texttt{)} ~\mathrm{d}x & + & ~~ x ~~ ~\mathrm{d}y & + & \mathrm{d}x ~\mathrm{d}y
\\
~~ y ~~ ~\mathrm{d}x & + & ~~ x ~~ ~\mathrm{d}y & + & \mathrm{d}x ~\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} y \texttt{)} ~\mathrm{d}x & + & \texttt{(} x \texttt{)} ~\mathrm{d}y
\\
~~ y ~~ ~\mathrm{d}x & + & \texttt{(} x \texttt{)} ~\mathrm{d}y
\\
\texttt{(} y \texttt{)} ~\mathrm{d}x & + & ~~ x ~~ ~\mathrm{d}y
\\
~~ y ~~ ~\mathrm{d}x & + & ~~ x ~~ ~\mathrm{d}y
\end{matrix}</math>
|-
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
| style="border-right:none" |
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
x
\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}x
\\
\mathrm{d}x
\end{matrix}</math>
| <math>\begin{matrix}
\mathrm{d}x
\\
\mathrm{d}x
\end{matrix}</math>
|-
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
| style="border-right:none" |
<math>\begin{matrix}
\texttt{(} x \texttt{,} ~~ y \texttt{)}
\\
\texttt{((} x \texttt{,} ~~ y \texttt{))}
\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}x & + & \mathrm{d}y
\\
\mathrm{d}x & + & \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x & + & \mathrm{d}y
\\
\mathrm{d}x & + & \mathrm{d}y
\end{matrix}</math>
|-
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
| style="border-right:none" |
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
y
\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}y
\\
\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}y
\\
\mathrm{d}y
\end{matrix}</math>
|-
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
| style="border-right:none" |
<math>\begin{matrix}
\texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)}
\\
\texttt{(} ~~ x ~~ \texttt{(} y \texttt{))}
\\
\texttt{((} x \texttt{)} ~~ y ~~ \texttt{)}
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
~~ y ~~ ~\mathrm{d}x & + & ~~ x ~~ ~\mathrm{d}y & + & \mathrm{d}x ~\mathrm{d}y
\\
\texttt{(} y \texttt{)} ~\mathrm{d}x & + & ~~ x ~~ ~\mathrm{d}y & + & \mathrm{d}x ~\mathrm{d}y
\\
~~ y ~~ ~\mathrm{d}x & + & \texttt{(} x \texttt{)} ~\mathrm{d}y & + & \mathrm{d}x ~\mathrm{d}y
\\
\texttt{(} y \texttt{)} ~\mathrm{d}x & + & \texttt{(} x \texttt{)} ~\mathrm{d}y & + & \mathrm{d}x ~\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
~~ y ~~ ~\mathrm{d}x & + & ~~ x ~~ ~\mathrm{d}y
\\
\texttt{(} y \texttt{)} ~\mathrm{d}x & + & ~~ x ~~ ~\mathrm{d}y
\\
~~ y ~~ ~\mathrm{d}x & + & \texttt{(} x \texttt{)} ~\mathrm{d}y
\\
\texttt{(} y \texttt{)} ~\mathrm{d}x & + & \texttt{(} x \texttt{)} ~\mathrm{d}y
\end{matrix}</math>
|-
| <math>f_{15}</math>
| style="border-right:none" | <math>\texttt{((} ~~ \texttt{))}</math>
| style="border-left:4px double black" | <math>0</math>
| <math>0</math>
|}

<br>

====Table A9. Tangent Proposition as Pointwise Linear Approximation====

<br>

{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\text{Table A9.} ~~ \text{Tangent Proposition}~ \mathrm{d}f = \text{Pointwise Linear Approximation to the Difference Map}~ \mathrm{D}f</math>
|- style="background:ghostwhite; height:40px"
| style="border-right:none" | <math>f</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}f =
\\[2pt]
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}^2\!f =
\\[2pt]
\partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y
\end{matrix}</math>
| <math>\mathrm{d}f|_{x \, y}</math>
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math>
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}</math>
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math>
|-
| style="border-right:none" | <math>f_0</math>
| style="border-left:4px double black" | <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
|-
| style="border-right:none" |
<math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
\\
~~ y ~~ \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & ~~ x ~~ \cdot \mathrm{d}y
\\
~~ y ~~ \cdot \mathrm{d}x & + & ~~ x ~~ \cdot \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x\;\mathrm{d}y
\\
\mathrm{d}x\;\mathrm{d}y
\\
\mathrm{d}x\;\mathrm{d}y
\\
\mathrm{d}x\;\mathrm{d}y
\end{matrix}</math>
| <math>\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}</math>
|-
| style="border-right:none" |
<math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
| <math>\begin{matrix}0\\0\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
|-
| style="border-right:none" |
<math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}0\\0\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math>
|-
| style="border-right:none" |
<math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}0\\0\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
|-
| style="border-right:none" |
<math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
~~ y ~~ \cdot \mathrm{d}x & + & ~~ x ~~ \cdot \mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & ~~ x ~~ \cdot \mathrm{d}y
\\
~~ y ~~ \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
\end{matrix}</math>
| <math>\begin{matrix}
\mathrm{d}x\;\mathrm{d}y
\\
\mathrm{d}x\;\mathrm{d}y
\\
\mathrm{d}x\;\mathrm{d}y
\\
\mathrm{d}x\;\mathrm{d}y
\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math>
|-
| style="border-right:none" | <math>f_{15}</math>
| style="border-left:4px double black" | <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
|}

<br>

====Table A10. Taylor Series Expansion D''f'' = d''f'' + d²''f''====

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" |
<math>\text{Table A10.} ~~ \text{Taylor Series Expansion}~ {\mathrm{D}f = \mathrm{d}f + \mathrm{d}^2\!f}</math>
|- style="background:ghostwhite; height:40px"
| style="border-right:none" | <math>f</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\mathrm{D}f
\\
= & \mathrm{d}f & + & \mathrm{d}^2\!f
\\
= & \partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y & + & \partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y
\end{matrix}</math>
| <math>\mathrm{d}f|_{x \, y}</math>
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math>
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}</math>
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math>
|-
| style="border-right:none" | <math>f_0</math>
| style="border-left:4px double black" | <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
|-
| style="border-right:none" | <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &
~~ 1 ~~ \cdot \mathrm{d}x\;\mathrm{d}y
\\
~~ y ~~ \cdot \mathrm{d}x & + &
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &
~~ 1 ~~ \cdot \mathrm{d}x\;\mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &
~~ x ~~ \cdot \mathrm{d}y & + &
~~ 1 ~~ \cdot \mathrm{d}x\;\mathrm{d}y
\\
~~ y ~~ \cdot \mathrm{d}x & + &
~~ x ~~ \cdot \mathrm{d}y & + &
~~ 1 ~~ \cdot \mathrm{d}x\;\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0
\end{matrix}</math>
|-
| style="border-right:none" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
~~ 1 ~~ \cdot \mathrm{d}x & + &
~~ 0 ~~ \cdot \mathrm{d}y & + &
~~ 0 ~~ \cdot \mathrm{d}x\;\mathrm{d}y
\\
~~ 1 ~~ \cdot \mathrm{d}x & + &
~~ 0 ~~ \cdot \mathrm{d}y & + &
~~ 0 ~~ \cdot \mathrm{d}x\;\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x\\\mathrm{d}x
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x\\\mathrm{d}x
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x\\\mathrm{d}x
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x\\\mathrm{d}x
\end{matrix}</math>
|-
| style="border-right:none" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
~~ 1 ~~ \cdot \mathrm{d}x & + &
~~ 1 ~~ \cdot \mathrm{d}y & + &
~~ 0 ~~ \cdot \mathrm{d}x\;\mathrm{d}y
\\
~~ 1 ~~ \cdot \mathrm{d}x & + &
~~ 1 ~~ \cdot \mathrm{d}y & + &
~~ 0 ~~ \cdot \mathrm{d}x\;\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
\end{matrix}</math>
|-
| style="border-right:none" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
~~ 0 ~~ \cdot \mathrm{d}x & + &
~~ 1 ~~ \cdot \mathrm{d}y & + &
~~ 0 ~~ \cdot \mathrm{d}x\;\mathrm{d}y
\\
~~ 0 ~~ \cdot \mathrm{d}x & + &
~~ 1 ~~ \cdot \mathrm{d}y & + &
~~ 0 ~~ \cdot \mathrm{d}x\;\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}y\\\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}y\\\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}y\\\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}y\\\mathrm{d}y
\end{matrix}</math>
|-
| style="border-right:none" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
~~ y ~~ \cdot \mathrm{d}x & + &
~~ x ~~ \cdot \mathrm{d}y & + &
~~ 1 ~~ \cdot \mathrm{d}x\;\mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &
~~ x ~~ \cdot \mathrm{d}y & + &
~~ 1 ~~ \cdot \mathrm{d}x\;\mathrm{d}y
\\
~~ y ~~ \cdot \mathrm{d}x & + &
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &
~~ 1 ~~ \cdot \mathrm{d}x\;\mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &
~~ 1 ~~ \cdot \mathrm{d}x\;\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
\end{matrix}</math>
|-
| style="border-right:none" | <math>f_{15}</math>
| style="border-left:4px double black" | <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
|}

<br>

====Table A11. Partial Differentials and Relative Differentials====

<br>

{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\text{Table A11.} ~~ \text{Partial Differentials and Relative Differentials}</math>
|- style="background:ghostwhite; height:50px"
|  
| <math>f</math>
| <math>\frac{\partial f}{\partial x}</math>
| <math>\frac{\partial f}{\partial y}</math>
|
<math>\begin{matrix}
\mathrm{d}f =
\\[2pt]
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y
\end{matrix}</math>
| <math>\left. \frac{\partial x}{\partial y} \right| f</math>
| <math>\left. \frac{\partial y}{\partial x} \right| f</math>
|-
| <math>f_0</math>
| <math>\texttt{(} ~ \texttt{)}</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
|-
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)} ~~ y ~~
\\
~~ x ~~ \texttt{(} y \texttt{)}
\\
x ~ ~ ~ y
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
y
\\
\texttt{(} y \texttt{)}
\\
y
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
\texttt{(} x \texttt{)}
\\
x
\\
x
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
\\
~~ y ~~ \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & ~~ x ~~ \cdot \mathrm{d}y
\\
~~ y ~~ \cdot \mathrm{d}x & + & ~~ x ~~ \cdot \mathrm{d}y
\end{matrix}</math>
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math>
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math>
|-
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
x
\end{matrix}</math>
| <math>\begin{matrix}1\\1\end{matrix}</math>
| <math>\begin{matrix}0\\0\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math>
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math>
|-
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} x \texttt{,} ~~ y \texttt{)}
\\
\texttt{((} x \texttt{,} ~~ y \texttt{))}
\end{matrix}</math>
| <math>\begin{matrix}1\\1\end{matrix}</math>
| <math>\begin{matrix}1\\1\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math>
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math>
|-
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
y
\end{matrix}</math>
| <math>\begin{matrix}0\\0\end{matrix}</math>
| <math>\begin{matrix}1\\1\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math>
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math>
|-
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)}
\\
\texttt{(} ~~ x ~~ \texttt{(} y \texttt{))}
\\
\texttt{((} x \texttt{)} ~~ y ~~ \texttt{)}
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\end{matrix}</math>
|
<math>\begin{matrix}
y
\\
\texttt{(} y \texttt{)}
\\
y
\\
\texttt{(} y \texttt{)}
\end{matrix}</math>
|
<math>\begin{matrix}
x
\\
x
\\
\texttt{(} x \texttt{)}
\\
\texttt{(} x \texttt{)}
\end{matrix}</math>
|
<math>\begin{matrix}
~~ y ~~ \cdot \mathrm{d}x & + & ~~ x ~~ \cdot \mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & ~~ x ~~ \cdot \mathrm{d}y
\\
~~ y ~~ \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
\end{matrix}</math>
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math>
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math>
|-
| <math>f_{15}</math>
| <math>\texttt{((} ~ \texttt{))}</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
|}

<br>

====Table A12. Detail of Calculation for the Difference Map====

<br>

{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:4px double black; border-left:4px double black; border-right:4px double black; border-top:4px double black; text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table A12.} ~~ \text{Detail of Calculation for}~ {\mathrm{E}f + f = \mathrm{D}f}</math>
|- style="background:ghostwhite"
| style="width:6%" |  
| style="width:14%; border-left:1px solid black" | <math>f</math>
| style="width:20%; border-left:4px double black" |
<math>\begin{array}{cr}
~ & \mathrm{E}f|_{\mathrm{d}x ~ \mathrm{d}y}
\\[4pt]
+ & f|_{\mathrm{d}x ~ \mathrm{d}y}
\\[4pt]
= & \mathrm{D}f|_{\mathrm{d}x ~ \mathrm{d}y}
\end{array}</math>
| style="width:20%; border-left:1px solid black" |
<math>\begin{array}{cr}
~ & \mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}
\\[4pt]
+ & f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}
\\[4pt]
= & \mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}
\end{array}</math>
| style="width:20%; border-left:1px solid black" |
<math>\begin{array}{cr}
~ & \mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}
\\[4pt]
+ & f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}
\\[4pt]
= & \mathrm{D}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}
\end{array}</math>
| style="width:20%; border-left:1px solid black" |
<math>\begin{array}{cr}
~ & \mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}
\\[4pt]
+ & f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}
\\[4pt]
= & \mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}
\end{array}</math>
|-
| style="border-top:4px double black" | <math>f_{0}</math>
| style="border-top:4px double black; border-left:1px solid black" | <math>0</math>
| style="border-top:4px double black; border-left:4px double black" | <math>0 ~+~ 0 ~=~ 0</math>
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0</math>
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0</math>
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0</math>
|-
| style="border-top:4px double black" | <math>f_{1}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\texttt{(} x \texttt{)(} y \texttt{)}</math>
| style="border-top:4px double black; border-left:4px double black" |
<math>\begin{matrix}
~ & ~ ~ ~ x ~ ~ ~ y ~ ~ ~
\\[4pt]
+ & ~~ \texttt{(} x \texttt{)(} y \texttt{)} ~~
\\[4pt]
= & \texttt{((} x \texttt{,} ~~ y \texttt{))}
\end{matrix}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~ ~ ~ x ~~ \texttt{(} y \texttt{)} ~~
\\[4pt]
+ & ~~ \texttt{(} x \texttt{)(} y \texttt{)} ~~
\\[4pt]
= & ~ ~ ~ ~ ~ ~ \texttt{(} y \texttt{)} ~~
\end{matrix}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~~ \texttt{(} x \texttt{)} ~~ y ~ ~ ~
\\[4pt]
+ & ~~ \texttt{(} x \texttt{)(} y \texttt{)} ~~
\\[4pt]
= & ~~ \texttt{(} x \texttt{)} ~ ~ ~ ~ ~ ~
\end{matrix}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~~ \texttt{(} x \texttt{)(} y \texttt{)} ~~
\\[4pt]
+ & ~~ \texttt{(} x \texttt{)(} y \texttt{)} ~~
\\[4pt]
= & 0
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{2}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\texttt{(} x \texttt{)} ~~ y ~~</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
~ & ~ ~ ~ x ~~ \texttt{(} y \texttt{)} ~~
\\[4pt]
+ & ~~ \texttt{(} x \texttt{)} ~~ y ~ ~ ~
\\[4pt]
= & ~~ \texttt{(} x \texttt{,} ~~ y \texttt{)} ~~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~ ~ ~ x ~ ~ ~ y ~ ~ ~
\\[4pt]
+ & ~~ \texttt{(} x \texttt{)} ~~ y ~ ~ ~
\\[4pt]
= & ~ ~ ~ ~~ ~ ~ ~ y ~ ~ ~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~~ \texttt{(} x \texttt{)(} y \texttt{)} ~~
\\[4pt]
+ & ~~ \texttt{(} x \texttt{)} ~~ y ~ ~ ~
\\[4pt]
= & ~~ \texttt{(} x \texttt{)} ~ ~ ~ ~ ~ ~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~~ \texttt{(} x \texttt{)} ~~ y ~ ~ ~
\\[4pt]
+ & ~~ \texttt{(} x \texttt{)} ~~ y ~ ~ ~
\\[4pt]
= & 0
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{4}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>~~ x ~~ \texttt{(} y \texttt{)}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
~ & ~~ \texttt{(} x \texttt{)} ~~ y ~ ~ ~
\\[4pt]
+ & ~ ~ ~ x ~~ \texttt{(} y \texttt{)} ~~
\\[4pt]
= & ~~ \texttt{(} x \texttt{,} ~~ y \texttt{)} ~~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~~ \texttt{(} x \texttt{)(} y \texttt{)} ~~
\\[4pt]
+ & ~ ~ ~ x ~~ \texttt{(} y \texttt{)} ~~
\\[4pt]
= & ~ ~ ~ ~ ~ ~ \texttt{(} y \texttt{)} ~~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~ ~ ~ x ~ ~ ~ y ~ ~ ~
\\[4pt]
+ & ~ ~ ~ x ~~ \texttt{(} y \texttt{)} ~~
\\[4pt]
= & ~ ~ ~ x ~ ~ ~ ~~ ~ ~ ~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~ ~ ~ x ~~ \texttt{(} y \texttt{)} ~~
\\[4pt]
+ & ~ ~ ~ x ~~ \texttt{(} y \texttt{)} ~~
\\[4pt]
= & 0
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{8}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>x ~ ~ ~ y</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
~ & ~~ \texttt{(} x \texttt{)(} y \texttt{)} ~~
\\[4pt]
+ & ~ ~ ~ x ~ ~ ~ y ~ ~ ~
\\[4pt]
= & \texttt{((} x \texttt{,} ~~ y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~~ \texttt{(} x \texttt{)} ~~ y ~ ~ ~
\\[4pt]
+ & ~ ~ ~ x ~ ~ ~ y ~ ~ ~
\\[4pt]
= & ~ ~ ~ ~~ ~ ~ ~ y ~ ~ ~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~ ~ ~ x ~~ \texttt{(} y \texttt{)} ~~
\\[4pt]
+ & ~ ~ ~ x ~ ~ ~ y ~ ~ ~
\\[4pt]
= & ~ ~ ~ x ~ ~ ~ ~~ ~ ~ ~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~ ~ ~ x ~ ~ ~ y ~ ~ ~
\\[4pt]
+ & ~ ~ ~ x ~ ~ ~ y ~ ~ ~
\\[4pt]
= & 0
\end{matrix}</math>
|-
| style="border-top:4px double black" | <math>f_{3}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\texttt{(} x \texttt{)}</math>
| style="border-top:4px double black; border-left:4px double black" |
<math>\begin{matrix}
~ & x
\\[4pt]
+ & \texttt{(} x \texttt{)}
\\[4pt]
= & 1
\end{matrix}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
~ & x
\\[4pt]
+ & \texttt{(} x \texttt{)}
\\[4pt]
= & 1
\end{matrix}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{(} x \texttt{)}
\\[4pt]
+ & \texttt{(} x \texttt{)}
\\[4pt]
= & 0
\end{matrix}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{(} x \texttt{)}
\\[4pt]
+ & \texttt{(} x \texttt{)}
\\[4pt]
= & 0
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{12}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>x</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
~ & \texttt{(} x \texttt{)}
\\[4pt]
+ & x
\\[4pt]
= & 1
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{(} x \texttt{)}
\\[4pt]
+ & x
\\[4pt]
= & 1
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~~ x ~~
\\[4pt]
+ & x
\\[4pt]
= & 0
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~~ x ~~
\\[4pt]
+ & x
\\[4pt]
= & 0
\end{matrix}</math>
|-
| style="border-top:4px double black" | <math>f_{6}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\texttt{(} x \texttt{,} ~~ y \texttt{)}</math>
| style="border-top:4px double black; border-left:4px double black" |
<math>\begin{matrix}
~ & ~~ \texttt{(} x \texttt{,} ~~ y \texttt{)} ~~
\\[4pt]
+ & ~~ \texttt{(} x \texttt{,} ~~ y \texttt{)} ~~
\\[4pt]
= & 0
\end{matrix}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{((} x \texttt{,} ~~ y \texttt{))}
\\[4pt]
+ & ~~ \texttt{(} x \texttt{,} ~~ y \texttt{)} ~~
\\[4pt]
= & 1
\end{matrix}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{((} x \texttt{,} ~~ y \texttt{))}
\\[4pt]
+ & ~~ \texttt{(} x \texttt{,} ~~ y \texttt{)} ~~
\\[4pt]
= & 1
\end{matrix}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~~ \texttt{(} x \texttt{,} ~~ y \texttt{)} ~~
\\[4pt]
+ & ~~ \texttt{(} x \texttt{,} ~~ y \texttt{)} ~~
\\[4pt]
= & 0
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{9}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\texttt{((} x \texttt{,} ~~ y \texttt{))}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
~ & \texttt{((} x \texttt{,} ~~ y \texttt{))}
\\[4pt]
+ & \texttt{((} x \texttt{,} ~~ y \texttt{))}
\\[4pt]
= & 0
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~~ \texttt{(} x \texttt{,} ~~ y \texttt{)} ~~
\\[4pt]
+ & \texttt{((} x \texttt{,} ~~ y \texttt{))}
\\[4pt]
= & 1
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~~ \texttt{(} x \texttt{,} ~~ y \texttt{)} ~~
\\[4pt]
+ & \texttt{((} x \texttt{,} ~~ y \texttt{))}
\\[4pt]
= & 1
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{((} x \texttt{,} ~~ y \texttt{))}
\\[4pt]
+ & \texttt{((} x \texttt{,} ~~ y \texttt{))}
\\[4pt]
= & 0
\end{matrix}</math>
|-
| style="border-top:4px double black" | <math>f_{5}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\texttt{(} y \texttt{)}</math>
| style="border-top:4px double black; border-left:4px double black" |
<math>\begin{matrix}
~ & y
\\[4pt]
+ & \texttt{(} y \texttt{)}
\\[4pt]
= & 1
\end{matrix}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{(} y \texttt{)}
\\[4pt]
+ & \texttt{(} y \texttt{)}
\\[4pt]
= & 0
\end{matrix}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
~ & y
\\[4pt]
+ & \texttt{(} y \texttt{)}
\\[4pt]
= & 1
\end{matrix}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{(} y \texttt{)}
\\[4pt]
+ & \texttt{(} y \texttt{)}
\\[4pt]
= & 0
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{10}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>y</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
~ & \texttt{(} y \texttt{)}
\\[4pt]
+ & y
\\[4pt]
= & 1
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~~ y ~~
\\[4pt]
+ & y
\\[4pt]
= & 0
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{(} y \texttt{)}
\\[4pt]
+ & y
\\[4pt]
= & 1
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~~ y ~~
\\[4pt]
+ & y
\\[4pt]
= & 0
\end{matrix}</math>
|-
| style="border-top:4px double black" | <math>f_{7}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)}</math>
| style="border-top:4px double black; border-left:4px double black" |
<math>\begin{matrix}
~ & \texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
+ & ~~ \texttt{(} x ~ ~ ~ y \texttt{)} ~~
\\[4pt]
= & \texttt{((} x \texttt{,} ~~ y \texttt{))}
\end{matrix}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{((} x \texttt{)} ~~ y \texttt{)} ~~
\\[4pt]
+ & ~~ \texttt{(} x ~ ~ ~ y \texttt{)} ~~
\\[4pt]
= & ~ ~ ~ ~~ ~ ~ ~ y ~ ~ ~
\end{matrix}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~~ \texttt{(} x ~~ \texttt{(} y \texttt{))}
\\[4pt]
+ & ~~ \texttt{(} x ~ ~ ~ y \texttt{)} ~~
\\[4pt]
= & ~ ~ ~ x ~ ~ ~ ~~ ~ ~ ~
\end{matrix}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~~ \texttt{(} x ~ ~ ~ y \texttt{)} ~~
\\[4pt]
+ & ~~ \texttt{(} x ~ ~ ~ y \texttt{)} ~~
\\[4pt]
= & 0
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{11}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\texttt{(} ~~ x ~~ \texttt{(} y \texttt{))}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
~ & \texttt{((} x \texttt{)} ~~ y \texttt{)} ~~
\\[4pt]
+ & ~~ \texttt{(} x ~~ \texttt{(} y \texttt{))}
\\[4pt]
= & ~~ \texttt{(} x \texttt{,} ~~ y \texttt{)} ~~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
+ & ~~ \texttt{(} x ~~ \texttt{(} y \texttt{))}
\\[4pt]
= & ~ ~ ~ ~ ~ ~ \texttt{(} y \texttt{)} ~~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~~ \texttt{(} x ~ ~ ~ y \texttt{)} ~~
\\[4pt]
+ & ~~ \texttt{(} x ~~ \texttt{(} y \texttt{))}
\\[4pt]
= & ~ ~ ~ x ~ ~ ~ ~~ ~ ~ ~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~~ \texttt{(} x ~~ \texttt{(} y \texttt{))}
\\[4pt]
+ & ~~ \texttt{(} x ~~ \texttt{(} y \texttt{))}
\\[4pt]
= & 0
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{13}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\texttt{((} x \texttt{)} ~~ y ~~ \texttt{)}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
~ & ~~ \texttt{(} x ~~ \texttt{(} y \texttt{))}
\\[4pt]
+ & \texttt{((} x \texttt{)} ~~ y \texttt{)} ~~
\\[4pt]
= & ~~ \texttt{(} x \texttt{,} ~~ y \texttt{)} ~~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~~ \texttt{(} x ~ ~ ~ y \texttt{)} ~~
\\[4pt]
+ & \texttt{((} x \texttt{)} ~~ y \texttt{)} ~~
\\[4pt]
= & ~ ~ ~ ~~ ~ ~ ~ y ~ ~ ~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
+ & \texttt{((} x \texttt{)} ~~ y \texttt{)} ~~
\\[4pt]
= & ~~ \texttt{(} x \texttt{)} ~ ~ ~ ~ ~ ~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{((} x \texttt{)} ~~ y \texttt{)} ~~
\\[4pt]
+ & \texttt{((} x \texttt{)} ~~ y \texttt{)} ~~
\\[4pt]
= & 0
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{14}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\texttt{((} x \texttt{)(} y \texttt{))}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
~ & ~~ \texttt{(} x ~ ~ ~ y \texttt{)} ~~
\\[4pt]
+ & \texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
= & \texttt{((} x \texttt{,} ~~ y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~~ \texttt{(} x ~~ \texttt{(} y \texttt{))}
\\[4pt]
+ & \texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
= & ~ ~ ~ ~ ~ ~ \texttt{(} y \texttt{)} ~~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{((} x \texttt{)} ~~ y \texttt{)} ~~
\\[4pt]
+ & \texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
= & ~~ \texttt{(} x \texttt{)} ~ ~ ~ ~ ~ ~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
+ & \texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
= & 0
\end{matrix}</math>
|-
| style="border-top:4px double black" | <math>f_{15}</math>
| style="border-top:4px double black; border-left:1px solid black" | <math>1</math>
| style="border-top:4px double black; border-left:4px double black" | <math>1 ~+~ 1 ~=~ 0</math>
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0</math>
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0</math>
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0</math>
|}

<br>

===Appendix 3. Computational Details===

====Operator Maps for the Logical Conjunction ''f''<sub>8</sub>⟨''u'', ''v''⟩====

=====Computation of ε''f''<sub>8</sub>=====

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F8.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{8}</math>
|
<math>\begin{array}{*{10}{l}}
\boldsymbol\varepsilon f_{8}
& = && f_{8} \langle u, v \rangle
\\[4pt]
& = && u v
\\[4pt]
& = && u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
& + & u v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
& + & u v \cdot \mathrm{d}u ~ \mathrm{d}v
\\[20pt]
\boldsymbol\varepsilon f_{8}
& = && u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v ~~
\\[4pt]
&& + & u v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & u v \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v ~~
\end{array}</math>
|}

<br>

=====Computation of E''f''<sub>8</sub>=====

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F8.2-i} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 1)}</math>
|
<math>\begin{array}{*{9}{l}}
\mathrm{E}f_{8}
& = & f_{8} \langle u + \mathrm{d}u, v + \mathrm{d}v \rangle
\\[4pt]
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}
\\[4pt]
& = & u v \cdot f_{8} \langle \texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)} \rangle
& + & u \texttt{(} v \texttt{)} \cdot f_{8} \langle \texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v \rangle
& + & \texttt{(} u \texttt{)} v \cdot f_{8} \langle \mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)} \rangle
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{8} \langle \mathrm{d}u, \mathrm{d}v \rangle
\\[4pt]
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\\[20pt]
\mathrm{E}f_{8}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
\\[4pt]
&&& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
\\[4pt]
&&&&& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
\\[4pt]
&&&&&&& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\end{array}</math>
|}

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F8.2-ii} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 2)}</math>
|
<math>\begin{array}{*{9}{c}}
\mathrm{E}f_{8}
& = & \{ u + \mathrm{d}u \} \cdot \{ v + \mathrm{d}v \}
\\[6pt]
& = & u \cdot v
& + & u \cdot \mathrm{d}v
& + & v \cdot \mathrm{d}u
& + & \mathrm{d}u \cdot \mathrm{d}v
\\[6pt]
\mathrm{E}f_{8}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\end{array}</math>
|}

<br>

=====Computation of D''f''<sub>8</sub>=====

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F8.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 1)}</math>
|
<math>\begin{array}{*{10}{l}}
\mathrm{D}f_{8}
& = && \mathrm{E}f_{8}
& + & \boldsymbol\varepsilon f_{8}
\\[4pt]
& = && f_{8} \langle u + \mathrm{d}u, v + \mathrm{d}v \rangle
& + & f_{8} \langle u, v \rangle
\\[4pt]
& = && \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}
& + & u v
\\[20pt]
\mathrm{D}f_{8}
& = && 0
& + & 0
& + & 0
& + & 0
\\[4pt]
&& + & u v \cdot ~~ \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v ~ ~ ~
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v ~~
& + & 0
& + & 0
\\[4pt]
&& + & u v \cdot ~ ~ ~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{} ~~
& + & 0
& + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & 0
\\[4pt]
&& + & u v \cdot ~ ~ ~ \mathrm{d}u ~ ~ ~ \mathrm{d}v ~ ~ ~
& + & 0
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~~ \mathrm{d}v ~~
\\[20pt]
\mathrm{D}f_{8}
& = && u v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v ~~
& + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~~ \mathrm{d}v ~~
\end{array}</math>
|}

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F8.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 2)}</math>
|
<math>\begin{array}{*{9}{l}}
\mathrm{D}f_{8}
& = & \boldsymbol\varepsilon f_{8}
& + & \mathrm{E}f_{8}
\\[6pt]
& = & f_{8} \langle u, v \rangle
& + & f_{8} \langle u + \mathrm{d}u, v + \mathrm{d}v \rangle
\\[6pt]
& = & u v
& + & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}
\\[6pt]
& = & 0
& + & u \cdot \mathrm{d}v
& + & v \cdot \mathrm{d}u
& + & \mathrm{d}u ~ \mathrm{d}v
\\[6pt]
\mathrm{D}f_{8}
& = & 0
& + & u \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
& + & v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v
\end{array}</math>
|}

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F8.3-iii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 3)}</math>
|
<math>\begin{array}{c*{9}{l}}
\mathrm{D}f_{8}
& = & \boldsymbol\varepsilon f_{8} ~+~ \mathrm{E}f_{8}
\\[20pt]
\boldsymbol\varepsilon f_{8}
& = & ~~ u ~~ v ~~ \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}
& + & ~~ u ~~ v ~~ \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & ~~ u ~~ v ~~ \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & ~ ~ ~ u ~ ~ ~ v ~ ~ ~ \cdot \mathrm{d}u ~~ \mathrm{d}v
\\[6pt]
\mathrm{E}f_{8}
& = & ~~ u ~~ v ~~ \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}
& + & ~~ u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & \texttt{(} u \texttt{)} v ~~ \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & ~~ \texttt{(} u \texttt{)(} v \texttt{)} ~~ \cdot \mathrm{d}u ~~ \mathrm{d}v
\\[20pt]
\mathrm{D}f_{8}
& = & ~ ~ ~ 0 ~ ~ ~ \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}
& + & ~ ~ ~ u ~ ~ ~ \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & ~ ~ ~ v ~ ~ ~ \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{((} u \texttt{,} ~~ v \texttt{))} \cdot \mathrm{d}u ~~ \mathrm{d}v
\end{array}</math>
|}

=====Computation of d''f''<sub>8</sub>=====

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F8.4} ~~ \text{Computation of}~ \mathrm{d}f_{8}</math>
|
<math>\begin{array}{c*{8}{l}}
\mathrm{D}f_{8}
& = & u v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\\[6pt]
\Downarrow
\\[6pt]
\mathrm{d}f_{8}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
\end{array}</math>
|}

<br>

=====Computation of r''f''<sub>8</sub>=====

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F8.5} ~~ \text{Computation of}~ \mathrm{r}f_{8}</math>
|
<math>\begin{array}{c*{8}{l}}
\mathrm{r}f_{8} & = & \mathrm{D}f_{8} ~+~ \mathrm{d}f_{8}
\\[20pt]
\mathrm{D}f_{8}
& = & u v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\\[6pt]
\mathrm{d}f_{8}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
\\[20pt]
\mathrm{r}f_{8}
& = & u v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\end{array}</math>
|}

<br>

=====Computation Summary for Conjunction=====

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F8.6} ~~ \text{Computation Summary for}~ f_{8} \langle u, v \rangle = u v</math>
|
<math>\begin{array}{c*{8}{l}}
\boldsymbol\varepsilon f_{8}
& = & u v \cdot 1
& + & u \texttt{(} v \texttt{)} \cdot 0
& + & \texttt{(} u \texttt{)} v \cdot 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
\\[6pt]
\mathrm{E}f_{8}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\\[6pt]
\mathrm{D}f_{8}
& = & u v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\\[6pt]
\mathrm{d}f_{8}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
\\[6pt]
\mathrm{r}f_{8}
& = & u v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\end{array}</math>
|}

<br>

====Operator Maps for the Logical Equality ''f''<sub>9</sub>⟨''u'', ''v''⟩====

=====Computation of ε''f''<sub>9</sub>=====

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F9.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{9}</math>
|
<math>\begin{array}{*{10}{l}}
\boldsymbol\varepsilon f_{9}
& = && f_{9} \langle u, v \rangle
\\[4pt]
& = && \texttt{((} u \texttt{,} ~~ v \texttt{))}
\\[4pt]
& = && u v \cdot f_{9} \langle 1, 1 \rangle
& + & u \texttt{(} v \texttt{)} \cdot f_{9} \langle 1, 0 \rangle
& + & \texttt{(} u \texttt{)} v \cdot f_{9} \langle 0, 1 \rangle
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{9} \langle 0, 0 \rangle
\\[4pt]
& = && u v & + & 0 & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)}
\\[20pt]
\boldsymbol\varepsilon f_{9}
& = && u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & 0
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
& + & 0
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
\\[4pt]
&& + & u v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & 0
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & u v \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
& + & 0
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
\end{array}</math>
|}

<br>

=====Computation of E''f''<sub>9</sub>=====

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F9.2} ~~ \text{Computation of}~ \mathrm{E}f_{9}</math>
|
<math>\begin{array}{*{10}{l}}
\mathrm{E}f_{9}
& = && f_{9} \langle u + \mathrm{d}u, v + \mathrm{d}v \rangle
\\[4pt]
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))}
\\[4pt]
& = && u v \cdot f_{9} \langle \texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)} \rangle
& + & u \texttt{(} v \texttt{)} \cdot f_{9} \langle \texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v \rangle
& + & \texttt{(} u \texttt{)} v \cdot f_{9} \langle \mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)} \rangle
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{9} \langle \mathrm{d}u, \mathrm{d}v \rangle
\\[4pt]
& = && u v \cdot \texttt{((} \mathrm{d}u \texttt{,} ~ \mathrm{d}v \texttt{))}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} ~ \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} ~ \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{,} ~ \mathrm{d}v \texttt{))}
\\[20pt]
\mathrm{E}f_{9}
& = && u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & 0
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & 0
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
& + & 0
\\[4pt]
&& + & 0
& + & u \texttt{(} v \texttt{)} \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & 0
\\[4pt]
&& + & u v \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
& + & 0
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
\end{array}</math>
|}

<br>

=====Computation of D''f''<sub>9</sub>=====

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F9.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 1)}</math>
|
<math>\begin{array}{*{10}{l}}
\mathrm{D}f_{9}
& = && \mathrm{E}f_{9}
& + & \boldsymbol\varepsilon f_{9}
\\[4pt]
& = && f_{9} \langle u + \mathrm{d}u, v + \mathrm{d}v \rangle
& + & f_{9} \langle u, v \rangle
\\[4pt]
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))}
& + & \texttt{((} u \texttt{,} v \texttt{))}
\\[20pt]
\mathrm{D}f_{9}
& = && 0
& + & 0
& + & 0
& + & 0
\\[4pt]
&& + & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
\\[4pt]
&& + & u v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & 0
& + & 0
& + & 0
& + & 0
\\[20pt]
\mathrm{D}f_{9}
& = && u v \cdot \texttt{(} \mathrm{d}u \texttt{,} ~~ \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} ~~ \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} ~~ \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} ~~ \mathrm{d}v \texttt{)}
\end{array}</math>
|}

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F9.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 2)}</math>
|
<math>\begin{array}{*{9}{l}}
\mathrm{D}f_{9}
& = & 0 \cdot \mathrm{d}u ~ \mathrm{d}v
& + & 1 \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
& + & 1 \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
& + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
\end{array}</math>
|}

<br>

=====Computation of d''f''<sub>9</sub>=====

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F9.4} ~~ \text{Computation of}~ \mathrm{d}f_{9}</math>
|
<math>\begin{array}{c*{8}{l}}
\mathrm{D}f_{9}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
\\[6pt]
\Downarrow
\\[6pt]
\mathrm{d}f_{9}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
\end{array}</math>
|}

<br>

=====Computation of r''f''<sub>9</sub>=====

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F9.5} ~~ \text{Computation of}~ \mathrm{r}f_{9}</math>
|
<math>\begin{array}{c*{8}{l}}
\mathrm{r}f_{9} & = & \mathrm{D}f_{9} ~+~ \mathrm{d}f_{9}
\\[20pt]
\mathrm{D}f_{9}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
\\[6pt]
\mathrm{d}f_{9}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
\\[20pt]
\mathrm{r}f_{9}
& = & u v \cdot 0
& + & u \texttt{(} v \texttt{)} \cdot 0
& + & \texttt{(} u \texttt{)} v \cdot 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
\end{array}</math>
|}

<br>

=====Computation Summary for Equality=====

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F9.6} ~~ \text{Computation Summary for}~ f_{9} \langle u, v \rangle = \texttt{((} u \texttt{,} v \texttt{))}</math>
|
<math>\begin{array}{c*{8}{l}}
\boldsymbol\varepsilon f_{9}
& = & u v \cdot 1
& + & u \texttt{(} v \texttt{)} \cdot 0
& + & \texttt{(} u \texttt{)} v \cdot 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1
\\[6pt]
\mathrm{E}f_{9}
& = & u v \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}
\\[6pt]
\mathrm{D}f_{9}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
\\[6pt]
\mathrm{d}f_{9}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
\\[6pt]
\mathrm{r}f_{9}
& = & u v \cdot 0
& + & u \texttt{(} v \texttt{)} \cdot 0
& + & \texttt{(} u \texttt{)} v \cdot 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
\end{array}</math>
|}

<br>

====Operator Maps for the Logical Implication ''f''<sub>11</sub>⟨''u'', ''v''⟩====

=====Computation of ε''f''<sub>11</sub>=====

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F11.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{11}</math>
|
<math>\begin{array}{*{10}{l}}
\boldsymbol\varepsilon f_{11}
& = && f_{11} \langle u, v \rangle
\\[4pt]
& = && \texttt{(} u \texttt{(} v \texttt{))}
\\[4pt]
& = && u v \cdot f_{11} \langle 1, 1 \rangle
& + & u \texttt{(} v \texttt{)} \cdot f_{11} \langle 1, 0 \rangle
& + & \texttt{(} u \texttt{)} v \cdot f_{11} \langle 0, 1 \rangle
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{11} \langle 0, 0 \rangle
\\[4pt]
& = && u v
& + & 0
& + & \texttt{(} u \texttt{)} v
& + & \texttt{(} u \texttt{)(} v \texttt{)}
\\[20pt]
\boldsymbol\varepsilon f_{11}
& = && u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & 0
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
& + & 0
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
\\[4pt]
&& + & u v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & 0
& + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & u v \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
& + & 0
& + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
\end{array}</math>
|}

<br>

=====Computation of E''f''<sub>11</sub>=====

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F11.2} ~~ \text{Computation of}~ \mathrm{E}f_{11}</math>
|
<math>\begin{array}{*{10}{l}}
\mathrm{E}f_{11}
& = && f_{11} \langle u + \mathrm{d}u, v + \mathrm{d}v \rangle
\\[4pt]
& = &&
\texttt{(}
\\
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}
\\
&&& \texttt{(}
\\
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
\\
&&& \texttt{))}
\\[4pt]
& = &&
u v
\cdot
\texttt{((} \mathrm{d}u \texttt{)((} \mathrm{d}v \texttt{)))}
& + &
u \texttt{(} v \texttt{)}
\cdot
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + &
\texttt{(} u \texttt{)} v
\cdot
\texttt{(} \mathrm{d}u \texttt{((} \mathrm{d}v \texttt{)))}
& + &
\texttt{(} u \texttt{)(} v \texttt{)}
\cdot
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}
\\[4pt]
& = &&
u v
\cdot
\texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}
& + &
u \texttt{(} v \texttt{)}
\cdot
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + &
\texttt{(} u \texttt{)} v
\cdot
\texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}
& + &
\texttt{(} u \texttt{)(} v \texttt{)}
\cdot
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}
\\[20pt]
\mathrm{E}f_{11}
& = && u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & 0
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & 0
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
\\[4pt]
&& + & u v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & 0
\\[4pt]
&& + & u v \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
\end{array}</math>
|}

<br>

=====Computation of D''f''<sub>11</sub>=====

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F11.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 1)}</math>
|
<math>\begin{array}{*{10}{l}}
\mathrm{D}f_{11}
& = && \mathrm{E}f_{11}
& + & \boldsymbol\varepsilon f_{11}
\\[4pt]
& = && f_{11} \langle u + \mathrm{d}u, v + \mathrm{d}v \rangle
& + & f_{11} \langle u, v \rangle
\\[4pt]
& = &&
\texttt{(} \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}
\texttt{(} \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
\texttt{))}
& + &
\texttt{(} u \texttt{(} v \texttt{))}
\\[20pt]
\mathrm{D}f_{11}
& = && 0
& + & 0
& + & 0
& + & 0
\\[4pt]
&& + & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot ~~ \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
& + & 0
& + & 0
\\[4pt]
&& + & 0
& + & u \texttt{(} v \texttt{)} \cdot ~ ~ ~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & 0
& + & u \texttt{(} v \texttt{)} \cdot ~ ~ ~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~~ \mathrm{d}v
& + & 0
\\[20pt]
\mathrm{D}f_{11}
& = && u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
\end{array}</math>
|}

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F11.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 2)}</math>
|
<math>\begin{array}{c*{9}{l}}
\mathrm{D}f_{11}
& = & \boldsymbol\varepsilon f_{11} ~+~ \mathrm{E}f_{11}
\\[20pt]
\boldsymbol\varepsilon f_{11}
& = & u v \cdot 1
& + & u \texttt{(} v \texttt{)} \cdot 0
& + & \texttt{(} u \texttt{)} v \cdot 1
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1
\\[6pt]
\mathrm{E}f_{11}
& = &
u v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}
& + &
u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + &
\texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u ~~ \mathrm{d}v \texttt{)}
& + &
\texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}
\\[20pt]
\mathrm{D}f_{11}
& = &
u v \cdot ~~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + &
u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + &
\texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~~ \mathrm{d}v
& + &
\texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
\end{array}</math>
|}

<br>

=====Computation of d''f''<sub>11</sub>=====

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F11.4} ~~ \text{Computation of}~ \mathrm{d}f_{11}</math>
|
<math>\begin{array}{c*{8}{l}}
\mathrm{D}f_{11}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
\\[6pt]
\Downarrow
\\[6pt]
\mathrm{d}f_{11}
& = & u v \cdot \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u
\end{array}</math>
|}

<br>

=====Computation of r''f''<sub>11</sub>=====

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F11.5} ~~ \text{Computation of}~ \mathrm{r}f_{11}</math>
|
<math>\begin{array}{c*{8}{l}}
\mathrm{r}f_{11} & = & \mathrm{D}f_{11} ~+~ \mathrm{d}f_{11}
\\[20pt]
\mathrm{D}f_{11}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
\\[6pt]
\mathrm{d}f_{11}
& = & u v \cdot \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u
\\[20pt]
\mathrm{r}f_{11}
& = & u v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\end{array}</math>
|}

<br>

=====Computation Summary for Implication=====

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F11.6} ~~ \text{Computation Summary for}~ f_{11} \langle u, v \rangle = \texttt{(} u \texttt{(} v \texttt{))}</math>
|
<math>\begin{array}{c*{8}{l}}
\boldsymbol\varepsilon f_{11}
& = & u v \cdot 1
& + & u \texttt{(} v \texttt{)} \cdot 0
& + & \texttt{(} u \texttt{)} v \cdot 1
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1
\\[6pt]
\mathrm{E}f_{11}
& = & u v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}
\\[6pt]
\mathrm{D}f_{11}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
\\[6pt]
\mathrm{d}f_{11}
& = & u v \cdot \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u
\\[6pt]
\mathrm{r}f_{11}
& = & u v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\end{array}</math>
|}

<br>

====Operator Maps for the Logical Disjunction ''f''<sub>14</sub>⟨''u'', ''v''⟩====

=====Computation of ε''f''<sub>14</sub>=====

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F14.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{14}</math>
|
<math>\begin{array}{*{10}{l}}
\boldsymbol\varepsilon f_{14}
& = && f_{14} \langle u, v \rangle
\\[4pt]
& = && \texttt{((} u \texttt{)(} v \texttt{))}
\\[4pt]
& = && u v \cdot f_{14} \langle 1, 1 \rangle
& + & u \texttt{(} v \texttt{)} \cdot f_{14} \langle 1, 0 \rangle
& + & \texttt{(} u \texttt{)} v \cdot f_{14} \langle 0, 1 \rangle
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{14} \langle 0, 0 \rangle
\\[4pt]
& = && u v
& + & u \texttt{(} v \texttt{)}
& + & \texttt{(} u \texttt{)} v
& + & 0
\\[20pt]
\boldsymbol\varepsilon f_{14}
& = && u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & 0
\\[4pt]
&& + & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
& + & 0
\\[4pt]
&& + & u v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & 0
\\[4pt]
&& + & u v \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
& + & 0
\end{array}</math>
|}

<br>

=====Computation of E''f''<sub>14</sub>=====

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F14.2} ~~ \text{Computation of}~ \mathrm{E}f_{14}</math>
|
<math>\begin{array}{*{10}{l}}
\mathrm{E}f_{14}
& = && f_{14} \langle u + \mathrm{d}u, v + \mathrm{d}v \rangle
\\[4pt]
& = &&
\texttt{((}
\\
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}
\\
&&& \texttt{)(}
\\
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
\\
&&& \texttt{))}
\\[4pt]
& = && u v \cdot
f_{14} \langle \texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)} \rangle
& + & u \texttt{(} v \texttt{)} \cdot f_{14} \langle \texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v \rangle
& + & \texttt{(} u \texttt{)} v \cdot f_{14} \langle \mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)} \rangle
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{14} \langle \mathrm{d}u, \mathrm{d}v \rangle
\\[4pt]
& = && u v \cdot \texttt{(} \mathrm{d}u ~~ \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
\\[20pt]
\mathrm{E}f_{14}
& = && u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & 0
\\[4pt]
&& + & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
\\[4pt]
&& + & u v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & 0
& + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & 0
& + & u \texttt{(} v \texttt{)} \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
\end{array}</math>
|}

<br>

=====Computation of D''f''<sub>14</sub>=====

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F14.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 1)}</math>
|
<math>\begin{array}{*{10}{l}}
\mathrm{D}f_{14}
& = && \mathrm{E}f_{14}
& + & \boldsymbol\varepsilon f_{14}
\\[4pt]
& = && f_{14} \langle u + \mathrm{d}u, v + \mathrm{d}v \rangle
& + & f_{14} \langle u, v \rangle
\\[4pt]
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{))((} v \texttt{,} \mathrm{d}v \texttt{)))}
& + & \texttt{((} u \texttt{)(} v \texttt{))}
\\[20pt]
\mathrm{D}f_{14}
& = && 0
& + & 0
& + & 0
& + & 0
\\[4pt]
&& + & 0
& + & 0
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
\\[4pt]
&& + & 0
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~ ~ ~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & u v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & 0
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~ ~ ~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
\\[20pt]
\mathrm{D}f_{14}
& = && u v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v
\texttt{))}
\end{array}</math>
|}

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F14.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 2)}</math>
|
<math>\begin{array}{*{9}{l}}
\mathrm{D}f_{14}
& = & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
\end{array}</math>
|}

<br>

=====Computation of d''f''<sub>14</sub>=====

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F14.4} ~~ \text{Computation of}~ \mathrm{d}f_{14}</math>
|
<math>\begin{array}{c*{8}{l}}
\mathrm{D}f_{14}
& = & u v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
\\[6pt]
\Downarrow
\\[6pt]
\mathrm{d}f_{14}
& = & u v \cdot 0
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
\end{array}</math>
|}

<br>

=====Computation of r''f''<sub>14</sub>=====

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F14.5} ~~ \text{Computation of}~ \mathrm{r}f_{14}</math>
|
<math>\begin{array}{c*{8}{l}}
\mathrm{r}f_{14} & = & \mathrm{D}f_{14} ~+~ \mathrm{d}f_{14}
\\[20pt]
\mathrm{D}f_{14}
& = & u v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
\\[6pt]
\mathrm{d}f_{14}
& = & u v \cdot 0
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
\\[20pt]
\mathrm{r}f_{14}
& = & u v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\end{array}</math>
|}

<br>

=====Computation Summary for Disjunction=====

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F14.6} ~~ \text{Computation Summary for}~ f_{14} \langle u, v \rangle = \texttt{((} u \texttt{)(} v \texttt{))}</math>
|
<math>\begin{array}{c*{8}{l}}
\boldsymbol\varepsilon f_{14}
& = & u v \cdot 1
& + & u \texttt{(} v \texttt{)} \cdot 1
& + & \texttt{(} u \texttt{)} v \cdot 1
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
\\[6pt]
\mathrm{E}f_{14}
& = & u v \cdot \texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
\\[6pt]
\mathrm{D}f_{14}
& = & u v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
\\[6pt]
\mathrm{d}f_{14}
& = & u v \cdot 0
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
\\[6pt]
\mathrm{r}f_{14}
& = & u v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\end{array}</math>
|}

<br>

===Appendix 4. Source Materials===

<br>

===Appendix 5. Various Definitions of the Tangent Vector===

<br>

----
<div align="center">
• [[Differential Logic and Dynamic Systems • Overview|Overview]]
• [[Differential Logic and Dynamic Systems • Part 1|Part 1]]
• [[Differential Logic and Dynamic Systems • Part 2|Part 2]]
• [[Differential Logic and Dynamic Systems • Part 3|Part 3]]
• [[Differential Logic and Dynamic Systems • Part 4|Part 4]]
• [[Differential Logic and Dynamic Systems • Part 5|Part 5]]
• [[Differential Logic and Dynamic Systems • Appendices|Appendices]]
• [[Differential Logic and Dynamic Systems • References|References]]
• [[Differential Logic and Dynamic Systems • Document History|Document History]]
•
</div>
----

[[Category:Adaptive systems]]
[[Category:Artificial intelligence]]
[[Category:Boolean algebra]]
[[Category:Boolean functions]]
[[Category:Category theory]]
[[Category:Combinatorics]]
[[Category:Computation theory]]
[[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:Logic]]
[[Category:Logical graphs]]
[[Category:Neural networks]]
[[Category:Peirce, Charles Sanders]]
[[Category:Semiotics]]
[[Category:Systems theory]]
[[Category:Visualization]]

Differential Logic and Dynamic Systems • Part 5

2026-04-10T16:12:15Z

Jon Awbrey: add Differential Logic and Dynamic Systems • Part 5

Differential Logic and Dynamic Systems • Part 4

2026-04-10T15:36:55Z

Jon Awbrey: add Differential Logic and Dynamic Systems • Part 4

User:Jon Awbrey/Figures and Tables 2

2026-03-12T13:58:03Z

Jon Awbrey: sub original User:Jon Awbrey/Figures and Tables 2

==Boolean Functions on Two Variables==

===Old Table===

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 7.} ~~ \text{Propositional Forms on Two Variables}</math>
|- style="height:40px; background:ghostwhite"
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math>
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math>
|- style="background:ghostwhite"
|  
| align="right" | <math>x\colon</math>
| <math>1~1~0~0</math>
|  
|  
|  
|- style="background:ghostwhite"
|  
| align="right" | <math>y\colon</math>
| <math>1~0~1~0</math>
|  
|  
|  
|-
| <math>f_{0}</math>
| <math>f_{0000}</math>
| <math>0~0~0~0</math>
| <math>(~)</math>
| <math>\text{false}</math>
| <math>0</math>
|-
| <math>f_{1}</math>
| <math>f_{0001}</math>
| <math>0~0~0~1</math>
| <math>(x)(y)</math>
| <math>\text{neither}~ x ~\text{nor}~ y</math>
| <math>\lnot x \land \lnot y</math>
|-
| <math>f_{2}</math>
| <math>f_{0010}</math>
| <math>0~0~1~0</math>
| <math>(x)\ y</math>
| <math>y ~\text{without}~ x</math>
| <math>\lnot x \land y</math>
|-
| <math>f_{3}</math>
| <math>f_{0011}</math>
| <math>0~0~1~1</math>
| <math>(x)</math>
| <math>\text{not}~ x</math>
| <math>\lnot x</math>
|-
| <math>f_{4}</math>
| <math>f_{0100}</math>
| <math>0~1~0~0</math>
| <math>x\ (y)</math>
| <math>x ~\text{without}~ y</math>
| <math>x \land \lnot y</math>
|-
| <math>f_{5}</math>
| <math>f_{0101}</math>
| <math>0~1~0~1</math>
| <math>(y)</math>
| <math>\text{not}~ y</math>
| <math>\lnot y</math>
|-
| <math>f_{6}</math>
| <math>f_{0110}</math>
| <math>0~1~1~0</math>
| <math>(x, y)</math>
| <math>x ~\text{not equal to}~ y</math>
| <math>x \ne y</math>
|-
| <math>f_{7}</math>
| <math>f_{0111}</math>
| <math>0~1~1~1</math>
| <math>(x\ y)</math>
| <math>\text{not both}~ x ~\text{and}~ y</math>
| <math>\lnot x \lor \lnot y</math>
|-
| <math>f_{8}</math>
| <math>f_{1000}</math>
| <math>1~0~0~0</math>
| <math>x\ y</math>
| <math>x ~\text{and}~ y</math>
| <math>x \land y</math>
|-
| <math>f_{9}</math>
| <math>f_{1001}</math>
| <math>1~0~0~1</math>
| <math>((x, y))</math>
| <math>x ~\text{equal to}~ y</math>
| <math>x = y</math>
|-
| <math>f_{10}</math>
| <math>f_{1010}</math>
| <math>1~0~1~0</math>
| <math>y</math>
| <math>y</math>
| <math>y</math>
|-
| <math>f_{11}</math>
| <math>f_{1011}</math>
| <math>1~0~1~1</math>
| <math>(x\ (y))</math>
| <math>\text{not}~ x ~\text{without}~ y</math>
| <math>x \Rightarrow y</math>
|-
| <math>f_{12}</math>
| <math>f_{1100}</math>
| <math>1~1~0~0</math>
| <math>x</math>
| <math>x</math>
| <math>x</math>
|-
| <math>f_{13}</math>
| <math>f_{1101}</math>
| <math>1~1~0~1</math>
| <math>((x)\ y)</math>
| <math>\text{not}~ y ~\text{without}~ x</math>
| <math>x \Leftarrow y</math>
|-
| <math>f_{14}</math>
| <math>f_{1110}</math>
| <math>1~1~1~0</math>
| <math>((x)(y))</math>
| <math>x ~\text{or}~ y</math>
| <math>x \lor y</math>
|-
| <math>f_{15}</math>
| <math>f_{1111}</math>
| <math>1~1~1~1</math>
| <math>((~))</math>
| <math>\text{true}</math>
| <math>1</math>
|}

===New Tables===

====Template====

<table align="center" cellpadding="10px" cellspacing ="0" style="border:1px solid black; font-size:medium; text-align:center;">

<caption style="height:2em;">
<math>\text{Boolean Functions on Two Variables}</math></caption>

<tr style="height:2em;">
<th style="border-bottom:2px solid black;"><math>\text{Boolean Function}</math></th>
<th style="border-bottom:2px solid black;"><math>\text{Linguistic Formula}</math></th>
<th style="border-bottom:2px solid black;"><math>\text{Entitative Graph}</math></th>
<th style="border-bottom:2px solid black;"><math>\text{Existential Graph}</math></th>
</tr>

<tr style="height:100px;">
<td style="vertical-align:middle;"><math>f_0</math></td>
<td style="vertical-align:middle;"><math>\text{false}</math></td>
<td style="vertical-align:middle;">[[File:Cactus Patch Root.jpg|32px]]</td>
<td style="vertical-align:middle;">[[File:Cactus Patch Stem.jpg|32px]]</td>
</tr>

<tr style="height:100px;">
<td style="vertical-align:middle;"><math>f_1</math></td>
<td style="vertical-align:middle;"><math>\text{neither}~ x ~\text{nor}~ y</math></td>
<td style="vertical-align:middle;"><math>f_1</math></td>
<td style="vertical-align:middle;"><math>f_1</math></td>
</tr>

<tr style="height:100px;">
<td style="vertical-align:middle;"><math>f_2</math></td>
<td style="vertical-align:middle;"><math>y ~\text{without}~ x</math></td>
<td style="vertical-align:middle;"><math>f_2</math></td>
<td style="vertical-align:middle;"><math>f_2</math></td>
</tr>

<tr style="height:100px;">
<td style="vertical-align:middle;"><math>f_3</math></td>
<td style="vertical-align:middle;"><math>\text{not}~ x</math></td>
<td style="vertical-align:middle;">[[File:Cactus Patch (X).jpg|32px]]</td>
<td style="vertical-align:middle;">[[File:Cactus Patch (X).jpg|32px]]</td>
</tr>

<tr style="height:100px;">
<td style="vertical-align:middle;"><math>f_4</math></td>
<td style="vertical-align:middle;"><math>x ~\text{without}~ y</math></td>
<td style="vertical-align:middle;"><math>f_4</math></td>
<td style="vertical-align:middle;"><math>f_4</math></td>
</tr>

<tr style="height:100px;">
<td style="vertical-align:middle;"><math>f_5</math></td>
<td style="vertical-align:middle;"><math>\text{not}~ y</math></td>
<td style="vertical-align:middle;">[[File:Cactus Patch (Y).jpg|32px]]</td>
<td style="vertical-align:middle;">[[File:Cactus Patch (Y).jpg|32px]]</td>
</tr>

<tr style="height:100px;">
<td style="vertical-align:middle;"><math>f_6</math></td>
<td style="vertical-align:middle;"><math>x ~\text{not equal to}~ y</math></td>
<td style="vertical-align:middle;"><math>f_6</math></td>
<td style="vertical-align:middle;">[[File:Cactus Patch (X,Y).jpg|64px]]</td>
</tr>

<tr style="height:100px;">
<td style="vertical-align:middle;border-bottom:1px solid black;">
<math>f_7</math></td>
<td style="vertical-align:middle;border-bottom:1px solid black;">
<math>\text{not both}~ x ~\text{and}~ y</math></td>
<td style="vertical-align:middle;border-bottom:1px solid black;">
<math>f_7</math></td>
<td style="vertical-align:middle;border-bottom:1px solid black;">
<math>f_7</math></td>
</tr>

<tr style="height:100px;">
<td style="vertical-align:middle;"><math>f_8</math></td>
<td style="vertical-align:middle;"><math>x ~\text{and}~ y</math></td>
<td style="vertical-align:middle;"><math>f_8</math></td>
<td style="vertical-align:middle;">[[File:Cactus Patch XY.jpg|32px]]</td>
</tr>

<tr style="height:100px;">
<td style="vertical-align:middle;"><math>f_9</math></td>
<td style="vertical-align:middle;"><math>x ~\text{equal to}~ y</math></td>
<td style="vertical-align:middle;">[[File:Cactus Patch (X,Y).jpg|64px]]</td>
<td style="vertical-align:middle;"><math>f_9</math></td>
</tr>

<tr style="height:100px;">
<td style="vertical-align:middle;"><math>f_{10}</math></td>
<td style="vertical-align:middle;"><math>y</math></td>
<td style="vertical-align:middle;">[[File:Cactus Patch Y.jpg|32px]]</td>
<td style="vertical-align:middle;">[[File:Cactus Patch Y.jpg|32px]]</td>
</tr>

<tr style="height:100px;">
<td style="vertical-align:middle;"><math>f_{11}</math></td>
<td style="vertical-align:middle;"><math>\text{not}~ x ~\text{without}~ y</math></td>
<td style="vertical-align:middle;"><math>f_{11}</math></td>
<td style="vertical-align:middle;"><math>f_{11}</math></td>
</tr>

<tr style="height:100px;">
<td style="vertical-align:middle;"><math>f_{12}</math></td>
<td style="vertical-align:middle;"><math>x</math></td>
<td style="vertical-align:middle;">[[File:Cactus Patch X.jpg|32px]]</td>
<td style="vertical-align:middle;">[[File:Cactus Patch X.jpg|32px]]</td>
</tr>

<tr style="height:100px;">
<td style="vertical-align:middle;"><math>f_{13}</math></td>
<td style="vertical-align:middle;"><math>\text{not}~ y ~\text{without}~ x</math></td>
<td style="vertical-align:middle;"><math>f_{13}</math></td>
<td style="vertical-align:middle;"><math>f_{13}</math></td>
</tr>

<tr style="height:100px;">
<td style="vertical-align:middle;"><math>f_{14}</math></td>
<td style="vertical-align:middle;"><math>x ~\text{or}~ y</math></td>
<td style="vertical-align:middle;"><math>f_{14}</math></td>
<td style="vertical-align:middle;"><math>f_{14}</math></td>
</tr>

<tr style="height:100px;">
<td style="vertical-align:middle;"><math>f_{15}</math></td>
<td style="vertical-align:middle;"><math>\text{true}</math></td>
<td style="vertical-align:middle;">[[File:Cactus Patch Stem.jpg|32px]]</td>
<td style="vertical-align:middle;">[[File:Cactus Patch Root.jpg|32px]]</td>
</tr>

</table>

====Entitative Interpretation====

====Existential Interpretation====

==Logical Cacti • Theme One Exposition==

Up till now we've been working to hammer out a two-edged sword of syntax, honing the syntax of ''painted and rooted cacti and expressions'' (PARCAE), and turning it to use in taming the syntax of two-level formal languages.

But the purpose of a logical syntax is to support a logical semantics, which means, for starters, to bear interpretation as sentential signs that can denote objective propositions about some universe of objects.

One of the difficulties that we face in this discussion is that the words ''interpretation'', ''meaning'', ''semantics'', and so on will have so many different meanings from one moment to the next of their use. A dedicated neologician might be able to think up distinctive names for all of the aspects of meaning and all of the approaches to them that will concern us here, but I will just have to do the best that I can with the common lot of ambiguous terms, leaving it to context and the intelligent interpreter to sort it out as much as possible.

As it happens, the language of cacti is so abstract that it can bear at least two different interpretations as logical sentences denoting logical propositions. The two interpretations that I know about are descended from the ones that C.S. Peirce called the ''entitative'' and the ''existential'' interpretations of his systems of graphical logics. For our present aims, I shall briefly introduce the alternatives and then quickly move to the existential interpretation of logical cacti.

===Existential Interpretation===

Table 13 illustrates the ''existential interpretation'' of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 13.} ~~ \text{Existential Interpretation}</math>
|- style="height:40px; background:#f0f0ff"
| <math>\text{Cactus Graph}</math>
| <math>\text{Cactus Expression}</math>
| <math>\text{Interpretation}</math>
|-
| height="100px" | [[File:Cactus Graph Node Big.jpg|24px]]
| <math>\mathrm{~}</math>
| <math>\mathrm{true}</math>
|-
| height="100px" | [[File:Cactus Graph Spike Big.jpg|24px]]
| <math>\texttt{(} ~ \texttt{)}</math>
| <math>\mathrm{false}</math>
|-
| height="100px" | [[File:Cactus A Big.jpg|20px]]
| <math>a</math>
| <math>a</math>
|-
| height="120px" | [[File:Cactus (A) Big.jpg|20px]]
| <math>\texttt{(} a \texttt{)}</math>
|
<math>\begin{matrix}
\tilde{a}
\\[2pt]
a^\prime
\\[2pt]
\lnot a
\\[2pt]
\mathrm{not}~ a
\end{matrix}</math>
|-
| height="100px" | [[File:Cactus ABC Big.jpg|50px]]
| <math>a~b~c</math>
|
<math>\begin{matrix}
a \land b \land c
\\[6pt]
a ~\mathrm{and}~ b ~\mathrm{and}~ c
\end{matrix}</math>
|-
| height="160px" | [[File:Cactus ((A)(B)(C)) Big.jpg|65px]]
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
|
<math>\begin{matrix}
a \lor b \lor c
\\[6pt]
a ~\mathrm{or}~ b ~\mathrm{or}~ c
\end{matrix}</math>
|-
| height="120px" | [[File:Cactus (A(B)) Big.jpg|60px]]
| <math>\texttt{(} a \texttt{(} b \texttt{))}</math>
|
<math>\begin{matrix}
a \Rightarrow b
\\[2pt]
a ~\mathrm{implies}~ b
\\[2pt]
\mathrm{if}~ a ~\mathrm{then}~ b
\\[2pt]
\mathrm{not}~ a ~\mathrm{without}~ b
\end{matrix}</math>
|-
| height="120px" | [[File:Cactus (A,B) Big.jpg|65px]]
| <math>\texttt{(} a, b \texttt{)}</math>
|
<math>\begin{matrix}
a + b
\\[2pt]
a \neq b
\\[2pt]
a ~\mathrm{exclusive~or}~ b
\\[2pt]
a ~\mathrm{not~equal~to}~ b
\end{matrix}</math>
|-
| height="160px" | [[File:Cactus ((A,B)) Big.jpg|65px]]
| <math>\texttt{((} a, b \texttt{))}</math>
|
<math>\begin{matrix}
a = b
\\[2pt]
a \iff b
\\[2pt]
a ~\mathrm{equals}~ b
\\[2pt]
a ~\mathrm{if~and~only~if}~ b
\end{matrix}</math>
|-
| height="120px" | [[File:Cactus (A,B,C) Big.jpg|65px]]
| <math>\texttt{(} a, b, c \texttt{)}</math>
|
<math>\begin{matrix}
\mathrm{just~one~of}
\\
a, b, c
\\
\mathrm{is~false}
\end{matrix}</math>
|-
| height="160px" | [[File:Cactus ((A),(B),(C)) Big.jpg|65px]]
| <math>\texttt{((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
|
<math>\begin{matrix}
\mathrm{just~one~of}
\\
a, b, c
\\
\mathrm{is~true}
\end{matrix}</math>
|-
| height="160px" | [[File:Cactus (A,(B),(C)) Big.jpg|65px]]
| <math>\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
|
<math>\begin{matrix}
\mathrm{genus}~ a ~\mathrm{of~species}~ b, c
\\[6pt]
\mathrm{partition}~ a ~\mathrm{into}~ b, c
\\[6pt]
\mathrm{pie}~ a ~\mathrm{of~slices}~ b, c
\end{matrix}</math>
|}

===Entitative Interpretation===

Table 14 illustrates the ''entitative interpretation'' of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 14.} ~~ \text{Entitative Interpretation}</math>
|- style="height:40px; background:#f0f0ff"
| <math>\text{Cactus Graph}</math>
| <math>\text{Cactus Expression}</math>
| <math>\text{Interpretation}</math>
|-
| height="100px" | [[File:Cactus Graph Node Big.jpg|24px]]
| <math>\mathrm{~}</math>
| <math>\mathrm{false}</math>
|-
| height="100px" | [[File:Cactus Graph Spike Big.jpg|24px]]
| <math>\texttt{(} ~ \texttt{)}</math>
| <math>\mathrm{true}</math>
|-
| height="100px" | [[File:Cactus A Big.jpg|20px]]
| <math>a</math>
| <math>a</math>
|-
| height="120px" | [[File:Cactus (A) Big.jpg|20px]]
| <math>\texttt{(} a \texttt{)}</math>
|
<math>\begin{matrix}
\tilde{a}
\\[2pt]
a^\prime
\\[2pt]
\lnot a
\\[2pt]
\mathrm{not}~ a
\end{matrix}</math>
|-
| height="100px" | [[File:Cactus ABC Big.jpg|50px]]
| <math>a~b~c</math>
|
<math>\begin{matrix}
a \lor b \lor c
\\[6pt]
a ~\mathrm{or}~ b ~\mathrm{or}~ c
\end{matrix}</math>
|-
| height="160px" | [[File:Cactus ((A)(B)(C)) Big.jpg|65px]]
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
|
<math>\begin{matrix}
a \land b \land c
\\[6pt]
a ~\mathrm{and}~ b ~\mathrm{and}~ c
\end{matrix}</math>
|-
| height="120px" | [[File:Cactus (A)B Big.jpg|35px]]
| <math>\texttt{(} a \texttt{)} b</math>
|
<math>\begin{matrix}
a \Rightarrow b
\\[2pt]
a ~\mathrm{implies}~ b
\\[2pt]
\mathrm{if}~ a ~\mathrm{then}~ b
\\[2pt]
\mathrm{not}~ a, ~\mathrm{or}~ b
\end{matrix}</math>
|-
| height="120px" | [[File:Cactus (A,B) Big.jpg|65px]]
| <math>\texttt{(} a, b \texttt{)}</math>
|
<math>\begin{matrix}
a = b
\\[2pt]
a \iff b
\\[2pt]
a ~\mathrm{equals}~ b
\\[2pt]
a ~\mathrm{if~and~only~if}~ b
\end{matrix}</math>
|-
| height="160px" | [[File:Cactus ((A,B)) Big.jpg|65px]]
| <math>\texttt{((} a, b \texttt{))}</math>
|
<math>\begin{matrix}
a + b
\\[2pt]
a \neq b
\\[2pt]
a ~\mathrm{exclusive~or}~ b
\\[2pt]
a ~\mathrm{not~equal~to}~ b
\end{matrix}</math>
|-
| height="120px" | [[File:Cactus (A,B,C) Big.jpg|65px]]
| <math>\texttt{(} a, b, c \texttt{)}</math>
|
<math>\begin{matrix}
\mathrm{not~just~one~of}
\\
a, b, c
\\
\mathrm{is~true}
\end{matrix}</math>
|-
| height="160px" | [[File:Cactus ((A,B,C)) Big.jpg|65px]]
| <math>\texttt{((} a, b, c \texttt{))}</math>
|
<math>\begin{matrix}
\mathrm{just~one~of}
\\
a, b, c
\\
\mathrm{is~true}
\end{matrix}</math>
|-
| height="200px" | [[File:Cactus (((A),B,C)) Big.jpg|65px]]
| <math>\texttt{(((} a \texttt{)}, b, c \texttt{))}</math>
|
<math>\begin{matrix}
\mathrm{genus}~ a ~\mathrm{of~species}~ b, c
\\[6pt]
\mathrm{partition}~ a ~\mathrm{into}~ b, c
\\[6pt]
\mathrm{pie}~ a ~\mathrm{of~slices}~ b, c
\end{matrix}</math>
|}

==Logical Graphs==

===Old Versions===

====Example 1====

{| align="center" cellpadding="10"
|
<pre>
o-----------------------------------------------------------o
| |
| o o o o o o |
| \| | | | | |
| o o o o o o o o o |
| \|/ \|/ |/ | |
| @ = @ = @ = @ |
| |
o-----------------------------------------------------------o
| |
| (()())(())() = (())(())() = (())() = ( ) |
| |
o-----------------------------------------------------------o
</pre>
|}

====Example 2====

{| align="center" cellpadding="10"
|
<pre>
o-------------------o-------------------o-------------------o
| Object | Sign | Interpretant |
o-------------------o-------------------o-------------------o
| | | |
| Falsity | "(()())(())()" | "(())(())()" |
| | | |
| Falsity | "(())(())()" | "(())()" |
| | | |
| Falsity | "(())()" | "()" |
| | | |
o-------------------o-------------------o-------------------o
</pre>
|}

====Example 3====

{| align="center" cellpadding="10"
|
<pre>
o-------------------o-------------------o-------------------o
| Object | Sign | Interpretant |
o-------------------o-------------------o-------------------o
| | | |
| Falsity | "(()())(())()" | "(()())(())()" |
| | | |
| Falsity | "(()())(())()" | "(())(())()" |
| | | |
| Falsity | "(()())(())()" | "(())()" |
| | | |
| Falsity | "(()())(())()" | "()" |
| | | |
o-------------------o-------------------o-------------------o
| | | |
| Falsity | "(())(())()" | "(()())(())()" |
| | | |
| Falsity | "(())(())()" | "(())(())()" |
| | | |
| Falsity | "(())(())()" | "(())()" |
| | | |
| Falsity | "(())(())()" | "()" |
| | | |
o-------------------o-------------------o-------------------o
| | | |
| Falsity | "(())()" | "(()())(())()" |
| | | |
| Falsity | "(())()" | "(())(())()" |
| | | |
| Falsity | "(())()" | "(())()" |
| | | |
| Falsity | "(())()" | "()" |
| | | |
o-------------------o-------------------o-------------------o
| | | |
| Falsity | "()" | "(()())(())()" |
| | | |
| Falsity | "()" | "(())(())()" |
| | | |
| Falsity | "()" | "(())()" |
| | | |
| Falsity | "()" | "()" |
| | | |
o-------------------o-------------------o-------------------o
</pre>
|}

====Example 4====

{| align="center" cellpadding="10"
|
<pre>
o-------------------o-------------------o-------------------o
| a | b | (a , b) |
o-------------------o-------------------o-------------------o
| | | |
| blank | blank | cross |
| | | |
| blank | cross | blank |
| | | |
| cross | blank | blank |
| | | |
| cross | cross | cross |
| | | |
o-------------------o-------------------o-------------------o
</pre>
|}

====Example 5====

{| align="center" cellpadding="10"
|
<pre>
o-------o-------o-------o-----------o
| a | b | c | (a, b, c) |
o-------o-------o-------o-----------o
| | | | |
| blank | blank | blank | cross |
| | | | |
| blank | blank | cross | blank |
| | | | |
| blank | cross | blank | blank |
| | | | |
| blank | cross | cross | cross |
| | | | |
| cross | blank | blank | blank |
| | | | |
| cross | blank | cross | cross |
| | | | |
| cross | cross | blank | cross |
| | | | |
| cross | cross | cross | cross |
| | | | |
o-------o-------o-------o-----------o
</pre>
|}

====Example 6====

{| align="center" cellpadding="10"
|
<pre>
o-------o-------o-------o-----------o
| a | b | c | (a, b, c) |
o-------o-------o-------o-----------o
| | | | |
| o | o | o | | |
| | | | |
| o | o | | | o |
| | | | |
| o | | | o | o |
| | | | |
| o | | | | | | |
| | | | |
| | | o | o | o |
| | | | |
| | | o | | | | |
| | | | |
| | | | | o | | |
| | | | |
| | | | | | | | |
| | | | |
o-------o-------o-------o-----------o
</pre>
|}

===New Versions===

====Example 1====

…

====Example 2a====

{| align="center" border="1" cellpadding="12" cellspacing="0" width="50%"
|- style="background:ghostwhite; height:40px"
| width="33%" | <math>\text{Object}</math>
| width="33%" | <math>\text{Sign}</math>
| width="33%" | <math>\text{Interpretant}</math>
|-
| <math>\mathrm{Falsity}</math>
| <math>{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}</math>
|-
| <math>\mathrm{Falsity}</math>
| <math>{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}</math>
|-
| <math>\mathrm{Falsity}</math>
| <math>{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}</math>
|}

====Example 2b====

{| align="center" border="1" cellpadding="12" cellspacing="0" width="50%"
|- style="background:ghostwhite; height:40px"
| width="33%" | <math>\text{Object}</math>
| width="33%" | <math>\text{Sign}</math>
| width="33%" | <math>\text{Interpretant}</math>
|-
| valign="bottom" |
<math>\begin{array}{l}
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\end{array}</math>
| valign="bottom" |
<math>\begin{array}{l}
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\end{array}</math>
| valign="bottom" |
<math>\begin{array}{l}
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\end{array}</math>
|}

====Example 3====

{| align="center" border="1" cellpadding="12" cellspacing="0" width="50%"
|- style="background:ghostwhite; height:40px"
| width="33%" | <math>\text{Object}</math>
| width="33%" | <math>\text{Sign}</math>
| width="33%" | <math>\text{Interpretant}</math>
|-
| valign="bottom" |
<math>\begin{array}{l}
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\end{array}</math>
| valign="bottom" |
<math>\begin{array}{l}
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\end{array}</math>
| valign="bottom" |
<math>\begin{array}{l}
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\end{array}</math>
|-
| valign="bottom" |
<math>\begin{array}{l}
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\end{array}</math>
| valign="bottom" |
<math>\begin{array}{l}
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\end{array}</math>
| valign="bottom" |
<math>\begin{array}{l}
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\end{array}</math>
|-
| valign="bottom" |
<math>\begin{array}{l}
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\end{array}</math>
| valign="bottom" |
<math>\begin{array}{l}
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\end{array}</math>
| valign="bottom" |
<math>\begin{array}{l}
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\end{array}</math>
|-
| valign="bottom" |
<math>\begin{array}{l}
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\end{array}</math>
| valign="bottom" |
<math>\begin{array}{l}
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\end{array}</math>
| valign="bottom" |
<math>\begin{array}{l}
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\end{array}</math>
|}

====Example 4====

{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:50%"
|- style="background:ghostwhite; height:40px"
| width="33%" | <math>a</math>
| width="33%" | <math>b</math>
| width="33%" | <math>\texttt{(} a \texttt{,} b \texttt{)}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\texttt{Blank}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Cross}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{Cross}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\end{matrix}</math>
|}

====Example 5====

{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:50%"
|- style="background:ghostwhite; height:40px"
| width="25%" | <math>a</math>
| width="25%" | <math>b</math>
| width="25%" | <math>c</math>
| width="25%" | <math>\texttt{(} a \texttt{,} b \texttt{,} \texttt{c} \texttt{)}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\texttt{Blank}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Cross}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{Blank}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Cross}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{Cross}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Cross}
\end{matrix}</math>
|}

====Example 6====

{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:50%"
|- style="background:ghostwhite; height:40px"
| width="25%" | <math>a</math>
| width="25%" | <math>b</math>
| width="25%" | <math>c</math>
| width="25%" | <math>\texttt{(} a \texttt{,} b \texttt{,} \texttt{c} \texttt{)}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\texttt{o}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{|}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{o}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{|}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{o}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{|}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{|}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{|}
\end{matrix}</math>
|}

User:Jon Awbrey/Figures and Tables 1

2026-03-12T13:54:13Z

Jon Awbrey: sub original User:Jon Awbrey/Figures and Tables 1

==Logic Syllabus==

===Related articles===

<table style="border:none;width:120%;">

<tr>
<td style="border:none;width:50%;"> ◾ [[Cactus Language]]</td>
<td style="border:none;width:50%;"> ◾ [[Propositions As Types]]</td>
</tr>

<tr>
<td style="border:none;width:50%;"> ◾ [[Futures Of Logical Graphs]]</td>
<td style="border:none;width:50%;"> ◾ [[Propositional Equation Reasoning Systems]]</td>
</tr>

<tr><td style="border:none;"></td></tr>

<tr>
<td style="border:none;width:50%;"> ◾ [[Correspondence Theory Of Truth]]</td>
<td style="border:none;width:50%;"> ◾ [[Pragmatic Theory Of Truth]]</td>
</tr>

<tr><td style="border:none;"></td></tr>

<tr>
<td style="border:none;width:50%;"> ◾ [[Differential Logic • Introduction]]</td>
<td style="border:none;width:50%;"> ◾ [[Introduction to Inquiry Driven Systems]]</td>
</tr>

<tr>
<td style="border:none;width:50%;"> ◾ [[Differential Propositional Calculus]]</td>
<td style="border:none;width:50%;"> ◾ [[Prospects for Inquiry Driven Systems]]</td>
</tr>

<tr>
<td style="border:none;width:50%;"> ◾ [[Differential Logic and Dynamic Systems]]</td>
<td style="border:none;width:50%;"> ◾ [[Inquiry Driven Systems • Inquiry Into Inquiry]]</td>
</tr>

</table>

==Logical Cacti • Theme One Exposition==

Up till now we've been working to hammer out a two-edged sword of syntax, honing the syntax of ''painted and rooted cacti and expressions'' (PARCAE), and turning it to use in taming the syntax of two-level formal languages.

But the purpose of a logical syntax is to support a logical semantics, which means, for starters, to bear interpretation as sentential signs that can denote objective propositions about some universe of objects.

One of the difficulties that we face in this discussion is that the words ''interpretation'', ''meaning'', ''semantics'', and so on will have so many different meanings from one moment to the next of their use. A dedicated neologician might be able to think up distinctive names for all of the aspects of meaning and all of the approaches to them that will concern us here, but I will just have to do the best that I can with the common lot of ambiguous terms, leaving it to context and the intelligent interpreter to sort it out as much as possible.

As it happens, the language of cacti is so abstract that it can bear at least two different interpretations as logical sentences denoting logical propositions. The two interpretations that I know about are descended from the ones that C.S. Peirce called the ''entitative'' and the ''existential'' interpretations of his systems of graphical logics. For our present aims, I shall briefly introduce the alternatives and then quickly move to the existential interpretation of logical cacti.

===Existential Interpretation===

Table 13 illustrates the ''existential interpretation'' of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 13.} ~~ \text{Existential Interpretation}</math>
|- style="height:40px; background:#f0f0ff"
| <math>\text{Cactus Graph}</math>
| <math>\text{Cactus Expression}</math>
| <math>\text{Interpretation}</math>
|-
| height="100px" | [[File:Cactus Graph Node Big.jpg|24px]]
| <math>\mathrm{~}</math>
| <math>\mathrm{true}</math>
|-
| height="100px" | [[File:Cactus Graph Spike Big.jpg|24px]]
| <math>\texttt{(} ~ \texttt{)}</math>
| <math>\mathrm{false}</math>
|-
| height="100px" | [[File:Cactus A Big.jpg|20px]]
| <math>a</math>
| <math>a</math>
|-
| height="120px" | [[File:Cactus (A) Big.jpg|20px]]
| <math>\texttt{(} a \texttt{)}</math>
|
<math>\begin{matrix}
\tilde{a}
\\[2pt]
a^\prime
\\[2pt]
\lnot a
\\[2pt]
\mathrm{not}~ a
\end{matrix}</math>
|-
| height="100px" | [[File:Cactus ABC Big.jpg|50px]]
| <math>a~b~c</math>
|
<math>\begin{matrix}
a \land b \land c
\\[6pt]
a ~\mathrm{and}~ b ~\mathrm{and}~ c
\end{matrix}</math>
|-
| height="160px" | [[File:Cactus ((A)(B)(C)) Big.jpg|65px]]
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
|
<math>\begin{matrix}
a \lor b \lor c
\\[6pt]
a ~\mathrm{or}~ b ~\mathrm{or}~ c
\end{matrix}</math>
|-
| height="120px" | [[File:Cactus (A(B)) Big.jpg|60px]]
| <math>\texttt{(} a \texttt{(} b \texttt{))}</math>
|
<math>\begin{matrix}
a \Rightarrow b
\\[2pt]
a ~\mathrm{implies}~ b
\\[2pt]
\mathrm{if}~ a ~\mathrm{then}~ b
\\[2pt]
\mathrm{not}~ a ~\mathrm{without}~ b
\end{matrix}</math>
|-
| height="120px" | [[File:Cactus (A,B) Big.jpg|65px]]
| <math>\texttt{(} a, b \texttt{)}</math>
|
<math>\begin{matrix}
a + b
\\[2pt]
a \neq b
\\[2pt]
a ~\mathrm{exclusive~or}~ b
\\[2pt]
a ~\mathrm{not~equal~to}~ b
\end{matrix}</math>
|-
| height="160px" | [[File:Cactus ((A,B)) Big.jpg|65px]]
| <math>\texttt{((} a, b \texttt{))}</math>
|
<math>\begin{matrix}
a = b
\\[2pt]
a \iff b
\\[2pt]
a ~\mathrm{equals}~ b
\\[2pt]
a ~\mathrm{if~and~only~if}~ b
\end{matrix}</math>
|-
| height="120px" | [[File:Cactus (A,B,C) Big.jpg|65px]]
| <math>\texttt{(} a, b, c \texttt{)}</math>
|
<math>\begin{matrix}
\mathrm{just~one~of}
\\
a, b, c
\\
\mathrm{is~false}
\end{matrix}</math>
|-
| height="160px" | [[File:Cactus ((A),(B),(C)) Big.jpg|65px]]
| <math>\texttt{((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
|
<math>\begin{matrix}
\mathrm{just~one~of}
\\
a, b, c
\\
\mathrm{is~true}
\end{matrix}</math>
|-
| height="160px" | [[File:Cactus (A,(B),(C)) Big.jpg|65px]]
| <math>\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
|
<math>\begin{matrix}
\mathrm{genus}~ a ~\mathrm{of~species}~ b, c
\\[6pt]
\mathrm{partition}~ a ~\mathrm{into}~ b, c
\\[6pt]
\mathrm{pie}~ a ~\mathrm{of~slices}~ b, c
\end{matrix}</math>
|}

===Entitative Interpretation===

Table 14 illustrates the ''entitative interpretation'' of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 14.} ~~ \text{Entitative Interpretation}</math>
|- style="height:40px; background:#f0f0ff"
| <math>\text{Cactus Graph}</math>
| <math>\text{Cactus Expression}</math>
| <math>\text{Interpretation}</math>
|-
| height="100px" | [[File:Cactus Graph Node Big.jpg|24px]]
| <math>\mathrm{~}</math>
| <math>\mathrm{false}</math>
|-
| height="100px" | [[File:Cactus Graph Spike Big.jpg|24px]]
| <math>\texttt{(} ~ \texttt{)}</math>
| <math>\mathrm{true}</math>
|-
| height="100px" | [[File:Cactus A Big.jpg|20px]]
| <math>a</math>
| <math>a</math>
|-
| height="120px" | [[File:Cactus (A) Big.jpg|20px]]
| <math>\texttt{(} a \texttt{)}</math>
|
<math>\begin{matrix}
\tilde{a}
\\[2pt]
a^\prime
\\[2pt]
\lnot a
\\[2pt]
\mathrm{not}~ a
\end{matrix}</math>
|-
| height="100px" | [[File:Cactus ABC Big.jpg|50px]]
| <math>a~b~c</math>
|
<math>\begin{matrix}
a \lor b \lor c
\\[6pt]
a ~\mathrm{or}~ b ~\mathrm{or}~ c
\end{matrix}</math>
|-
| height="160px" | [[File:Cactus ((A)(B)(C)) Big.jpg|65px]]
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
|
<math>\begin{matrix}
a \land b \land c
\\[6pt]
a ~\mathrm{and}~ b ~\mathrm{and}~ c
\end{matrix}</math>
|-
| height="120px" | [[File:Cactus (A)B Big.jpg|35px]]
| <math>\texttt{(} a \texttt{)} b</math>
|
<math>\begin{matrix}
a \Rightarrow b
\\[2pt]
a ~\mathrm{implies}~ b
\\[2pt]
\mathrm{if}~ a ~\mathrm{then}~ b
\\[2pt]
\mathrm{not}~ a, ~\mathrm{or}~ b
\end{matrix}</math>
|-
| height="120px" | [[File:Cactus (A,B) Big.jpg|65px]]
| <math>\texttt{(} a, b \texttt{)}</math>
|
<math>\begin{matrix}
a = b
\\[2pt]
a \iff b
\\[2pt]
a ~\mathrm{equals}~ b
\\[2pt]
a ~\mathrm{if~and~only~if}~ b
\end{matrix}</math>
|-
| height="160px" | [[File:Cactus ((A,B)) Big.jpg|65px]]
| <math>\texttt{((} a, b \texttt{))}</math>
|
<math>\begin{matrix}
a + b
\\[2pt]
a \neq b
\\[2pt]
a ~\mathrm{exclusive~or}~ b
\\[2pt]
a ~\mathrm{not~equal~to}~ b
\end{matrix}</math>
|-
| height="120px" | [[File:Cactus (A,B,C) Big.jpg|65px]]
| <math>\texttt{(} a, b, c \texttt{)}</math>
|
<math>\begin{matrix}
\mathrm{not~just~one~of}
\\
a, b, c
\\
\mathrm{is~true}
\end{matrix}</math>
|-
| height="160px" | [[File:Cactus ((A,B,C)) Big.jpg|65px]]
| <math>\texttt{((} a, b, c \texttt{))}</math>
|
<math>\begin{matrix}
\mathrm{just~one~of}
\\
a, b, c
\\
\mathrm{is~true}
\end{matrix}</math>
|-
| height="200px" | [[File:Cactus (((A),B,C)) Big.jpg|65px]]
| <math>\texttt{(((} a \texttt{)}, b, c \texttt{))}</math>
|
<math>\begin{matrix}
\mathrm{genus}~ a ~\mathrm{of~species}~ b, c
\\[6pt]
\mathrm{partition}~ a ~\mathrm{into}~ b, c
\\[6pt]
\mathrm{pie}~ a ~\mathrm{of~slices}~ b, c
\end{matrix}</math>
|}

==Logical Graphs==

===Old Versions===

====Example 1====

{| align="center" cellpadding="10"
|
<pre>
o-----------------------------------------------------------o
| |
| o o o o o o |
| \| | | | | |
| o o o o o o o o o |
| \|/ \|/ |/ | |
| @ = @ = @ = @ |
| |
o-----------------------------------------------------------o
| |
| (()())(())() = (())(())() = (())() = ( ) |
| |
o-----------------------------------------------------------o
</pre>
|}

====Example 2====

{| align="center" cellpadding="10"
|
<pre>
o-------------------o-------------------o-------------------o
| Object | Sign | Interpretant |
o-------------------o-------------------o-------------------o
| | | |
| Falsity | "(()())(())()" | "(())(())()" |
| | | |
| Falsity | "(())(())()" | "(())()" |
| | | |
| Falsity | "(())()" | "()" |
| | | |
o-------------------o-------------------o-------------------o
</pre>
|}

====Example 3====

{| align="center" cellpadding="10"
|
<pre>
o-------------------o-------------------o-------------------o
| Object | Sign | Interpretant |
o-------------------o-------------------o-------------------o
| | | |
| Falsity | "(()())(())()" | "(()())(())()" |
| | | |
| Falsity | "(()())(())()" | "(())(())()" |
| | | |
| Falsity | "(()())(())()" | "(())()" |
| | | |
| Falsity | "(()())(())()" | "()" |
| | | |
o-------------------o-------------------o-------------------o
| | | |
| Falsity | "(())(())()" | "(()())(())()" |
| | | |
| Falsity | "(())(())()" | "(())(())()" |
| | | |
| Falsity | "(())(())()" | "(())()" |
| | | |
| Falsity | "(())(())()" | "()" |
| | | |
o-------------------o-------------------o-------------------o
| | | |
| Falsity | "(())()" | "(()())(())()" |
| | | |
| Falsity | "(())()" | "(())(())()" |
| | | |
| Falsity | "(())()" | "(())()" |
| | | |
| Falsity | "(())()" | "()" |
| | | |
o-------------------o-------------------o-------------------o
| | | |
| Falsity | "()" | "(()())(())()" |
| | | |
| Falsity | "()" | "(())(())()" |
| | | |
| Falsity | "()" | "(())()" |
| | | |
| Falsity | "()" | "()" |
| | | |
o-------------------o-------------------o-------------------o
</pre>
|}

====Example 4====

{| align="center" cellpadding="10"
|
<pre>
o-------------------o-------------------o-------------------o
| a | b | (a , b) |
o-------------------o-------------------o-------------------o
| | | |
| blank | blank | cross |
| | | |
| blank | cross | blank |
| | | |
| cross | blank | blank |
| | | |
| cross | cross | cross |
| | | |
o-------------------o-------------------o-------------------o
</pre>
|}

====Example 5====

{| align="center" cellpadding="10"
|
<pre>
o-------o-------o-------o-----------o
| a | b | c | (a, b, c) |
o-------o-------o-------o-----------o
| | | | |
| blank | blank | blank | cross |
| | | | |
| blank | blank | cross | blank |
| | | | |
| blank | cross | blank | blank |
| | | | |
| blank | cross | cross | cross |
| | | | |
| cross | blank | blank | blank |
| | | | |
| cross | blank | cross | cross |
| | | | |
| cross | cross | blank | cross |
| | | | |
| cross | cross | cross | cross |
| | | | |
o-------o-------o-------o-----------o
</pre>
|}

====Example 6====

{| align="center" cellpadding="10"
|
<pre>
o-------o-------o-------o-----------o
| a | b | c | (a, b, c) |
o-------o-------o-------o-----------o
| | | | |
| o | o | o | | |
| | | | |
| o | o | | | o |
| | | | |
| o | | | o | o |
| | | | |
| o | | | | | | |
| | | | |
| | | o | o | o |
| | | | |
| | | o | | | | |
| | | | |
| | | | | o | | |
| | | | |
| | | | | | | | |
| | | | |
o-------o-------o-------o-----------o
</pre>
|}

===New Versions===

====Example 1====

…

====Example 2a====

{| align="center" border="1" cellpadding="12" cellspacing="0" width="50%"
|- style="background:ghostwhite; height:40px"
| width="33%" | <math>\text{Object}</math>
| width="33%" | <math>\text{Sign}</math>
| width="33%" | <math>\text{Interpretant}</math>
|-
| <math>\mathrm{Falsity}</math>
| <math>{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}</math>
|-
| <math>\mathrm{Falsity}</math>
| <math>{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}</math>
|-
| <math>\mathrm{Falsity}</math>
| <math>{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}</math>
|}

====Example 2b====

{| align="center" border="1" cellpadding="12" cellspacing="0" width="50%"
|- style="background:ghostwhite; height:40px"
| width="33%" | <math>\text{Object}</math>
| width="33%" | <math>\text{Sign}</math>
| width="33%" | <math>\text{Interpretant}</math>
|-
| valign="bottom" |
<math>\begin{array}{l}
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\end{array}</math>
| valign="bottom" |
<math>\begin{array}{l}
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\end{array}</math>
| valign="bottom" |
<math>\begin{array}{l}
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\end{array}</math>
|}

====Example 3====

{| align="center" border="1" cellpadding="12" cellspacing="0" width="50%"
|- style="background:ghostwhite; height:40px"
| width="33%" | <math>\text{Object}</math>
| width="33%" | <math>\text{Sign}</math>
| width="33%" | <math>\text{Interpretant}</math>
|-
| valign="bottom" |
<math>\begin{array}{l}
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\end{array}</math>
| valign="bottom" |
<math>\begin{array}{l}
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\end{array}</math>
| valign="bottom" |
<math>\begin{array}{l}
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\end{array}</math>
|-
| valign="bottom" |
<math>\begin{array}{l}
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\end{array}</math>
| valign="bottom" |
<math>\begin{array}{l}
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\end{array}</math>
| valign="bottom" |
<math>\begin{array}{l}
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\end{array}</math>
|-
| valign="bottom" |
<math>\begin{array}{l}
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\end{array}</math>
| valign="bottom" |
<math>\begin{array}{l}
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\end{array}</math>
| valign="bottom" |
<math>\begin{array}{l}
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\end{array}</math>
|-
| valign="bottom" |
<math>\begin{array}{l}
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\end{array}</math>
| valign="bottom" |
<math>\begin{array}{l}
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\end{array}</math>
| valign="bottom" |
<math>\begin{array}{l}
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\end{array}</math>
|}

====Example 4====

{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:50%"
|- style="background:ghostwhite; height:40px"
| width="33%" | <math>a</math>
| width="33%" | <math>b</math>
| width="33%" | <math>\texttt{(} a \texttt{,} b \texttt{)}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\texttt{Blank}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Cross}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{Cross}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\end{matrix}</math>
|}

====Example 5====

{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:50%"
|- style="background:ghostwhite; height:40px"
| width="25%" | <math>a</math>
| width="25%" | <math>b</math>
| width="25%" | <math>c</math>
| width="25%" | <math>\texttt{(} a \texttt{,} b \texttt{,} \texttt{c} \texttt{)}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\texttt{Blank}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Cross}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{Blank}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Cross}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{Cross}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Cross}
\end{matrix}</math>
|}

====Example 6====

{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:50%"
|- style="background:ghostwhite; height:40px"
| width="25%" | <math>a</math>
| width="25%" | <math>b</math>
| width="25%" | <math>c</math>
| width="25%" | <math>\texttt{(} a \texttt{,} b \texttt{,} \texttt{c} \texttt{)}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\texttt{o}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{|}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{o}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{|}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{o}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{|}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{|}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{|}
\end{matrix}</math>
|}

User:Jon Awbrey/Figures and Tables 54

2026-03-12T13:48:06Z

Jon Awbrey: + User:Jon Awbrey/Figures and Tables 54

==Syntax and Semantics of a Calculus for Propositional Logic==

===PNG===

Table 1 collects a sample of basic propositional forms as expressed in terms of cactus language connectives.

{| align="center" style="font-size:large; text-align:center"
|+ height="30px" | <math>\text{Table 1. Syntax and Semantics of a Calculus for Propositional Logic}</math>
|-
| [[File:Syntax and Semantics of a Calculus for Propositional Logic 4.0.png|600px]]
|}

===Wiki + LaTeX + JPG===

Table 1 outlines a notation for propositional calculus based on two types of logical connectives, both of variable <math>k</math>-ary scope.

<br>

{| align="center" border="1" cellspacing="0" style="font-size:large;text-align:center;width:60%"
|+ style="height:30px" |
<math>\text{Table 1. Syntax and Semantics of a Calculus for Propositional Logic}</math>
|- style="height:40px; background:#f0f0ff"
| <math>\text{Graph}</math>
| <math>\text{Expression}</math>
| <math>\text{Interpretation}</math>
| <math>\text{Other Notations}</math>
|-
| height="100px" | [[File:Rooted Node.jpg|20px]]
| <math>~</math>
| <math>\mathrm{true}</math>
| <math>1</math>
|-
| height="100px" | [[File:Rooted Edge.jpg|20px]]
| <math>\texttt{(}~\texttt{)}</math>
| <math>\mathrm{false}</math>
| <math>0</math>
|-
| height="100px" | [[File:Cactus A Big.jpg|20px]]
| <math>a</math>
| <math>a</math>
| <math>a</math>
|-
| height="120px" | [[File:Cactus (A) Big.jpg|20px]]
| <math>\texttt{(} a \texttt{)}</math>
| <math>\mathrm{not}~ a</math>
| <math>\lnot a \quad \bar{a} \quad \tilde{a} \quad a^\prime</math>
|-
| height="100px" | [[File:Cactus ABC Big.jpg|50px]]
| <math>a ~ b ~ c</math>
| <math>a ~\mathrm{and}~ b ~\mathrm{and}~ c</math>
| <math>a \land b \land c</math>
|-
| height="160px" | [[File:Cactus ((A)(B)(C)) Big.jpg|65px]]
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
| <math>a ~\mathrm{or}~ b ~\mathrm{or}~ c</math>
| <math>a \lor b \lor c</math>
|-
| height="120px" | [[File:Cactus (A(B)) Big.jpg|60px]]
| <math>\texttt{(} a \texttt{(} b \texttt{))}</math>
|
<math>\begin{matrix}
a ~\mathrm{implies}~ b
\\[6pt]
\mathrm{if}~ a ~\mathrm{then}~ b
\end{matrix}</math>
| <math>a \Rightarrow b</math>
|-
| height="120px" | [[File:Cactus (A,B) Big.jpg|65px]]
| <math>\texttt{(} a \texttt{,} b \texttt{)}</math>
|
<math>\begin{matrix}
a ~\mathrm{not~equal~to}~ b
\\[6pt]
a ~\mathrm{exclusive~or}~ b
\end{matrix}</math>
|
<math>\begin{matrix}
a \neq b
\\[6pt]
a + b
\end{matrix}</math>
|-
| height="160px" | [[File:Cactus ((A,B)) Big.jpg|65px]]
| <math>\texttt{((} a \texttt{,} b \texttt{))}</math>
|
<math>\begin{matrix}
a ~\mathrm{is~equal~to}~ b
\\[6pt]
a ~\mathrm{if~and~only~if}~ b
\end{matrix}</math>
|
<math>\begin{matrix}
a = b
\\[6pt]
a \Leftrightarrow b
\end{matrix}</math>
|-
| height="120px" | [[File:Cactus (A,B,C) Big.jpg|65px]]
| <math>\texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}</math>
|
<math>\begin{matrix}
\mathrm{just~one~of}
\\
a, b, c
\\
\mathrm{is~false}.
\end{matrix}</math>
|
<math>\begin{matrix}
& \bar{a} ~ b ~ c
\\
\lor & a ~ \bar{b} ~ c
\\
\lor & a ~ b ~ \bar{c}
\end{matrix}</math>
|-
| height="160px" | [[File:Cactus ((A),(B),(C)) Big.jpg|65px]]
| <math>\texttt{((} a \texttt{),(} b \texttt{),(} c \texttt{))}</math>
|
<math>\begin{matrix}
\mathrm{just~one~of}
\\
a, b, c
\\
\mathrm{is~true}.
\\[6pt]
\mathrm{partition~all}
\\
\mathrm{into}~ a, b, c.
\end{matrix}</math>
|
<math>\begin{matrix}
& a ~ \bar{b} ~ \bar{c}
\\
\lor & \bar{a} ~ b ~ \bar{c}
\\
\lor & \bar{a} ~ \bar{b} ~ c
\end{matrix}</math>
|-
| height="160px" | [[File:Cactus (A,(B,C)) Big.jpg|90px]]
| <math>\texttt{(} a \texttt{,(} b \texttt{,} c \texttt{))}</math>
|
<math>\begin{matrix}
\mathrm{oddly~many~of}
\\
a, b, c
\\
\mathrm{are~true}.
\end{matrix}</math>
|
<p><math>a + b + c</math></p>
<br>
<p><math>\begin{matrix}
& a ~ b ~ c
\\
\lor & a ~ \bar{b} ~ \bar{c}
\\
\lor & \bar{a} ~ b ~ \bar{c}
\\
\lor & \bar{a} ~ \bar{b} ~ c
\end{matrix}</math></p>
|-
| height="160px" | [[File:Cactus (X,(A),(B),(C)) Big.jpg|90px]]
| <math>\texttt{(} x \texttt{,(} a \texttt{),(} b \texttt{),(} c \texttt{))}</math>
|
<math>\begin{matrix}
\mathrm{partition}~ x
\\
\mathrm{into}~ a, b, c.
\\[6pt]
\mathrm{genus}~ x ~\mathrm{comprises}
\\
\mathrm{species}~ a, b, c.
\end{matrix}</math>
|
<math>\begin{matrix}
& \bar{x} ~ \bar{a} ~ \bar{b} ~ \bar{c}
\\
\lor & x ~ a ~ \bar{b} ~ \bar{c}
\\
\lor & x ~ \bar{a} ~ b ~ \bar{c}
\\
\lor & x ~ \bar{a} ~ \bar{b} ~ c
\end{matrix}</math>
|}

<br>

===Alt Text Work Area===

Table 1 is the first of several “Rosetta Stones” we'll use in this discussion to translate between different languages for the same subject matters. The present Table displays equivalent expressions for frequently encountered propositional forms in four notations for propositional calculus. The Table has four columns, labeled “Graph”, “Expression”, “Interpretation”, and “Other Notations”, respectively.

• Column 1 “Graph” exhibits a logical graph for a commonly occurring propositional form.

• Column 2 “Expression” exhibits the text string transcription of the graph in Column 1.

• Column 3 “Interpretation” gives one or more verbal formulas for the graph in Column 1.

• Column 4 “Other Notations” shows several ways of notating the graph's logical meaning.

{|
| [[File:Rooted Node.jpg|20px]] || ~ || true || 1
|}

{|
| [[File:Rooted Edge.jpg|20px]] || ( ) || false || 0
|}

{|
| [[File:Cactus A Big.jpg|20px]] || a || a || a
|}

{|
| [[File:Cactus (A) Big.jpg|20px]] || ( a ) || not a || ¬a ā ã a′
|}

{|
| [[File:Cactus ABC Big.jpg|50px]] || abc || a and b and c || a &and; b &and; c
|}

{|
| [[File:Cactus ((A)(B)(C)) Big.jpg|65px]] || ((a)(b)(c)) || a or b or c || a &or; b &or; c
|}

{|
| [[File:Cactus (A(B)) Big.jpg|60px]]
| ( a ( b ))
|
{|
| a implies b
|-
| if a then b
|}
| a ⇒ b
|}

{|
| [[File:Cactus (A,B) Big.jpg|65px]]
| ( a , b )
|
{|
| a not equal to b
|-
| a exclusive or b
|}
|
{|
| a ≠ b
|-
| a + b
|}
|}

{|
| [[File:Cactus ((A,B)) Big.jpg|65px]]
| (( a , b ))
|
{|
| a equal to b
|-
| a if and only if b
|}
|
{|
| a = b
|-
| a ⇔ b
|}
|}

{|
| [[File:Cactus (A,B,C) Big.jpg|65px]]
| ( a , b , c )
| just one of a, b, c is false.
|
{|
| ¬a &and; b &and; c
|-
| &or; a &and; ¬b &and; c
|-
| &or; a &and; b &and; ¬c
|}
|}

{|
| [[File:Cactus ((A),(B),(C)) Big.jpg|65px]]
| (( a ),( b ),( c ))
|
{|
| just one of a, b, c is true.
|-
| partition all into a, b, c.
|}
|
{|
| a &and; ¬b &and; ¬c
|-
| &or; ¬a &and; b &and; ¬c
|-
| &or; ¬a &and; ¬b &and; c
|}
|}

{|
| [[File:Cactus (A,(B,C)) Big.jpg|90px]]
| ( a ,( b , c ))
| oddly many of a, b, c are true.
|
{|
|    a + b + c
|-
{|
|    a &and; b &and; c
|-
| &or; a &and; ¬b &and; ¬c
|-
| &or; ¬a &and; b &and; ¬c
|-
| &or; ¬a &and; ¬b &and; c
|}
|}

{|
| [[File:Cactus (X,(A),(B),(C)) Big.jpg|90px]]
| ( x ,( a ),( b ),( c ))
|
{|
| partition x into a, b, c.
|-
| genus x of species a, b, c.
|}
|
{|
| ¬x &and; ¬a &and; ¬b &and; ¬c
|-
| &or; x &and; a &and; ¬b &and; ¬c
|-
| &or; x &and; ¬a &and; b &and; ¬c
|-
| &or; x &and; ¬a &and; ¬b &and; c
|}
|}

==Mathstodon Versions==

===Differential Logic and Dynamic Systems • Overview===
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overview

❝Stand and unfold yourself.❞
— Hamlet • Francisco • 1.1.2

In modeling intelligent systems, whether we are trying to understand a natural system or engineer an artificial system, there has long been a tension or trade-off between dynamic paradigms and symbolic paradigms. Dynamic models take their cue from physics, using quantitative measures and differential equations to model the evolution of a system’s state through time. Symbolic models use logical methods to describe systems and their agents in qualitative terms, deriving logical consequences of a system’s description or an agent’s state of information. Logic-based systems have tended to be static in character, largely because we have lacked a proper logical analogue of differential calculus. The work laid out in this report is intended to address that lack.

This article develops a differential extension of propositional calculus and applies it to the analysis of dynamic systems whose states are described in qualitative logical terms.

The work pursued here is coordinated with a parallel application focusing on neural network systems but the dependencies are arranged to make the present article the main and the more self-contained work, to serve as a conceptual frame and a technical background for the network project.

===Differential Logic and Dynamic Systems • Review and Transition 1===
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1#Review_and_Transition

This note continues a previous discussion on the problem of dealing with change and diversity in logic-based intelligent systems. It is useful to begin by summarizing essential material from previous reports.

Table 1 outlines a notation for propositional calculus based on two types of logical connectives, both of variable $k$-ary scope.

• A bracketed list of propositional expressions in the form $\texttt{(} e_1 \texttt{,} e_2 \texttt{,} \ldots \texttt{,} e_{k-1} \texttt{,} e_k \texttt{)}$ indicates that exactly one of the propositions $e_1, e_2, \ldots, e_{k-1}, e_k$ is false.

• A concatenation of propositional expressions in the form $e_1 ~ e_2 ~ \ldots ~ e_{k-1} ~ e_k$ indicates that all of the propositions $e_1, e_2, \ldots, e_{k-1}, e_k$ are true, in other words, that their logical conjunction is true.

All other propositional connectives can be obtained in a very efficient style of representation through combinations of these two forms. Strictly speaking, the concatenation form is dispensable in light of the bracketed form but it is convenient to maintain it as an abbreviation of more complicated bracket expressions.

==Review and Transition (OEIS Version)==

This note continues a previous discussion on the problem of dealing with change and diversity in logic-based intelligent systems.  It is useful to begin by summarizing essential material from previous reports.

Table 1 outlines a notation for propositional calculus based on two types of logical connectives, both of variable <math>k</math>-ary scope.

* A bracketed list of propositional expressions in the form <math>\texttt{(} e_1 \texttt{,} e_2 \texttt{,} \ldots \texttt{,} e_{k-1} \texttt{,} e_k \texttt{)}</math> indicates that exactly one of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> is false.

* A concatenation of propositional expressions in the form <math>e_1 ~ e_2 ~ \ldots ~ e_{k-1} ~ e_k</math> indicates that all of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> are true, in other words, that their [[logical conjunction]] is true.

All other propositional connectives can be obtained in a very efficient style of representation through combinations of these two forms.  Strictly speaking, the concatenation form is dispensable in light of the bracketed form but it is convenient to maintain it as an abbreviation of more complicated bracket expressions.

{| align="center" style="font-size:larger; text-align:center; width:100%"
| height="20px" | <math>\text{Table 1. Syntax and Semantics of a Calculus for Propositional Logic}</math>
|-
| [[File:Syntax and Semantics of a Calculus for Propositional Logic 4.0.png|600px]]
|}

Table 1 is the first of several “Rosetta Stones” we'll use in this discussion to translate between different languages for the same subject matters.  In this case the Table displays equivalent expressions for simple examples of propositional forms in four notations for propositional calculus.

* Column 1 shows the logical graphs used to represent a number of simple propositional forms.
* Column 2 shows the traverse strings corresponding to the logical graphs in Column 1.
* Column 3 interprets the graph and string by means of conventional verbal formulas.
* Column 4 translates the interpretation into a number of symbolic notations.

This treatment of propositional logic is derived from the work of C.S. Peirce [P1, P2], who gave this approach an extensive development in his graphical systems of predicate, relational, and modal logic [Rob]. More recently, these ideas were revived and supplemented in an alternative interpretation by George Spencer-Brown [SpB]. Both of these authors used other forms of enclosure where I use parentheses, but the structural topologies of expression and the functional varieties of interpretation are fundamentally the same.

While working with expressions solely in propositional calculus, it is easiest to use plain parentheses for logical connectives. In contexts where parentheses are needed for other purposes “teletype” parentheses <math>\texttt{(} \ldots \texttt{)}</math> or barred parentheses <math>(\!| \ldots |\!)</math> may be used for logical operators.

The briefest expression for logical truth is the empty word, usually denoted by <math>{}^{\backprime\backprime} \boldsymbol\varepsilon {}^{\prime\prime}</math> or <math>{}^{\backprime\backprime} \boldsymbol\lambda {}^{\prime\prime}</math> in formal languages, where it forms the identity element for concatenation. To make it visible in this text, it may be denoted by the equivalent expression <math>{}^{\backprime\backprime} \texttt{((} ~ \texttt{))} {}^{\prime\prime},</math> or, especially if operating in an algebraic context, by a simple <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}.</math> Also when working in an algebraic mode, the plus sign <math>{}^{\backprime\backprime} + {}^{\prime\prime}</math> may be used for [[exclusive disjunction]]. For example, we have the following paraphrases of algebraic expressions by bracket expressions:

{| align="center" cellpadding="6" style="text-align:center"
|
<math>\begin{matrix}
x + y ~=~ \texttt{(} x, y \texttt{)}
\\[6pt]
x + y + z ~=~ \texttt{((} x, y \texttt{)}, z \texttt{)} ~=~ \texttt{(} x, \texttt{(} y, z \texttt{))}
\end{matrix}</math>
|}

It is important to note that the last expressions are not equivalent to the triple bracket <math>\texttt{(} x, y, z \texttt{)}.</math>

<b>Note.</b> The usage that one often sees, of a plus sign "<math>+</math>" to represent inclusive disjunction, and the reference to this operation as ''boolean addition'', is a misnomer on at least two counts. Boole used the plus sign to represent exclusive disjunction (at any rate, an operation of aggregation restricted in its logical interpretation to cases where the represented sets are disjoint (Boole, 32)), as any mathematician with a sensitivity to the ring and field properties of algebra would do:

<blockquote>
The expression <math>x + y</math> seems indeed uninterpretable, unless it be assumed that the things represented by <math>x</math> and the things represented by <math>y</math> are entirely separate; that they embrace no individuals in common. (Boole, 66).
</blockquote>

It was only later that Peirce and Jevons treated inclusive disjunction as a fundamental operation, but these authors, with a respect for the algebraic properties that were already associated with the plus sign, used a variety of other symbols for inclusive disjunction (Sty, 177, 189). It seems to have been Schröder who later reassigned the plus sign to inclusive disjunction (Sty, 208). Additional information, discussion, and references can be found in (Boole) and (Sty, 177–263). Aside from these historical points, which never really count against a current practice that has gained a life of its own, this usage does have a further disadvantage of cutting or confounding the lines of communication between algebra and logic. For this reason, it will be avoided here.

User:Jon Awbrey/Figures and Tables 53

2026-03-12T13:45:19Z

Jon Awbrey: + User:Jon Awbrey/Figures and Tables 53 + modify use of tilde "~" spaces for this wiki

==Propositional Forms on Two Variables (Index Order)==

===PNG===

{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"
|+ style="height:25px; font-size:large" | <math>\text{Table A1. Propositional Forms on Two Variables (Index Order)}</math>
|-
| [[File:Propositional Forms on Two Variables (Index Order) 1.0.png|600px]]
|}

===LaTeX===

{| align="center" border="1" cellspacing="0" style="font-size:larger; text-align:center; width:80%"
|+ style="height:30px; font-size:large" | <math>\text{Table A1. Propositional Forms on Two Variables (Index Order)}</math>
|- style="background:ghostwhite"
| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math>
| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math>
|- style="background:ghostwhite"
|  
| align="right" | <math>p\colon</math>
| <math>1~1~0~0</math>
|   ||   ||  
|- style="background:ghostwhite"
|  
| align="right" | <math>q\colon</math>
| <math>1~0~1~0</math>
|   ||   ||  
|-
|
<math>\begin{matrix}
f_{0}\\f_{1}\\f_{2}\\f_{3}\\f_{4}\\f_{5}\\f_{6}\\f_{7}
\end{matrix}</math>
|
<math>\begin{matrix}
f_{0000}\\f_{0001}\\f_{0010}\\f_{0011}\\f_{0100}\\f_{0101}\\f_{0110}\\f_{0111}
\end{matrix}</math>
|
<math>\begin{matrix}
0~0~0~0\\0~0~0~1\\0~0~1~0\\0~0~1~1\\0~1~0~0\\0~1~0~1\\0~1~1~0\\0~1~1~1
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} ~~ \texttt{)}
\\
\texttt{(} p \texttt{)(} q \texttt{)}
\\
\texttt{(} p \texttt{)} ~~ q ~~
\\
\texttt{(} p \texttt{)} ~ ~ ~ ~
\\
~~ p ~~ \texttt{(} q \texttt{)}
\\
~ ~ ~ ~ \texttt{(} q \texttt{)}
\\
\texttt{(} p \texttt{,} ~~ q \texttt{)}
\\
\texttt{(} p ~ ~ ~ q \texttt{)}
\end{matrix}</math>
|
<math>\begin{matrix}
\text{false}
\\
\text{neither}~ p ~\text{nor}~ q
\\
q ~\text{without}~ p
\\
\text{not}~ p
\\
p ~\text{without}~ q
\\
\text{not}~ q
\\
p ~\text{not equal to}~ q
\\
\text{not both}~ p ~\text{and}~ q
\end{matrix}</math>
|
<math>\begin{matrix}
0
\\
\lnot p \land \lnot q
\\
\lnot p \land q
\\
\lnot p
\\
p \land \lnot q
\\
\lnot q
\\
p \ne q
\\
\lnot p \lor \lnot q
\end{matrix}</math>
|-
|
<math>\begin{matrix}
f_{8}\\f_{9}\\f_{10}\\f_{11}\\f_{12}\\f_{13}\\f_{14}\\f_{15}
\end{matrix}</math>
|
<math>\begin{matrix}
f_{1000}\\f_{1001}\\f_{1010}\\f_{1011}\\f_{1100}\\f_{1101}\\f_{1110}\\f_{1111}
\end{matrix}</math>
|
<math>\begin{matrix}
1~0~0~0\\1~0~0~1\\1~0~1~0\\1~0~1~1\\1~1~0~0\\1~1~0~1\\1~1~1~0\\1~1~1~1
\end{matrix}</math>
|
<math>\begin{matrix}
p ~ ~ ~ q
\\
\texttt{((} p \texttt{,} ~~ q \texttt{))}
\\
~ ~ ~ ~ q
\\
~~ \texttt{(} p ~~ \texttt{(} q \texttt{))}
\\
p ~ ~ ~ ~
\\
\texttt{((} p \texttt{)} ~~ q \texttt{)} ~~
\\
\texttt{((} p \texttt{)(} q \texttt{))}
\\
\texttt{((} ~~ \texttt{))}
\end{matrix}</math>
|
<math>\begin{matrix}
p ~\text{and}~ q
\\
p ~\text{equal to}~ q
\\
q
\\
\text{not}~ p ~\text{without}~ q
\\
p
\\
\text{not}~ q ~\text{without}~ p
\\
p ~\text{or}~ q
\\
\text{true}
\end{matrix}</math>
|
<math>\begin{matrix}
p \land q
\\
p = q
\\
q
\\
p \Rightarrow q
\\
p
\\
p \Leftarrow q
\\
p \lor q
\\
1
\end{matrix}</math>
|}

==Propositional Forms on Two Variables (Orbit Order)==

===PNG===

{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"
|+ style="height:25px; font-size:large" | <math>\text{Table A2. Propositional Forms on Two Variables (Orbit Order)}</math>
|-
| [[File:Propositional Forms on Two Variables (Orbit Order) 1.0.png|600px]]
|}

===LaTeX===

{| align="center" border="1" cellspacing="0" style="font-size:larger; text-align:center; width:80%"
|+ style="height:30px; font-size:large" | <math>\text{Table A2. Propositional Forms on Two Variables (Orbit Order)}</math>
|- style="background:ghostwhite"
| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math>
| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math>
|- style="background:ghostwhite"
|  
| align="right" | <math>p\colon</math>
| <math>1~1~0~0</math>
|   ||   ||  
|- style="background:ghostwhite"
|  
| align="right" | <math>q\colon</math>
| <math>1~0~1~0</math>
|   ||   ||  
|-
| <math>f_{0}</math>
| <math>f_{0000}</math>
| <math>0~0~0~0</math>
| <math>\texttt{(} ~~ \texttt{)}</math>
| <math>\text{false}</math>
| <math>0</math>
|-
|
<math>\begin{matrix}
f_{1}\\f_{2}\\f_{4}\\f_{8}
\end{matrix}</math>
|
<math>\begin{matrix}
f_{0001}\\f_{0010}\\f_{0100}\\f_{1000}
\end{matrix}</math>
|
<math>\begin{matrix}
0~0~0~1\\0~0~1~0\\0~1~0~0\\1~0~0~0
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} p \texttt{)(} q \texttt{)}
\\
\texttt{(} p \texttt{)} ~~ q ~~
\\
~~ p ~~ \texttt{(} q \texttt{)}
\\
p ~ ~ ~ q
\end{matrix}</math>
|
<math>\begin{matrix}
\text{neither}~ p ~\text{nor}~ q
\\
q ~\text{without}~ p
\\
p ~\text{without}~ q
\\
p ~\text{and}~ q
\end{matrix}</math>
|
<math>\begin{matrix}
\lnot p \land \lnot q
\\
\lnot p \land q
\\
p \land \lnot q
\\
p \land q
\end{matrix}</math>
|-
|
<math>\begin{matrix}
f_{3}\\f_{12}
\end{matrix}</math>
|
<math>\begin{matrix}
f_{0011}\\f_{1100}
\end{matrix}</math>
|
<math>\begin{matrix}
0~0~1~1\\1~1~0~0
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} p \texttt{)}
\\
p
\end{matrix}</math>
|
<math>\begin{matrix}
\text{not}~ p
\\
p
\end{matrix}</math>
|
<math>\begin{matrix}
\lnot p
\\
p
\end{matrix}</math>
|-
|
<math>\begin{matrix}
f_{6}\\f_{9}
\end{matrix}</math>
|
<math>\begin{matrix}
f_{0110}\\f_{1001}
\end{matrix}</math>
|
<math>\begin{matrix}
0~1~1~0\\1~0~0~1
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} p \texttt{,} ~~ q \texttt{)}
\\
\texttt{((} p \texttt{,} ~~ q \texttt{))}
\end{matrix}</math>
|
<math>\begin{matrix}
p ~\text{not equal to}~ q
\\
p ~\text{equal to}~ q
\end{matrix}</math>
|
<math>\begin{matrix}
p \ne q
\\
p = q
\end{matrix}</math>
|-
|
<math>\begin{matrix}
f_{5}\\f_{10}
\end{matrix}</math>
|
<math>\begin{matrix}
f_{0101}\\f_{1010}
\end{matrix}</math>
|
<math>\begin{matrix}
0~1~0~1\\1~0~1~0
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} q \texttt{)}
\\
q
\end{matrix}</math>
|
<math>\begin{matrix}
\text{not}~ q
\\
q
\end{matrix}</math>
|
<math>\begin{matrix}
\lnot q
\\
q
\end{matrix}</math>
|-
|
<math>\begin{matrix}
f_{7}\\f_{11}\\f_{13}\\f_{14}
\end{matrix}</math>
|
<math>\begin{matrix}
f_{0111}\\f_{1011}\\f_{1101}\\f_{1110}
\end{matrix}</math>
|
<math>\begin{matrix}
0~1~1~1\\1~0~1~1\\1~1~0~1\\1~1~1~0
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} p ~ ~ ~ q \texttt{)}
\\
~~ \texttt{(} p ~~ \texttt{(} q \texttt{))}
\\
\texttt{((} p \texttt{)} ~~ q \texttt{)} ~~
\\
\texttt{((} p \texttt{)(} q \texttt{))}
\end{matrix}</math>
|
<math>\begin{matrix}
\text{not both}~ p ~\text{and}~ q
\\
\text{not}~ p ~\text{without}~ q
\\
\text{not}~ q ~\text{without}~ p
\\
p ~\text{or}~ q
\end{matrix}</math>
|
<math>\begin{matrix}
\lnot p \lor \lnot q
\\
p \Rightarrow q
\\
p \Leftarrow q
\\
p \lor q
\end{matrix}</math>
|-
| <math>f_{15}</math>
| <math>f_{1111}</math>
| <math>1~1~1~1</math>
| <math>\texttt{((} ~~ \texttt{))}</math>
| <math>\text{true}</math>
| <math>1</math>
|}

==Ef Expanded over Ordinary Variables==

===Version 3.0===

====PNG====

{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"
|+ style="height:25px; font-size:large" | <math>\text{Table A3.}~ \mathrm{E}f ~\text{Expanded over Ordinary Variables}~ \{ p, q \}</math>
|-
| [[File:Ef Expanded Over Ordinary Variables p, q 3.0.png|800px]]
|}

===Version 2.0===

====PNG====

{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"
|+ style="height:25px; font-size:large" | <math>\text{Table A3.}~ \mathrm{E}f ~\text{Expanded over Ordinary Variables}~ \{ p, q \}</math>
|-
| [[File:Ef Expanded Over Ordinary Variables p, q 2.0.png|800px]]
|}

====LaTeX====

{| align="center" cellpadding="4" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; font-size:larger; text-align:center; width:90%"
|+ style="height:30px; font-size:large" | <math>\text{Table A3.}~ \mathrm{E}f ~\text{Expanded over Ordinary Variables}~ \{ p, q \}</math>
|- style="background:ghostwhite"
| style="width:5%; border-bottom:1px solid black" |  
| style="width:15%; border-bottom:1px solid black; border-left:1px solid black" | <math>f</math>
| style="width:20%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{E}f</math>
| style="width:15%; border-bottom:1px solid black; border-left:4px double black" |
<math>\mathrm{E}f|_{pq}</math>
| style="width:15%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{E}f|_{p \texttt{(} q \texttt{)}}</math>
| style="width:15%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{E}f|_{\texttt{(} p \texttt{)} q}</math>
| style="width:15%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{E}f|_{\texttt{(} p \texttt{)(} q \texttt{)}}</math>
|-
| style="border-top:1px solid black" | <math>f_{0}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{1}\\f_{2}\\f_{4}\\f_{8}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{)(} q \texttt{)}
\\
\texttt{(} p \texttt{)} ~~ q ~~
\\
~~ p ~~ \texttt{(} q \texttt{)}
\\
p ~ ~ ~ q
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} p \texttt{,} \mathrm{d}p \texttt{))((} q \texttt{,} \mathrm{d}q \texttt{))}
\\
\texttt{((} p \texttt{,} \mathrm{d}p \texttt{))} ~~ \texttt{(} q \texttt{,} \mathrm{d}q \texttt{)} ~~
\\
~~ \texttt{(} p \texttt{,} \mathrm{d}p \texttt{)} ~~ \texttt{((} q \texttt{,} \mathrm{d}q \texttt{))}
\\
\texttt{(} p \texttt{,} \mathrm{d}p \texttt{)} ~ ~ ~ \texttt{(} q \texttt{,} \mathrm{d}q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}p ~ ~ ~ \mathrm{d}q
\\
~~ \mathrm{d}p ~~ \texttt{(} \mathrm{d}q \texttt{)}
\\
\texttt{(} \mathrm{d}p \texttt{)} ~~ \mathrm{d}q ~~
\\
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~~ \mathrm{d}p ~~ \texttt{(} \mathrm{d}q \texttt{)}
\\
\mathrm{d}p ~ ~ ~ \mathrm{d}q
\\
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)}
\\
\texttt{(} \mathrm{d}p \texttt{)} ~~ \mathrm{d}q ~~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}p \texttt{)} ~~ \mathrm{d}q ~~
\\
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)}
\\
\mathrm{d}p ~ ~ ~ \mathrm{d}q
\\
~~ \mathrm{d}p ~~ \texttt{(} \mathrm{d}q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)}
\\
\texttt{(} \mathrm{d}p \texttt{)} ~~ \mathrm{d}q ~~
\\
~~ \mathrm{d}p ~~ \texttt{(} \mathrm{d}q \texttt{)}
\\
\mathrm{d}p ~ ~ ~ \mathrm{d}q
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{3}\\f_{12}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{)}
\\
p
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} p \texttt{,} \mathrm{d}p \texttt{))}
\\
\texttt{(} p \texttt{,} \mathrm{d}p \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}p
\\
\texttt{(} \mathrm{d}p \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\mathrm{d}p
\\
\texttt{(} \mathrm{d}p \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}p \texttt{)}
\\
\mathrm{d}p
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}p \texttt{)}
\\
\mathrm{d}p
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{6}\\f_{9}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{,} ~~ q \texttt{)}
\\
\texttt{((} p \texttt{,} ~~ q \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} p \texttt{,} \mathrm{d}p \texttt{),} ~~ \texttt{(} q \texttt{,} \mathrm{d}q \texttt{))}
\\
\texttt{(((} p \texttt{,} \mathrm{d}p \texttt{),} ~~ \texttt{(} q \texttt{,} \mathrm{d}q \texttt{)))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}p \texttt{,} ~~ \mathrm{d}q \texttt{)}
\\
\texttt{((} \mathrm{d}p \texttt{,} ~~ \mathrm{d}q \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} \mathrm{d}p \texttt{,} ~~ \mathrm{d}q \texttt{))}
\\
\texttt{(} \mathrm{d}p \texttt{,} ~~ \mathrm{d}q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} \mathrm{d}p \texttt{,} ~~ \mathrm{d}q \texttt{))}
\\
\texttt{(} \mathrm{d}p \texttt{,} ~~ \mathrm{d}q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}p \texttt{,} ~~ \mathrm{d}q \texttt{)}
\\
\texttt{((} \mathrm{d}p \texttt{,} ~~ \mathrm{d}q \texttt{))}
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{5}\\f_{10}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} q \texttt{)}
\\
q
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} q \texttt{,} \mathrm{d}q \texttt{))}
\\
\texttt{(} q \texttt{,} \mathrm{d}q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}q
\\
\texttt{(} \mathrm{d}q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}q \texttt{)}
\\
\mathrm{d}q
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\mathrm{d}q
\\
\texttt{(} \mathrm{d}q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}q \texttt{)}
\\
\mathrm{d}q
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{7}\\f_{11}\\f_{13}\\f_{14}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ p ~ ~ ~ q ~~ \texttt{)}
\\
\texttt{(} ~~ p ~~ \texttt{(} q \texttt{))}
\\
\texttt{((} p \texttt{)} ~~ q ~~ \texttt{)}
\\
\texttt{((} p \texttt{)(} q \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ \texttt{(} p \texttt{,} \mathrm{d}p \texttt{)} ~ ~ ~
\texttt{(} q \texttt{,} \mathrm{d}q \texttt{)} ~~ \texttt{)}
\\
\texttt{(} ~~ \texttt{(} p \texttt{,} \mathrm{d}p \texttt{)} ~~ \texttt{((} q \texttt{,} \mathrm{d}q \texttt{)))}
\\
\texttt{(((} p \texttt{,} \mathrm{d}p \texttt{))} ~~ \texttt{(} q \texttt{,} \mathrm{d}q \texttt{)} ~~ \texttt{)}
\\
\texttt{(((} p \texttt{,} \mathrm{d}p \texttt{))((} q \texttt{,} \mathrm{d}q \texttt{)))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{))}
\\
\texttt{((} \mathrm{d}p \texttt{)} ~~ \mathrm{d}q ~~ \texttt{)}
\\
\texttt{(} ~~ \mathrm{d}p ~~ \texttt{(} \mathrm{d}q \texttt{))}
\\
\texttt{(} ~~ \mathrm{d}p ~ ~ ~ \mathrm{d}q ~~ \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} \mathrm{d}p \texttt{)} ~~ \mathrm{d}q ~~ \texttt{)}
\\
\texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{))}
\\
\texttt{(} ~~ \mathrm{d}p ~ ~ ~ \mathrm{d}q ~~ \texttt{)}
\\
\texttt{(} ~~ \mathrm{d}p ~~ \texttt{(} \mathrm{d}q \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ \mathrm{d}p ~~ \texttt{(} \mathrm{d}q \texttt{))}
\\
\texttt{(} ~~ \mathrm{d}p ~ ~ ~ \mathrm{d}q ~~ \texttt{)}
\\
\texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{))}
\\
\texttt{((} \mathrm{d}p \texttt{)} ~~ \mathrm{d}q ~~ \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ \mathrm{d}p ~ ~ ~ \mathrm{d}q ~~ \texttt{)}
\\
\texttt{(} ~~ \mathrm{d}p ~~ \texttt{(} \mathrm{d}q \texttt{))}
\\
\texttt{((} \mathrm{d}p \texttt{)} ~~ \mathrm{d}q ~~ \texttt{)}
\\
\texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{))}
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{15}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>1</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
|}

===Version 1.0===

====PNG====

{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"
|+ style="height:25px; font-size:large" | <math>\text{Table A3.}~ \mathrm{E}f ~\text{Expanded over Ordinary Variables}~ \{ p, q \}</math>
|-
| [[File:Ef Expanded Over Ordinary Variables p, q 1.0.png|720px]]
|}

====LaTeX====

{| align="center" cellpadding="4" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; font-size:larger; text-align:center; width:80%"
|+ style="height:30px; font-size:large" | <math>\text{Table A3.}~ \mathrm{E}f ~\text{Expanded over Ordinary Variables}~ \{ p, q \}</math>
|- style="background:ghostwhite"
| style="width:10%; border-bottom:1px solid black" |  
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f</math>
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |
<math>\mathrm{E}f|_{pq}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{E}f|_{p \texttt{(} q \texttt{)}}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{E}f|_{\texttt{(} p \texttt{)} q}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{E}f|_{\texttt{(} p \texttt{)(} q \texttt{)}}</math>
|-
| style="border-top:1px solid black" | <math>f_{0}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{1}\\f_{2}\\f_{4}\\f_{8}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{)(} q \texttt{)}
\\
\texttt{(} p \texttt{)} ~~ q ~~
\\
~~ p ~~ \texttt{(} q \texttt{)}
\\
p ~ ~ ~ q
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}p ~ ~ ~ \mathrm{d}q
\\
~~ \mathrm{d}p ~~ \texttt{(} \mathrm{d}q \texttt{)}
\\
\texttt{(} \mathrm{d}p \texttt{)} ~~ \mathrm{d}q ~~
\\
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~~ \mathrm{d}p ~~ \texttt{(} \mathrm{d}q \texttt{)}
\\
\mathrm{d}p ~ ~ ~ \mathrm{d}q
\\
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)}
\\
\texttt{(} \mathrm{d}p \texttt{)} ~~ \mathrm{d}q ~~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}p \texttt{)} ~~ \mathrm{d}q ~~
\\
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)}
\\
\mathrm{d}p ~ ~ ~ \mathrm{d}q
\\
~~ \mathrm{d}p ~~ \texttt{(} \mathrm{d}q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)}
\\
\texttt{(} \mathrm{d}p \texttt{)} ~~ \mathrm{d}q ~~
\\
~~ \mathrm{d}p ~~ \texttt{(} \mathrm{d}q \texttt{)}
\\
\mathrm{d}p ~ ~ ~ \mathrm{d}q
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{3}\\f_{12}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{)}
\\
p
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}p
\\
\texttt{(} \mathrm{d}p \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\mathrm{d}p
\\
\texttt{(} \mathrm{d}p \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}p \texttt{)}
\\
\mathrm{d}p
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}p \texttt{)}
\\
\mathrm{d}p
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{6}\\f_{9}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{,} ~~ q \texttt{)}
\\
\texttt{((} p \texttt{,} ~~ q \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}p \texttt{,} ~~ \mathrm{d}q \texttt{)}
\\
\texttt{((} \mathrm{d}p \texttt{,} ~~ \mathrm{d}q \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} \mathrm{d}p \texttt{,} ~~ \mathrm{d}q \texttt{))}
\\
\texttt{(} \mathrm{d}p \texttt{,} ~~ \mathrm{d}q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} \mathrm{d}p \texttt{,} ~~ \mathrm{d}q \texttt{))}
\\
\texttt{(} \mathrm{d}p \texttt{,} ~~ \mathrm{d}q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}p \texttt{,} ~~ \mathrm{d}q \texttt{)}
\\
\texttt{((} \mathrm{d}p \texttt{,} ~~ \mathrm{d}q \texttt{))}
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{5}\\f_{10}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} q \texttt{)}
\\
q
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}q
\\
\texttt{(} \mathrm{d}q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}q \texttt{)}
\\
\mathrm{d}q
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\mathrm{d}q
\\
\texttt{(} \mathrm{d}q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}q \texttt{)}
\\
\mathrm{d}q
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{7}\\f_{11}\\f_{13}\\f_{14}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ p ~ ~ ~ q ~~ \texttt{)}
\\
\texttt{(} ~~ p ~~ \texttt{(} q \texttt{))}
\\
\texttt{((} p \texttt{)} ~~ q ~~ \texttt{)}
\\
\texttt{((} p \texttt{)(} q \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{))}
\\
\texttt{((} \mathrm{d}p \texttt{)} ~~ \mathrm{d}q ~~ \texttt{)}
\\
\texttt{(} ~~ \mathrm{d}p ~~ \texttt{(} \mathrm{d}q \texttt{))}
\\
\texttt{(} ~~ \mathrm{d}p ~ ~ ~ \mathrm{d}q ~~ \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} \mathrm{d}p \texttt{)} ~~ \mathrm{d}q ~~ \texttt{)}
\\
\texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{))}
\\
\texttt{(} ~~ \mathrm{d}p ~ ~ ~ \mathrm{d}q ~~ \texttt{)}
\\
\texttt{(} ~~ \mathrm{d}p ~~ \texttt{(} \mathrm{d}q \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ \mathrm{d}p ~~ \texttt{(} \mathrm{d}q \texttt{))}
\\
\texttt{(} ~~ \mathrm{d}p ~ ~ ~ \mathrm{d}q ~~ \texttt{)}
\\
\texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{))}
\\
\texttt{((} \mathrm{d}p \texttt{)} ~~ \mathrm{d}q ~~ \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ \mathrm{d}p ~ ~ ~ \mathrm{d}q ~~ \texttt{)}
\\
\texttt{(} ~~ \mathrm{d}p ~~ \texttt{(} \mathrm{d}q \texttt{))}
\\
\texttt{((} \mathrm{d}p \texttt{)} ~~ \mathrm{d}q ~~ \texttt{)}
\\
\texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{))}
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{15}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>1</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
|}

==Ef Expanded over Differential Variables==

===Version 3.0===

====PNG====

{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"
|+ style="height:25px; font-size:large" | <math>\text{Table A4.}~ \mathrm{E}f ~\text{Expanded over Differential Variables}~ \{ \mathrm{d}p, \mathrm{d}q \}</math>
|-
| [[File:Ef Expanded Over Differential Variables dp, dq 3.0.png|800px]]
|}

===Version 2.0===

====PNG====

{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"
|+ style="height:25px; font-size:large" | <math>\text{Table A4.}~ \mathrm{E}f ~\text{Expanded over Differential Variables}~ \{ \mathrm{d}p, \mathrm{d}q \}</math>
|-
| [[File:Ef Expanded Over Differential Variables dp, dq 2.0.png|800px]]
|}

====LaTeX====

{| align="center" cellpadding="4" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; font-size:larger; text-align:center; width:90%"
|+ style="height:30px; font-size:large" | <math>\text{Table A4.}~ \mathrm{E}f ~\text{Expanded over Differential Variables}~ \{ \mathrm{d}p, \mathrm{d}q \}</math>
|- style="background:ghostwhite"
| style="width:5%; border-bottom:1px solid black" |  
| style="width:15%; border-bottom:1px solid black; border-left:1px solid black" | <math>f</math>
| style="width:20%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{E}f</math>
| style="width:15%; border-bottom:1px solid black; border-left:4px double black" |
<math>\begin{matrix}\mathrm{T}_{11}f\\\mathrm{E}f|_{\mathrm{d}p ~ \mathrm{d}q}\end{matrix}</math>
| style="width:15%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}\mathrm{T}_{10}f\\\mathrm{E}f|_{\mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)}}\end{matrix}</math>
| style="width:15%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}\mathrm{T}_{01}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q}\end{matrix}</math>
| style="width:15%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}\mathrm{T}_{00}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)}}\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{0}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{1}\\f_{2}\\f_{4}\\f_{8}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{)(} q \texttt{)}
\\
\texttt{(} p \texttt{)} ~~ q ~~
\\
~~ p ~~ \texttt{(} q \texttt{)}
\\
p ~ ~ ~ q
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} p \texttt{,} \mathrm{d}p \texttt{))((} q \texttt{,} \mathrm{d}q \texttt{))}
\\
\texttt{((} p \texttt{,} \mathrm{d}p \texttt{))} ~~ \texttt{(} q \texttt{,} \mathrm{d}q \texttt{)} ~~
\\
~~ \texttt{(} p \texttt{,} \mathrm{d}p \texttt{)} ~~ \texttt{((} q \texttt{,} \mathrm{d}q \texttt{))}
\\
\texttt{(} p \texttt{,} \mathrm{d}p \texttt{)} ~ ~ ~ \texttt{(} q \texttt{,} \mathrm{d}q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
p ~ ~ ~ q
\\
~~ p ~~ \texttt{(} q \texttt{)}
\\
\texttt{(} p \texttt{)} ~~ q ~~
\\
\texttt{(} p \texttt{)(} q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~~ p ~~ \texttt{(} q \texttt{)}
\\
p ~ ~ ~ q
\\
\texttt{(} p \texttt{)(} q \texttt{)}
\\
\texttt{(} p \texttt{)} ~~ q ~~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{)} ~~ q ~~
\\
\texttt{(} p \texttt{)(} q \texttt{)}
\\
p ~ ~ ~ q
\\
~~ p ~~ \texttt{(} q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{)(} q \texttt{)}
\\
\texttt{(} p \texttt{)} ~~ q ~~
\\
~~ p ~~ \texttt{(} q \texttt{)}
\\
p ~ ~ ~ q
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{3}\\f_{12}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{)}
\\
p
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} p \texttt{,} \mathrm{d}p \texttt{))}
\\
\texttt{(} p \texttt{,} \mathrm{d}p \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
p
\\
\texttt{(} p \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
p
\\
\texttt{(} p \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{)}
\\
p
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{)}
\\
p
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{6}\\f_{9}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{,} ~~ q \texttt{)}
\\
\texttt{((} p \texttt{,} ~~ q \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} p \texttt{,} \mathrm{d}p \texttt{),} ~~ \texttt{(} q \texttt{,} \mathrm{d}q \texttt{))}
\\
\texttt{(((} p \texttt{,} \mathrm{d}p \texttt{),} ~~ \texttt{(} q \texttt{,} \mathrm{d}q \texttt{)))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{(} p \texttt{,} ~~ q \texttt{)}
\\
\texttt{((} p \texttt{,} ~~ q \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} p \texttt{,} ~~ q \texttt{))}
\\
\texttt{(} p \texttt{,} ~~ q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} p \texttt{,} ~~ q \texttt{))}
\\
\texttt{(} p \texttt{,} ~~ q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{,} ~~ q \texttt{)}
\\
\texttt{((} p \texttt{,} ~~ q \texttt{))}
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{5}\\f_{10}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} q \texttt{)}
\\
q
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} q \texttt{,} \mathrm{d}q \texttt{))}
\\
\texttt{(} q \texttt{,} \mathrm{d}q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
q
\\
\texttt{(} q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} q \texttt{)}
\\
q
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
q
\\
\texttt{(} q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} q \texttt{)}
\\
q
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{7}\\f_{11}\\f_{13}\\f_{14}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ p ~ ~ ~ q ~~ \texttt{)}
\\
\texttt{(} ~~ p ~~ \texttt{(} q \texttt{))}
\\
\texttt{((} p \texttt{)} ~~ q ~~ \texttt{)}
\\
\texttt{((} p \texttt{)(} q \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ \texttt{(} p \texttt{,} \mathrm{d}p \texttt{)} ~ ~ ~
\texttt{(} q \texttt{,} \mathrm{d}q \texttt{)} ~~ \texttt{)}
\\
\texttt{(} ~~ \texttt{(} p \texttt{,} \mathrm{d}p \texttt{)} ~~ \texttt{((} q \texttt{,} \mathrm{d}q \texttt{)))}
\\
\texttt{(((} p \texttt{,} \mathrm{d}p \texttt{))} ~~ \texttt{(} q \texttt{,} \mathrm{d}q \texttt{)} ~~ \texttt{)}
\\
\texttt{(((} p \texttt{,} \mathrm{d}p \texttt{))((} q \texttt{,} \mathrm{d}q \texttt{)))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{((} p \texttt{)(} q \texttt{))}
\\
\texttt{((} p \texttt{)} ~~ q ~~ \texttt{)}
\\
\texttt{(} ~~ p ~~ \texttt{(} q \texttt{))}
\\
\texttt{(} ~~ p ~ ~ ~ q ~~ \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} p \texttt{)} ~~ q ~~ \texttt{)}
\\
\texttt{((} p \texttt{)(} q \texttt{))}
\\
\texttt{(} ~~ p ~ ~ ~ q ~~ \texttt{)}
\\
\texttt{(} ~~ p ~~ \texttt{(} q \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ p ~~ \texttt{(} q \texttt{))}
\\
\texttt{(} ~~ p ~ ~ ~ q ~~ \texttt{)}
\\
\texttt{((} p \texttt{)(} q \texttt{))}
\\
\texttt{((} p \texttt{)} ~~ q ~~ \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ p ~ ~ ~ q ~~ \texttt{)}
\\
\texttt{(} ~~ p ~~ \texttt{(} q \texttt{))}
\\
\texttt{((} p \texttt{)} ~~ q ~~ \texttt{)}
\\
\texttt{((} p \texttt{)(} q \texttt{))}
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{15}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>1</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
|- style="background:ghostwhite"
| style="border-top:1px solid black" colspan="3" | <math>\text{Fixed Point Total}</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>4</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>4</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>4</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>16</math>
|}

===Version 1.0===

====PNG====

{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"
|+ style="height:25px; font-size:large" | <math>\text{Table A4.}~ \mathrm{E}f ~\text{Expanded over Differential Variables}~ \{ \mathrm{d}p, \mathrm{d}q \}</math>
|-
| [[File:Ef Expanded Over Differential Variables dp, dq 1.0.png|720px]]
|}

====LaTeX====

{| align="center" cellpadding="4" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; font-size:larger; text-align:center; width:80%"
|+ style="height:30px; font-size:large" | <math>\text{Table A4.}~ \mathrm{E}f ~\text{Expanded over Differential Variables}~ \{ \mathrm{d}p, \mathrm{d}q \}</math>
|- style="background:ghostwhite"
| style="width:10%; border-bottom:1px solid black" |  
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f</math>
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |
<math>\begin{matrix}\mathrm{T}_{11}f\\\mathrm{E}f|_{\mathrm{d}p ~ \mathrm{d}q}\end{matrix}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}\mathrm{T}_{10}f\\\mathrm{E}f|_{\mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)}}\end{matrix}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}\mathrm{T}_{01}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q}\end{matrix}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}\mathrm{T}_{00}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)}}\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{0}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{1}\\f_{2}\\f_{4}\\f_{8}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{)(} q \texttt{)}
\\
\texttt{(} p \texttt{)} ~~ q ~~
\\
~~ p ~~ \texttt{(} q \texttt{)}
\\
p ~ ~ ~ q
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
p ~ ~ ~ q
\\
~~ p ~~ \texttt{(} q \texttt{)}
\\
\texttt{(} p \texttt{)} ~~ q ~~
\\
\texttt{(} p \texttt{)(} q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~~ p ~~ \texttt{(} q \texttt{)}
\\
p ~ ~ ~ q
\\
\texttt{(} p \texttt{)(} q \texttt{)}
\\
\texttt{(} p \texttt{)} ~~ q ~~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{)} ~~ q ~~
\\
\texttt{(} p \texttt{)(} q \texttt{)}
\\
p ~ ~ ~ q
\\
~~ p ~~ \texttt{(} q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{)(} q \texttt{)}
\\
\texttt{(} p \texttt{)} ~~ q ~~
\\
~~ p ~~ \texttt{(} q \texttt{)}
\\
p ~ ~ ~ q
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{3}\\f_{12}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{)}
\\
p
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
p
\\
\texttt{(} p \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
p
\\
\texttt{(} p \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{)}
\\
p
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{)}
\\
p
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{6}\\f_{9}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{,} ~~ q \texttt{)}
\\
\texttt{((} p \texttt{,} ~~ q \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{(} p \texttt{,} ~~ q \texttt{)}
\\
\texttt{((} p \texttt{,} ~~ q \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} p \texttt{,} ~~ q \texttt{))}
\\
\texttt{(} p \texttt{,} ~~ q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} p \texttt{,} ~~ q \texttt{))}
\\
\texttt{(} p \texttt{,} ~~ q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{,} ~~ q \texttt{)}
\\
\texttt{((} p \texttt{,} ~~ q \texttt{))}
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{5}\\f_{10}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} q \texttt{)}
\\
q
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
q
\\
\texttt{(} q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} q \texttt{)}
\\
q
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
q
\\
\texttt{(} q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} q \texttt{)}
\\
q
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{7}\\f_{11}\\f_{13}\\f_{14}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ p ~ ~ ~ q ~~ \texttt{)}
\\
\texttt{(} ~~ p ~~ \texttt{(} q \texttt{))}
\\
\texttt{((} p \texttt{)} ~~ q ~~ \texttt{)}
\\
\texttt{((} p \texttt{)(} q \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{((} p \texttt{)(} q \texttt{))}
\\
\texttt{((} p \texttt{)} ~~ q ~~ \texttt{)}
\\
\texttt{(} ~~ p ~~ \texttt{(} q \texttt{))}
\\
\texttt{(} ~~ p ~ ~ ~ q ~~ \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} p \texttt{)} ~~ q ~~ \texttt{)}
\\
\texttt{((} p \texttt{)(} q \texttt{))}
\\
\texttt{(} ~~ p ~ ~ ~ q ~~ \texttt{)}
\\
\texttt{(} ~~ p ~~ \texttt{(} q \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ p ~~ \texttt{(} q \texttt{))}
\\
\texttt{(} ~~ p ~ ~ ~ q ~~ \texttt{)}
\\
\texttt{((} p \texttt{)(} q \texttt{))}
\\
\texttt{((} p \texttt{)} ~~ q ~~ \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ p ~ ~ ~ q ~~ \texttt{)}
\\
\texttt{(} ~~ p ~~ \texttt{(} q \texttt{))}
\\
\texttt{((} p \texttt{)} ~~ q ~~ \texttt{)}
\\
\texttt{((} p \texttt{)(} q \texttt{))}
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{15}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>1</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
|- style="background:ghostwhite"
| style="border-top:1px solid black" colspan="2" | <math>\text{Fixed Point Total}</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>4</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>4</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>4</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>16</math>
|}

==Df Expanded over Ordinary Variables==

===Version 2.0===

====PNG====

====LaTeX====

===Version 1.0===

====PNG====

{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"
|+ style="height:25px; font-size:large" | <math>\text{Table A5.}~ \mathrm{D}f ~\text{Expanded over Ordinary Variables}~ \{ p, q \}</math>
|-
| [[File:Df Expanded Over Ordinary Variables p, q 1.0.png|720px]]
|}

====LaTeX====

{| align="center" cellpadding="4" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; font-size:larger; text-align:center; width:80%"
|+ style="height:30px; font-size:large" | <math>\text{Table A5.}~ \mathrm{D}f ~\text{Expanded over Ordinary Variables}~ \{ p, q \}</math>
|- style="background:ghostwhite"
| style="width:10%; border-bottom:1px solid black" |  
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f</math>
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |
<math>\mathrm{D}f|_{pq}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{D}f|_{p \texttt{(} q \texttt{)}}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{D}f|_{\texttt{(} p \texttt{)} q}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{D}f|_{\texttt{(} p \texttt{)(} q \texttt{)}}</math>
|-
| style="border-top:1px solid black" | <math>f_{0}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{)(} q \texttt{)}
\\
\texttt{(} p \texttt{)} ~~ q ~~
\\
~~ p ~~ \texttt{(} q \texttt{)}
\\
p ~ ~ ~ q
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}p ~ ~ ~ \mathrm{d}q
\\
~~ \mathrm{d}p ~~ \texttt{(} \mathrm{d}q \texttt{)}
\\
\texttt{(} \mathrm{d}p \texttt{)} ~~ \mathrm{d}q ~~
\\
\texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~~ \mathrm{d}p ~~ \texttt{(} \mathrm{d}q \texttt{)}
\\
\mathrm{d}p ~ ~ ~ \mathrm{d}q
\\
\texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{))}
\\
\texttt{(} \mathrm{d}p \texttt{)} ~~ \mathrm{d}q ~~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}p \texttt{)} ~~ \mathrm{d}q ~~
\\
\texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{))}
\\
\mathrm{d}p ~ ~ ~ \mathrm{d}q
\\
~~ \mathrm{d}p ~~ \texttt{(} \mathrm{d}q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{))}
\\
\texttt{(} \mathrm{d}p \texttt{)} ~~ \mathrm{d}q ~~
\\
~~ \mathrm{d}p ~~ \texttt{(} \mathrm{d}q \texttt{)}
\\
\mathrm{d}p ~ ~ ~ \mathrm{d}q
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{)}
\\
p
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}p
\\
\mathrm{d}p
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\mathrm{d}p
\\
\mathrm{d}p
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\mathrm{d}p
\\
\mathrm{d}p
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\mathrm{d}p
\\
\mathrm{d}p
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{,} ~~ q \texttt{)}
\\
\texttt{((} p \texttt{,} ~~ q \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}p \texttt{,} ~~ \mathrm{d}q \texttt{)}
\\
\texttt{(} \mathrm{d}p \texttt{,} ~~ \mathrm{d}q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}p \texttt{,} ~~ \mathrm{d}q \texttt{)}
\\
\texttt{(} \mathrm{d}p \texttt{,} ~~ \mathrm{d}q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}p \texttt{,} ~~ \mathrm{d}q \texttt{)}
\\
\texttt{(} \mathrm{d}p \texttt{,} ~~ \mathrm{d}q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}p \texttt{,} ~~ \mathrm{d}q \texttt{)}
\\
\texttt{(} \mathrm{d}p \texttt{,} ~~ \mathrm{d}q \texttt{)}
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} q \texttt{)}
\\
q
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}q
\\
\mathrm{d}q
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\mathrm{d}q
\\
\mathrm{d}q
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\mathrm{d}q
\\
\mathrm{d}q
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\mathrm{d}q
\\
\mathrm{d}q
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ p ~ ~ ~ q ~~ \texttt{)}
\\
\texttt{(} ~~ p ~~ \texttt{(} q \texttt{))}
\\
\texttt{((} p \texttt{)} ~~ q ~~ \texttt{)}
\\
\texttt{((} p \texttt{)(} q \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{))}
\\
\texttt{(} \mathrm{d}p \texttt{)} ~~ \mathrm{d}q ~~
\\
~~ \mathrm{d}p ~~ \texttt{(} \mathrm{d}q \texttt{)}
\\
\mathrm{d}p ~ ~ ~ \mathrm{d}q
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}p \texttt{)} ~~ \mathrm{d}q ~~
\\
\texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{))}
\\
\mathrm{d}p ~ ~ ~ \mathrm{d}q
\\
~~ \mathrm{d}p ~~ \texttt{(} \mathrm{d}q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~~ \mathrm{d}p ~~ \texttt{(} \mathrm{d}q \texttt{)}
\\
\mathrm{d}p ~ ~ ~ \mathrm{d}q
\\
\texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{))}
\\
\texttt{(} \mathrm{d}p \texttt{)} ~~ \mathrm{d}q ~~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\mathrm{d}p ~ ~ ~ \mathrm{d}q
\\
~~ \mathrm{d}p ~~ \texttt{(} \mathrm{d}q \texttt{)}
\\
\texttt{(} \mathrm{d}p \texttt{)} ~~ \mathrm{d}q ~~
\\
\texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{))}
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{15}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
|}

==Df Expanded over Differential Variables==

===Version 2.0===

====PNG====

====LaTeX====

===Version 1.0===

====PNG====

{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"
|+ style="height:25px; font-size:large" | <math>\text{Table A6.}~ \mathrm{D}f ~\text{Expanded over Differential Variables}~ \{ \mathrm{d}p, \mathrm{d}q \}</math>
|-
| [[File:Df Expanded Over Differential Variables dp, dq 1.0.png|720px]]
|}

====LaTeX====

{| align="center" cellpadding="4" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; font-size:larger; text-align:center; width:80%"
|+ style="height:30px; font-size:large" | <math>\text{Table A6.}~ \mathrm{D}f ~\text{Expanded over Differential Variables}~ \{ \mathrm{d}p, \mathrm{d}q \}</math>
|- style="background:ghostwhite"
| style="width:10%; border-bottom:1px solid black" |  
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f</math>
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |
<math>\mathrm{D}f|_{\mathrm{d}p ~ \mathrm{d}q}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{D}f|_{\mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)}}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{D}f|_{\texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{D}f|_{\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)}}</math>
|-
| style="border-top:1px solid black" | <math>f_{0}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{1}\\f_{2}\\f_{4}\\f_{8}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{)(} q \texttt{)}
\\
\texttt{(} p \texttt{)} ~~ q ~~
\\
~~ p ~~ \texttt{(} q \texttt{)}
\\
p ~ ~ ~ q
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{((} p \texttt{,} ~~ q \texttt{))}
\\
\texttt{(} p \texttt{,} ~~ q \texttt{)}
\\
\texttt{(} p \texttt{,} ~~ q \texttt{)}
\\
\texttt{((} p \texttt{,} ~~ q \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} q \texttt{)}
\\
q
\\
\texttt{(} q \texttt{)}
\\
q
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{)}
\\
\texttt{(} p \texttt{)}
\\
p
\\
p
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{)}
\\
p
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}1\\1\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}1\\1\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{,} ~~ q \texttt{)}
\\
\texttt{((} p \texttt{,} ~~ q \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}0\\0\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}1\\1\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}1\\1\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} q \texttt{)}
\\
q
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}1\\1\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}1\\1\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ p ~ ~ ~ q ~~ \texttt{)}
\\
\texttt{(} ~~ p ~~ \texttt{(} q \texttt{))}
\\
\texttt{((} p \texttt{)} ~~ q ~~ \texttt{)}
\\
\texttt{((} p \texttt{)(} q \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{((} p \texttt{,} ~~ q \texttt{))}
\\
\texttt{(} p \texttt{,} ~~ q \texttt{)}
\\
\texttt{(} p \texttt{,} ~~ q \texttt{)}
\\
\texttt{((} p \texttt{,} ~~ q \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
q
\\
\texttt{(} q \texttt{)}
\\
q
\\
\texttt{(} q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
p
\\
p
\\
\texttt{(} p \texttt{)}
\\
\texttt{(} p \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{15}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
|}

==Operational Representation==

If you think that I linger in the realm of logical difference calculus out of sheer vacillation about getting down to the differential proper, it is probably out of a prior expectation that you derive from the art or the long-engrained practice of real analysis. But the fact is that ordinary calculus only rushes on to the sundry orders of approximation because the strain of comprehending the full import of <math>\mathrm{E}</math> and <math>\mathrm{D}</math> at once overwhelms its discrete and finite powers to grasp them. But here, in the fully serene idylls of [[zeroth order logic]], we find ourselves fit with the compass of a wit that is all we'd ever need to explore their effects with care.

So let us do just that.

I will first rationalize the novel grouping of propositional forms in the last set of Tables, as that will extend a gentle invitation to the mathematical subject of ''group theory'', and demonstrate its relevance to differential logic in a strikingly apt and useful way. The data for that account is contained in Table A4.

<br>

{| align="center" cellpadding="4" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; font-size:larger; text-align:center; width:90%"
|+ style="height:30px; font-size:large" | <math>\text{Table A4.} ~~ \mathrm{E}f ~\text{Expanded over Differential Variables}~ \{ \mathrm{d}p, \mathrm{d}q \}</math>
|- style="background:ghostwhite"
| style="width:5%; border-bottom:1px solid black" |  
| style="width:15%; border-bottom:1px solid black; border-left:1px solid black" | <math>f</math>
| style="width:20%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{E}f</math>
| style="width:15%; border-bottom:1px solid black; border-left:4px double black" |
<math>\begin{matrix}\mathrm{T}_{11}f\\\mathrm{E}f|_{\mathrm{d}p ~ \mathrm{d}q}\end{matrix}</math>
| style="width:15%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}\mathrm{T}_{10}f\\\mathrm{E}f|_{\mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)}}\end{matrix}</math>
| style="width:15%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}\mathrm{T}_{01}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q}\end{matrix}</math>
| style="width:15%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}\mathrm{T}_{00}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)}}\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{0}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{1}\\f_{2}\\f_{4}\\f_{8}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{)(} q \texttt{)}
\\
\texttt{(} p \texttt{)} ~~ q ~~
\\
~~ p ~~ \texttt{(} q \texttt{)}
\\
p ~ ~ ~ q
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} p \texttt{,} \mathrm{d}p \texttt{))((} q \texttt{,} \mathrm{d}q \texttt{))}
\\
\texttt{((} p \texttt{,} \mathrm{d}p \texttt{))} ~~ \texttt{(} q \texttt{,} \mathrm{d}q \texttt{)} ~~
\\
~~ \texttt{(} p \texttt{,} \mathrm{d}p \texttt{)} ~~ \texttt{((} q \texttt{,} \mathrm{d}q \texttt{))}
\\
\texttt{(} p \texttt{,} \mathrm{d}p \texttt{)} ~ ~ ~ \texttt{(} q \texttt{,} \mathrm{d}q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
p ~ ~ ~ q
\\
~~ p ~~ \texttt{(} q \texttt{)}
\\
\texttt{(} p \texttt{)} ~~ q ~~
\\
\texttt{(} p \texttt{)(} q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~~ p ~~ \texttt{(} q \texttt{)}
\\
p ~ ~ ~ q
\\
\texttt{(} p \texttt{)(} q \texttt{)}
\\
\texttt{(} p \texttt{)} ~~ q ~~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{)} ~~ q ~~
\\
\texttt{(} p \texttt{)(} q \texttt{)}
\\
p ~ ~ ~ q
\\
~~ p ~~ \texttt{(} q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{)(} q \texttt{)}
\\
\texttt{(} p \texttt{)} ~~ q ~~
\\
~~ p ~~ \texttt{(} q \texttt{)}
\\
p ~ ~ ~ q
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{3}\\f_{12}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{)}
\\
p
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} p \texttt{,} \mathrm{d}p \texttt{))}
\\
\texttt{(} p \texttt{,} \mathrm{d}p \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
p
\\
\texttt{(} p \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
p
\\
\texttt{(} p \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{)}
\\
p
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{)}
\\
p
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{6}\\f_{9}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{,} ~~ q \texttt{)}
\\
\texttt{((} p \texttt{,} ~~ q \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} p \texttt{,} \mathrm{d}p \texttt{),} ~~ \texttt{(} q \texttt{,} \mathrm{d}q \texttt{))}
\\
\texttt{(((} p \texttt{,} \mathrm{d}p \texttt{),} ~~ \texttt{(} q \texttt{,} \mathrm{d}q \texttt{)))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{(} p \texttt{,} ~~ q \texttt{)}
\\
\texttt{((} p \texttt{,} ~~ q \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} p \texttt{,} ~~ q \texttt{))}
\\
\texttt{(} p \texttt{,} ~~ q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} p \texttt{,} ~~ q \texttt{))}
\\
\texttt{(} p \texttt{,} ~~ q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} p \texttt{,} ~~ q \texttt{)}
\\
\texttt{((} p \texttt{,} ~~ q \texttt{))}
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{5}\\f_{10}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} q \texttt{)}
\\
q
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} q \texttt{,} \mathrm{d}q \texttt{))}
\\
\texttt{(} q \texttt{,} \mathrm{d}q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
q
\\
\texttt{(} q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} q \texttt{)}
\\
q
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
q
\\
\texttt{(} q \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} q \texttt{)}
\\
q
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{7}\\f_{11}\\f_{13}\\f_{14}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ p ~ ~ ~ q ~~ \texttt{)}
\\
\texttt{(} ~~ p ~~ \texttt{(} q \texttt{))}
\\
\texttt{((} p \texttt{)} ~~ q ~~ \texttt{)}
\\
\texttt{((} p \texttt{)(} q \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ \texttt{(} p \texttt{,} \mathrm{d}p \texttt{)} ~ ~ ~
\texttt{(} q \texttt{,} \mathrm{d}q \texttt{)} ~~ \texttt{)}
\\
\texttt{(} ~~ \texttt{(} p \texttt{,} \mathrm{d}p \texttt{)} ~~ \texttt{((} q \texttt{,} \mathrm{d}q \texttt{)))}
\\
\texttt{(((} p \texttt{,} \mathrm{d}p \texttt{))} ~~ \texttt{(} q \texttt{,} \mathrm{d}q \texttt{)} ~~ \texttt{)}
\\
\texttt{(((} p \texttt{,} \mathrm{d}p \texttt{))((} q \texttt{,} \mathrm{d}q \texttt{)))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{((} p \texttt{)(} q \texttt{))}
\\
\texttt{((} p \texttt{)} ~~ q ~~ \texttt{)}
\\
\texttt{(} ~~ p ~~ \texttt{(} q \texttt{))}
\\
\texttt{(} ~~ p ~ ~ ~ q ~~ \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} p \texttt{)} ~~ q ~~ \texttt{)}
\\
\texttt{((} p \texttt{)(} q \texttt{))}
\\
\texttt{(} ~~ p ~ ~ ~ q ~~ \texttt{)}
\\
\texttt{(} ~~ p ~~ \texttt{(} q \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ p ~~ \texttt{(} q \texttt{))}
\\
\texttt{(} ~~ p ~ ~ ~ q ~~ \texttt{)}
\\
\texttt{((} p \texttt{)(} q \texttt{))}
\\
\texttt{((} p \texttt{)} ~~ q ~~ \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ p ~ ~ ~ q ~~ \texttt{)}
\\
\texttt{(} ~~ p ~~ \texttt{(} q \texttt{))}
\\
\texttt{((} p \texttt{)} ~~ q ~~ \texttt{)}
\\
\texttt{((} p \texttt{)(} q \texttt{))}
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{15}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>1</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
|- style="background:ghostwhite"
| style="border-top:1px solid black" colspan="3" | <math>\text{Fixed Point Total}</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>4</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>4</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>4</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>16</math>
|}

<br>

The shift operator <math>\mathrm{E}</math> can be understood as enacting a substitution operation on the propositional form <math>f(p, q)</math> that is given as its argument. In our present focus on propositional forms that involve two variables, we have the following type specifications and definitions:

{| align="center" cellpadding="6" width="90%"
|
<math>\begin{array}{lcl}
\mathrm{E} ~:~ (X \to \mathbb{B})
& \to &
(\mathrm{E}X \to \mathbb{B})
\\[6pt]
\mathrm{E} ~:~ f(p, q)
& \mapsto &
\mathrm{E}f(p, q, \mathrm{d}p, \mathrm{d}q)
\\[6pt]
\mathrm{E}f(p, q, \mathrm{d}p, \mathrm{d}q)
& = &
f(p + \mathrm{d}p, q + \mathrm{d}q)
\\[6pt]
& = &
f( \texttt{(} p, \mathrm{d}p \texttt{)}, \texttt{(} q, \mathrm{d}q \texttt{)} )
\end{array}</math>
|}

Evaluating <math>\mathrm{E}f</math> at particular values of <math>\mathrm{d}p</math> and <math>\mathrm{d}q,</math> for example, <math>\mathrm{d}p = i</math> and <math>\mathrm{d}q = j,</math> where <math>i</math> and <math>j</math> are values in <math>\mathbb{B},</math> produces the following result:

{| align="center" cellpadding="6" width="90%"
|
<math>\begin{array}{lclcl}
\mathrm{E}_{ij}
& : &
(X \to \mathbb{B})
& \to &
(X \to \mathbb{B})
\\[6pt]
\mathrm{E}_{ij}
& : &
f
& \mapsto &
\mathrm{E}_{ij}f
\\[6pt]
\mathrm{E}_{ij}f
& = &
\mathrm{E}f|_{\mathrm{d}p = i, \mathrm{d}q = j}
& = &
f(p + i, q + j)
\\[6pt]
& &
& = &
f( \texttt{(} p, i \texttt{)}, \texttt{(} q, j \texttt{)} )
\end{array}</math>
|}

The notation is a little awkward, but the data of Table A3 should make the sense clear. The important thing to observe is that <math>\mathrm{E}_{ij}</math> has the effect of transforming each proposition <math>f : X \to \mathbb{B}</math> into a proposition <math>f^\prime : X \to \mathbb{B}.</math> As it happens, the action of each <math>\mathrm{E}_{ij}</math> is one-to-one and onto, so the gang of four operators <math>\{ \mathrm{E}_{ij} : i, j \in \mathbb{B} \}</math> is an example of what is called a ''transformation group'' on the set of sixteen propositions. Bowing to a longstanding local and linear tradition, I will therefore redub the four elements of this group as <math>\mathrm{T}_{00}, \mathrm{T}_{01}, \mathrm{T}_{10}, \mathrm{T}_{11},</math> to bear in mind their transformative character, or nature, as the case may be. Abstractly viewed, this group of order four has the following operation table:

<br>

{| align="center" cellpadding="0" 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:60%"
|- style="height:50px"
| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" |
<math>\cdot</math>
| width="22%" style="border-bottom:1px solid black" |
<math>\mathrm{T}_{00}</math>
| width="22%" style="border-bottom:1px solid black" |
<math>\mathrm{T}_{01}</math>
| width="22%" style="border-bottom:1px solid black" |
<math>\mathrm{T}_{10}</math>
| width="22%" style="border-bottom:1px solid black" |
<math>\mathrm{T}_{11}</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\mathrm{T}_{00}</math>
| <math>\mathrm{T}_{00}</math>
| <math>\mathrm{T}_{01}</math>
| <math>\mathrm{T}_{10}</math>
| <math>\mathrm{T}_{11}</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\mathrm{T}_{01}</math>
| <math>\mathrm{T}_{01}</math>
| <math>\mathrm{T}_{00}</math>
| <math>\mathrm{T}_{11}</math>
| <math>\mathrm{T}_{10}</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\mathrm{T}_{10}</math>
| <math>\mathrm{T}_{10}</math>
| <math>\mathrm{T}_{11}</math>
| <math>\mathrm{T}_{00}</math>
| <math>\mathrm{T}_{01}</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\mathrm{T}_{11}</math>
| <math>\mathrm{T}_{11}</math>
| <math>\mathrm{T}_{10}</math>
| <math>\mathrm{T}_{01}</math>
| <math>\mathrm{T}_{00}</math>
|}

<br>

It happens that there are just two possible groups of 4 elements. One is the cyclic group <math>Z_4</math> (from German ''Zyklus''), which this is not. The other is the Klein four-group <math>V_4</math> (from German ''Vier''), which this is.

More concretely viewed, the group as a whole pushes the set of sixteen propositions around in such a way that they fall into seven natural classes, called ''orbits''. One says that the orbits are preserved by the action of the group. There is an ''Orbit Lemma'' of immense utility to “those who count” which, depending on your upbringing, you may associate with the names of Burnside, Cauchy, Frobenius, or some subset or superset of these three, vouching that the number of orbits is equal to the mean number of fixed points, in other words, the total number of points (in our case, propositions) that are left unmoved by the separate operations, divided by the order of the group. In this instance, <math>\mathrm{T}_{00}</math> operates as the group identity, fixing all 16 propositions, while the other three group elements fix 4 propositions each, and so we get:   Number of Orbits  =  (4 + 4 + 4 + 16) ÷ 4  =  7.   Amazing!

{| align="center" cellpadding="0" cellspacing="0" width="90%"
|
<p>Consider what effects that might ''conceivably'' have practical bearings you ''conceive'' the objects of your ''conception'' to have. Then, your ''conception'' of those effects is the whole of your ''conception'' of the object.</p>
|-
| align="right" | — Charles Sanders Peirce, “Issues of Pragmaticism”, (CP 5.438)
|}

One other subject that it would be opportune to mention at this point, while we have an object example of a mathematical group fresh in mind, is the relationship between the pragmatic maxim and what are commonly known in mathematics as ''representation principles''. As it turns out, with regard to its formal characteristics, the pragmatic maxim unites the aspects of a representation principle with the attributes of what would ordinarily be known as a ''closure principle''. We will consider the form of closure that is invoked by the pragmatic maxim on another occasion, focusing here and now on the topic of group representations.

Let us return to the example of the ''four-group'' <math>V_4.</math> We encountered this group in one of its concrete representations, namely, as a ''transformation group'' that acts on a set of objects, in this case a set of sixteen functions or propositions. Forgetting about the set of objects that the group transforms among themselves, we may take the abstract view of the group's operational structure, for example, in the form of the group operation table copied here:

<br>

{| align="center" cellpadding="0" 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:60%"
|- style="height:50px"
| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" |
<math>\cdot</math>
| width="22%" style="border-bottom:1px solid black" |
<math>\mathrm{e}</math>
| width="22%" style="border-bottom:1px solid black" |
<math>\mathrm{f}</math>
| width="22%" style="border-bottom:1px solid black" |
<math>\mathrm{g}</math>
| width="22%" style="border-bottom:1px solid black" |
<math>\mathrm{h}</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\mathrm{e}</math>
| <math>\mathrm{e}</math>
| <math>\mathrm{f}</math>
| <math>\mathrm{g}</math>
| <math>\mathrm{h}</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\mathrm{f}</math>
| <math>\mathrm{f}</math>
| <math>\mathrm{e}</math>
| <math>\mathrm{h}</math>
| <math>\mathrm{g}</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\mathrm{g}</math>
| <math>\mathrm{g}</math>
| <math>\mathrm{h}</math>
| <math>\mathrm{e}</math>
| <math>\mathrm{f}</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\mathrm{h}</math>
| <math>\mathrm{h}</math>
| <math>\mathrm{g}</math>
| <math>\mathrm{f}</math>
| <math>\mathrm{e}</math>
|}

<br>

This table is abstractly the same as, or isomorphic to, the versions with the <math>\mathrm{E}_{ij}</math> operators and the <math>\mathrm{T}_{ij}</math> transformations that we took up earlier. That is to say, the story is the same, only the names have been changed. An abstract group can have a variety of significantly and superficially different representations. But even after we have long forgotten the details of any particular representation there is a type of concrete representations, called ''regular representations'', that are always readily available, as they can be generated from the mere data of the abstract operation table itself.

To see how a regular representation is constructed from the abstract operation table, select a group element from the top margin of the Table, and “consider its effects” on each of the group elements as they are listed along the left margin. We may record these effects as Peirce usually did, as a ''logical aggregate'' of elementary dyadic relatives, that is, as a logical disjunction or boolean sum whose terms represent the ordered pairs of <math>\mathrm{input} : \mathrm{output}</math> transactions that are produced by each group element in turn. This forms one of the two possible ''regular representations'' of the group, in this case the one that is called the ''post-regular representation'' or the ''right regular representation''. It has long been conventional to organize the terms of this logical aggregate in the form of a matrix:

Reading “<math>+</math>” as a logical disjunction:

{| align="center" cellpadding="6" width="90%"
| align="center" |
<math>\begin{matrix}
\mathrm{G}
& = & \mathrm{e}
& + & \mathrm{f}
& + & \mathrm{g}
& + & \mathrm{h}
\end{matrix}</math>
|}

And so, by expanding effects, we get:

{| align="center" cellpadding="6" width="90%"
| align="center" |
<math>\begin{matrix}
\mathrm{G}
& = & \mathrm{e}:\mathrm{e}
& + & \mathrm{f}:\mathrm{f}
& + & \mathrm{g}:\mathrm{g}
& + & \mathrm{h}:\mathrm{h}
\\[4pt]
& + & \mathrm{e}:\mathrm{f}
& + & \mathrm{f}:\mathrm{e}
& + & \mathrm{g}:\mathrm{h}
& + & \mathrm{h}:\mathrm{g}
\\[4pt]
& + & \mathrm{e}:\mathrm{g}
& + & \mathrm{f}:\mathrm{h}
& + & \mathrm{g}:\mathrm{e}
& + & \mathrm{h}:\mathrm{f}
\\[4pt]
& + & \mathrm{e}:\mathrm{h}
& + & \mathrm{f}:\mathrm{g}
& + & \mathrm{g}:\mathrm{f}
& + & \mathrm{h}:\mathrm{e}
\end{matrix}</math>
|}

More on the pragmatic maxim as a representation principle later.

The above-mentioned fact about the regular representations of a group is universally known as Cayley's Theorem, typically stated in the following form:

{| align="center" cellpadding="6" width="90%"
| Every group is isomorphic to a subgroup of <math>\mathrm{Aut}(X),</math> the group of automorphisms of a suitably chosen set <math>X</math>.
|}

There is a considerable generalization of these regular representations to a broad class of relational algebraic systems in Peirce's earliest papers. The crux of the whole idea is this:

{| align="center" cellpadding="6" width="90%"
| Contemplate the effects of the symbol whose meaning you wish to investigate as they play out on all the stages of conduct where you can imagine that symbol playing a role.
|}

This idea of contextual definition by way of conduct transforming operators is basically the same as Jeremy Bentham's notion of ''paraphrasis'', a “method of accounting for fictions by explaining various purported terms away” (Quine, in Van Heijenoort, ''From Frege to Gödel'', p. 216). Today we'd call these constructions ''term models''. This, again, is the big idea behind Schönfinkel's combinators <math>\mathrm{S}, \mathrm{K}, \mathrm{I},</math> and hence of lambda calculus, and I reckon you know where that leads.

The next few excursions in this series will provide a scenic tour of various ideas in group theory that will turn out to be of constant guidance in several of the settings that are associated with our topic.

Let me return to Peirce's early papers on the algebra of relatives to pick up the conventions that he used there, and then rewrite my account of regular representations in a way that conforms to those.

Peirce describes the action of an “elementary dual relative” in this way:

{| align="center" cellpadding="6" width="90%"
| Elementary simple relatives are connected together in systems of four. For if <math>\mathrm{A}\!:\!\mathrm{B}</math> be taken to denote the elementary relative which multiplied into <math>\mathrm{B}</math> gives <math>\mathrm{A},</math> then this relation existing as elementary, we have the four elementary relatives
|-
| align="center" | <math>\mathrm{A}\!:\!\mathrm{A} \qquad \mathrm{A}\!:\!\mathrm{B} \qquad \mathrm{B}\!:\!\mathrm{A} \qquad \mathrm{B}\!:\!\mathrm{B}.</math>
|-
| C.S. Peirce, ''Collected Papers'', CP 3.123.
|}

Peirce is well aware that it is not at all necessary to arrange the elementary relatives of a relation into arrays, matrices, or tables, but when he does so he tends to prefer organizing 2-adic relations in the following manner:

{| align="center" cellpadding="6" width="90%"
| align="center" |
<math>\begin{bmatrix}
a\!:\!a & a\!:\!b & a\!:\!c
\\
b\!:\!a & b\!:\!b & b\!:\!c
\\
c\!:\!a & c\!:\!b & c\!:\!c
\end{bmatrix}</math>
|}

For example, given the set <math>X = \{ a, b, c \},</math> suppose that we have the 2-adic relative term <math>\mathit{m} = {}^{\backprime\backprime}\, \text{marker for}\, \underline{~ ~ ~}\, {}^{\prime\prime}</math> and the associated 2-adic relation <math>M \subseteq X \times X,</math> the general pattern of whose common structure is represented by the following matrix:

{| align="center" cellpadding="6" width="90%"
| align="center" |
<math>
M \quad = \quad
\begin{bmatrix}
M_{aa}(a\!:\!a) & M_{ab}(a\!:\!b) & M_{ac}(a\!:\!c)
\\
M_{ba}(b\!:\!a) & M_{bb}(b\!:\!b) & M_{bc}(b\!:\!c)
\\
M_{ca}(c\!:\!a) & M_{cb}(c\!:\!b) & M_{cc}(c\!:\!c)
\end{bmatrix}
</math>
|}

For at least a little while longer, I will keep explicit the distinction between a ''relative term'' like <math>\mathit{m}</math> and a ''relation'' like <math>M \subseteq X \times X,</math> but it is best to view both these entities as involving different applications of the same information, and so we could just as easily write the following form:

{| align="center" cellpadding="6" width="90%"
| align="center" |
<math>
m \quad = \quad
\begin{bmatrix}
m_{aa}(a\!:\!a) & m_{ab}(a\!:\!b) & m_{ac}(a\!:\!c)
\\
m_{ba}(b\!:\!a) & m_{bb}(b\!:\!b) & m_{bc}(b\!:\!c)
\\
m_{ca}(c\!:\!a) & m_{cb}(c\!:\!b) & m_{cc}(c\!:\!c)
\end{bmatrix}
</math>
|}

By way of making up a concrete example, let us say that <math>\mathit{m}</math> or <math>M</math> is given as follows:

{| align="center" cellpadding="6" width="90%"
| align="center" |
<math>\begin{array}{l}
a ~\text{is a marker for}~ a
\\
a ~\text{is a marker for}~ b
\\
b ~\text{is a marker for}~ b
\\
b ~\text{is a marker for}~ c
\\
c ~\text{is a marker for}~ c
\\
c ~\text{is a marker for}~ a
\end{array}</math>
|}

In sum, then, the relative term <math>\mathit{m}</math> and the relation <math>M</math> are both represented by the following matrix:

{| align="center" cellpadding="6" width="90%"
| align="center" |
<math>\begin{bmatrix}
1 \cdot (a\!:\!a) & 1 \cdot (a\!:\!b) & 0 \cdot (a\!:\!c)
\\
0 \cdot (b\!:\!a) & 1 \cdot (b\!:\!b) & 1 \cdot (b\!:\!c)
\\
1 \cdot (c\!:\!a) & 0 \cdot (c\!:\!b) & 1 \cdot (c\!:\!c)
\end{bmatrix}</math>
|}

I think this much will serve to fix notation and set up the remainder of the discussion.

In Peirce's time, and even in some circles of mathematics today, the information indicated by the elementary relatives <math>(i\!:\!j),</math> as the indices <math>i, j</math> range over the universe of discourse, would be referred to as the ''umbral elements'' of the algebraic operation represented by the matrix, though I seem to recall that Peirce preferred to call these terms the “ingredients”. When this ordered basis is understood well enough, one will tend to drop any mention of it from the matrix itself, leaving us nothing but these bare bones:

{| align="center" cellpadding="6" width="90%"
| align="center" |
<math>
M \quad = \quad
\begin{bmatrix}
1 & 1 & 0
\\
0 & 1 & 1
\\
1 & 0 & 1
\end{bmatrix}
</math>
|}

The various representations of <math>M</math> are nothing more than alternative ways of specifying its basic ingredients, namely, the following aggregate of elementary relatives:

{| align="center" cellpadding="6" width="90%"
| align="center" |
<math>\begin{array}{*{13}{c}}
M
& = & a\!:\!a
& + & b\!:\!b
& + & c\!:\!c
& + & a\!:\!b
& + & b\!:\!c
& + & c\!:\!a
\end{array}</math>
|}

Recognizing that <math>a\!:\!a + b\!:\!b + c\!:\!c</math> is the identity transformation otherwise known as <math>\mathit{1},</math> the 2-adic relative term <math>m = {}^{\backprime\backprime}\, \text{marker for}\, \underline{~ ~ ~}\, {}^{\prime\prime}</math> can be parsed as an element <math>\mathit{1} + a\!:\!b + b\!:\!c + c\!:\!a</math> of the so-called ''group ring'', all of which makes this element just a special sort of linear transformation.

Up to this point, we are still reading the elementary relatives of the form <math>i\!:\!j</math> in the way that Peirce read them in logical contexts:  <math>i</math> is the relate, <math>j</math> is the correlate, and in our current example <math>i\!:\!j,</math> or more exactly, <math>m_{ij} = 1,</math> is taken to say that <math>i</math> is a marker for <math>j.</math>  This is the mode of reading that we call “multiplying on the left”.

In the algebraic, permutational, or transformational contexts of application, however, Peirce converts to the alternative mode of reading, although still calling <math>i</math> the relate and <math>j</math> the correlate, the elementary relative <math>i\!:\!j</math> now means that <math>i</math> gets changed into <math>j.</math> In this scheme of reading, the transformation <math>a\!:\!b + b\!:\!c + c\!:\!a</math> is a permutation of the aggregate <math>\mathbf{1} = a + b + c,</math> or what we would now call the set <math>\{ a, b, c \},</math> in particular, it is the permutation that is otherwise notated as follows:

{| align="center" cellpadding="6"
|
<math>\begin{Bmatrix}
a & b & c
\\
b & c & a
\end{Bmatrix}</math>
|}

This is consistent with the convention that Peirce uses in the paper “On a Class of Multiple Algebras” (CP 3.324–327).

We've been exploring the applications of a certain technique for clarifying abstruse concepts, a rough-cut version of the pragmatic maxim that I've been accustomed to refer to as the ''operationalization'' of ideas. The basic idea is to replace the question of ''What it is'', which modest people comprehend is far beyond their powers to answer definitively any time soon, with the question of ''What it does'', which most people know at least a modicum about.

In the case of regular representations of groups we found a non-plussing surplus of answers to sort our way through. So let us track back one more time to see if we can learn any lessons that might carry over to more realistic cases.

Here is is the operation table of <math>V_4</math> once again:

<br>

{| align="center" cellpadding="0" 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:60%"
|+ style="height:30px; font-size:large" | <math>\text{Klein Four-Group}~ V_4</math>
|- style="height:50px"
| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" |
<math>\cdot</math>
| width="22%" style="border-bottom:1px solid black" |
<math>\mathrm{e}</math>
| width="22%" style="border-bottom:1px solid black" |
<math>\mathrm{f}</math>
| width="22%" style="border-bottom:1px solid black" |
<math>\mathrm{g}</math>
| width="22%" style="border-bottom:1px solid black" |
<math>\mathrm{h}</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\mathrm{e}</math>
| <math>\mathrm{e}</math>
| <math>\mathrm{f}</math>
| <math>\mathrm{g}</math>
| <math>\mathrm{h}</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\mathrm{f}</math>
| <math>\mathrm{f}</math>
| <math>\mathrm{e}</math>
| <math>\mathrm{h}</math>
| <math>\mathrm{g}</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\mathrm{g}</math>
| <math>\mathrm{g}</math>
| <math>\mathrm{h}</math>
| <math>\mathrm{e}</math>
| <math>\mathrm{f}</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\mathrm{h}</math>
| <math>\mathrm{h}</math>
| <math>\mathrm{g}</math>
| <math>\mathrm{f}</math>
| <math>\mathrm{e}</math>
|}

<br>

A group operation table is really just a device for recording a certain 3-adic relation, to be specific, the set of triples of the form <math>(x, y, z)</math> satisfying the equation <math>x \cdot y = z.</math>

In the case of <math>V_4 = (G, \cdot),</math> where <math>G</math> is the ''underlying set'' <math>\{ \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h} \},</math> we have the 3-adic relation <math>L(V_4) \subseteq G \times G \times G</math> whose triples are listed below:

{| align="center" cellpadding="6" width="90%"
| align="center" |
<math>\begin{matrix}
(\mathrm{e}, \mathrm{e}, \mathrm{e}) &
(\mathrm{e}, \mathrm{f}, \mathrm{f}) &
(\mathrm{e}, \mathrm{g}, \mathrm{g}) &
(\mathrm{e}, \mathrm{h}, \mathrm{h})
\\[6pt]
(\mathrm{f}, \mathrm{e}, \mathrm{f}) &
(\mathrm{f}, \mathrm{f}, \mathrm{e}) &
(\mathrm{f}, \mathrm{g}, \mathrm{h}) &
(\mathrm{f}, \mathrm{h}, \mathrm{g})
\\[6pt]
(\mathrm{g}, \mathrm{e}, \mathrm{g}) &
(\mathrm{g}, \mathrm{f}, \mathrm{h}) &
(\mathrm{g}, \mathrm{g}, \mathrm{e}) &
(\mathrm{g}, \mathrm{h}, \mathrm{f})
\\[6pt]
(\mathrm{h}, \mathrm{e}, \mathrm{h}) &
(\mathrm{h}, \mathrm{f}, \mathrm{g}) &
(\mathrm{h}, \mathrm{g}, \mathrm{f}) &
(\mathrm{h}, \mathrm{h}, \mathrm{e})
\end{matrix}</math>
|}

It is part of the definition of a group that the 3-adic relation <math>L \subseteq G^3</math> is actually a function <math>L : G \times G \to G.</math> It is from this functional perspective that we can see an easy way to derive the two regular representations. Since we have a function of the type <math>L : G \times G \to G,</math> we can define a couple of substitution operators:

{| align="center" cellpadding="6" width="90%"
| valign="top" | 1.
| <math>\mathrm{Sub}(x, (\underline{~~}, y))</math> puts any specified <math>x</math> into the empty slot of the rheme <math>(\underline{~~}, y),</math> with the effect of producing the saturated rheme <math>(x, y)</math> that evaluates to <math>xy.</math>
|-
| valign="top" | 2.
| <math>\mathrm{Sub}(x, (y, \underline{~~}))</math> puts any specified <math>x</math> into the empty slot of the rheme <math>(y, \underline{~~}),</math> with the effect of producing the saturated rheme <math>(y, x)</math> that evaluates to <math>yx.</math>
|}

In (1) we consider the effects of each <math>x</math> in its practical bearing on contexts of the form <math>(\underline{~~}, y),</math> as <math>y</math> ranges over <math>G,</math> and the effects are such that <math>x</math> takes <math>(\underline{~~}, y)</math> into <math>xy,</math> for <math>y</math> in <math>G,</math> all of which is notated as <math>x = \{ (y : xy) ~|~ y \in G \}.</math> The pairs <math>(y : xy)</math> can be found by picking an <math>x</math> from the left margin of the group operation table and considering its effects on each <math>y</math> in turn as these run across the top margin. This aspect of pragmatic definition we recognize as the regular ante-representation:

{| align="center" cellpadding="6" width="90%"
| align="center" |
<math>\begin{matrix}
\mathrm{e}
& = & \mathrm{e}\!:\!\mathrm{e}
& + & \mathrm{f}\!:\!\mathrm{f}
& + & \mathrm{g}\!:\!\mathrm{g}
& + & \mathrm{h}\!:\!\mathrm{h}
\\[4pt]
\mathrm{f}
& = & \mathrm{e}\!:\!\mathrm{f}
& + & \mathrm{f}\!:\!\mathrm{e}
& + & \mathrm{g}\!:\!\mathrm{h}
& + & \mathrm{h}\!:\!\mathrm{g}
\\[4pt]
\mathrm{g}
& = & \mathrm{e}\!:\!\mathrm{g}
& + & \mathrm{f}\!:\!\mathrm{h}
& + & \mathrm{g}\!:\!\mathrm{e}
& + & \mathrm{h}\!:\!\mathrm{f}
\\[4pt]
\mathrm{h}
& = & \mathrm{e}\!:\!\mathrm{h}
& + & \mathrm{f}\!:\!\mathrm{g}
& + & \mathrm{g}\!:\!\mathrm{f}
& + & \mathrm{h}\!:\!\mathrm{e}
\end{matrix}</math>
|}

In (2) we consider the effects of each <math>x</math> in its practical bearing on contexts of the form <math>(y, \underline{~~}),</math> as <math>y</math> ranges over <math>G,</math> and the effects are such that <math>x</math> takes <math>(y, \underline{~~})</math> into <math>yx,</math> for <math>y</math> in <math>G,</math> all of which is notated as <math>x = \{ (y : yx) ~|~ y \in G \}.</math> The pairs <math>(y : yx)</math> can be found by picking an <math>x</math> from the top margin of the group operation table and considering its effects on each <math>y</math> in turn as these run down the left margin. This aspect of pragmatic definition we recognize as the regular post-representation:

{| align="center" cellpadding="6" width="90%"
| align="center" |
<math>\begin{matrix}
\mathrm{e}
& = & \mathrm{e}\!:\!\mathrm{e}
& + & \mathrm{f}\!:\!\mathrm{f}
& + & \mathrm{g}\!:\!\mathrm{g}
& + & \mathrm{h}\!:\!\mathrm{h}
\\[4pt]
\mathrm{f}
& = & \mathrm{e}\!:\!\mathrm{f}
& + & \mathrm{f}\!:\!\mathrm{e}
& + & \mathrm{g}\!:\!\mathrm{h}
& + & \mathrm{h}\!:\!\mathrm{g}
\\[4pt]
\mathrm{g}
& = & \mathrm{e}\!:\!\mathrm{g}
& + & \mathrm{f}\!:\!\mathrm{h}
& + & \mathrm{g}\!:\!\mathrm{e}
& + & \mathrm{h}\!:\!\mathrm{f}
\\[4pt]
\mathrm{h}
& = & \mathrm{e}\!:\!\mathrm{h}
& + & \mathrm{f}\!:\!\mathrm{g}
& + & \mathrm{g}\!:\!\mathrm{f}
& + & \mathrm{h}\!:\!\mathrm{e}
\end{matrix}</math>
|}

If the ante-rep looks the same as the post-rep, now that I'm writing them in the same dialect, that is because <math>V_4</math> is abelian (commutative), and so the two representations have the very same effects on each point of their bearing.

So long as we're in the neighborhood, we might as well take in some more of the sights, for instance, the smallest example of a non-abelian (non-commutative) group. This is a group of six elements, say, <math>G = \{ \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}, \mathrm{i}, \mathrm{j} \},</math> with no relation to any other employment of these six symbols being implied, of course, and it can be most easily represented as the permutation group on a set of three letters, say, <math>X = \{ a, b, c \},</math> usually notated as <math>G = \mathrm{Sym}(X)</math> or more abstractly and briefly, as <math>\mathrm{Sym}(3)</math> or <math>S_3.</math> The next Table shows the intended correspondence between abstract group elements and the permutation or substitution operations in <math>\mathrm{Sym}(X).</math>

<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px; font-size:large" |
<math>\text{Permutation Substitutions in}~ \mathrm{Sym} \{ a, b, c \}</math>
|- style="height:40px; background:#f0f0ff"
| width="16%" | <math>\mathrm{e}</math>
| width="16%" | <math>\mathrm{f}</math>
| width="16%" | <math>\mathrm{g}</math>
| width="16%" | <math>\mathrm{h}</math>
| width="16%" | <math>\mathrm{i}</math>
| width="16%" | <math>\mathrm{j}</math>
|-
|
<math>\begin{matrix}
a & b & c
\\[3pt]
\downarrow & \downarrow & \downarrow
\\[6pt]
a & b & c
\end{matrix}</math>
|
<math>\begin{matrix}
a & b & c
\\[3pt]
\downarrow & \downarrow & \downarrow
\\[6pt]
c & a & b
\end{matrix}</math>
|
<math>\begin{matrix}
a & b & c
\\[3pt]
\downarrow & \downarrow & \downarrow
\\[6pt]
b & c & a
\end{matrix}</math>
|
<math>\begin{matrix}
a & b & c
\\[3pt]
\downarrow & \downarrow & \downarrow
\\[6pt]
a & c & b
\end{matrix}</math>
|
<math>\begin{matrix}
a & b & c
\\[3pt]
\downarrow & \downarrow & \downarrow
\\[6pt]
c & b & a
\end{matrix}</math>
|
<math>\begin{matrix}
a & b & c
\\[3pt]
\downarrow & \downarrow & \downarrow
\\[6pt]
b & a & c
\end{matrix}</math>
|}

<br>

Here is the operation table for <math>S_3,</math> given in abstract fashion:

{| align="center" cellpadding="10" style="text-align:center"
| <math>\text{Symmetric Group}~ S_3</math>
|-
| [[File:Symmetric Group S(3).jpg|500px]]
|}

By the way, we will meet with the symmetric group <math>S_3</math> again when we return to take up the study of Peirce's early paper “On a Class of Multiple Algebras” (CP 3.324–327), and also his late unpublished work “The Simplest Mathematics” (1902) (CP 4.227–323), with particular reference to the section that treats of “Trichotomic Mathematics” (CP 4.307–323).

By way of collecting a short-term pay-off for all the work that we did on the regular representations of the Klein 4-group <math>V_4,</math> let us write out as quickly as possible in ''relative form'' a minimal budget of representations for the symmetric group on three letters, <math>\mathrm{Sym}(3).</math> After doing the usual bit of compare and contrast among the various representations, we will have enough concrete material beneath our abstract belts to tackle a few of the presently obscured details of Peirce's early “Algebra + Logic” papers.

Writing the permutations or substitutions of <math>\mathrm{Sym} \{ a, b, c \}</math> in relative form generates what is generally thought of as a ''natural representation'' of <math>S_3.</math>

{| align="center" cellpadding="10" width="90%"
| align="center" |
<math>\begin{matrix}
\mathrm{e}
& = & a\!:\!a
& + & b\!:\!b
& + & c\!:\!c
\\[4pt]
\mathrm{f}
& = & a\!:\!c
& + & b\!:\!a
& + & c\!:\!b
\\[4pt]
\mathrm{g}
& = & a\!:\!b
& + & b\!:\!c
& + & c\!:\!a
\\[4pt]
\mathrm{h}
& = & a\!:\!a
& + & b\!:\!c
& + & c\!:\!b
\\[4pt]
\mathrm{i}
& = & a\!:\!c
& + & b\!:\!b
& + & c\!:\!a
\\[4pt]
\mathrm{j}
& = & a\!:\!b
& + & b\!:\!a
& + & c\!:\!c
\end{matrix}</math>
|}

I have without stopping to think about it written out this natural representation of <math>S_3</math> in the style that comes most naturally to me, to wit, the “right” way, whereby an ordered pair configured as <math>x\!:\!y</math> constitutes the turning of <math>x</math> into <math>y.</math> It is possible that the next time we check in with CSP we will have to adjust our sense of direction, but that will be an easy enough bridge to cross when we come to it.

To construct the regular representations of <math>S_3,</math> we begin with the data of its operation table:

{| align="center" cellpadding="10" style="text-align:center"
| <math>\text{Symmetric Group}~ S_3</math>
|-
| [[File:Symmetric Group S(3).jpg|500px]]
|}

Just by way of staying clear about what we are doing, let's return to the recipe that we worked out before:

It is part of the definition of a group that the 3-adic relation <math>L \subseteq G^3</math> is actually a function <math>L : G \times G \to G.</math> It is from this functional perspective that we can see an easy way to derive the two regular representations.

Since we have a function of the type <math>L : G \times G \to G,</math> we can define a couple of substitution operators:

{| align="center" cellpadding="10" width="90%"
| valign="top" | 1.
| <math>\mathrm{Sub}(x, (\underline{~~}, y))</math> puts any specified <math>x</math> into the empty slot of the rheme <math>(\underline{~~}, y),</math> with the effect of producing the saturated rheme <math>(x, y)</math> that evaluates to <math>xy.</math>
|-
| valign="top" | 2.
| <math>\mathrm{Sub}(x, (y, \underline{~~}))</math> puts any specified <math>x</math> into the empty slot of the rheme <math>(y, \underline{~~}),</math> with the effect of producing the saturated rheme <math>(y, x)</math> that evaluates to <math>yx.</math>
|}

In (1) we consider the effects of each <math>x</math> in its practical bearing on contexts of the form <math>(\underline{~~}, y),</math> as <math>y</math> ranges over <math>G,</math> and the effects are such that <math>x</math> takes <math>(\underline{~~}, y)</math> into <math>xy,</math> for <math>y</math> in <math>G,</math> all of which is notated as <math>x = \{ (y : xy) ~|~ y \in G \}.</math> The pairs <math>(y : xy)</math> can be found by picking an <math>x</math> from the left margin of the group operation table and considering its effects on each <math>y</math> in turn as these run along the right margin. This produces the ''regular ante-representation'' of <math>S_3,</math> like so:

{| align="center" cellpadding="10" style="text-align:center"
|
<math>\begin{array}{*{13}{c}}
\mathrm{e}
& = & \mathrm{e}\!:\!\mathrm{e}
& + & \mathrm{f}\!:\!\mathrm{f}
& + & \mathrm{g}\!:\!\mathrm{g}
& + & \mathrm{h}\!:\!\mathrm{h}
& + & \mathrm{i}\!:\!\mathrm{i}
& + & \mathrm{j}\!:\!\mathrm{j}
\\[4pt]
\mathrm{f}
& = & \mathrm{e}\!:\!\mathrm{f}
& + & \mathrm{f}\!:\!\mathrm{g}
& + & \mathrm{g}\!:\!\mathrm{e}
& + & \mathrm{h}\!:\!\mathrm{j}
& + & \mathrm{i}\!:\!\mathrm{h}
& + & \mathrm{j}\!:\!\mathrm{i}
\\[4pt]
\mathrm{g}
& = & \mathrm{e}\!:\!\mathrm{g}
& + & \mathrm{f}\!:\!\mathrm{e}
& + & \mathrm{g}\!:\!\mathrm{f}
& + & \mathrm{h}\!:\!\mathrm{i}
& + & \mathrm{i}\!:\!\mathrm{j}
& + & \mathrm{j}\!:\!\mathrm{h}
\\[4pt]
\mathrm{h}
& = & \mathrm{e}\!:\!\mathrm{h}
& + & \mathrm{f}\!:\!\mathrm{i}
& + & \mathrm{g}\!:\!\mathrm{j}
& + & \mathrm{h}\!:\!\mathrm{e}
& + & \mathrm{i}\!:\!\mathrm{f}
& + & \mathrm{j}\!:\!\mathrm{g}
\\[4pt]
\mathrm{i}
& = & \mathrm{e}\!:\!\mathrm{i}
& + & \mathrm{f}\!:\!\mathrm{j}
& + & \mathrm{g}\!:\!\mathrm{h}
& + & \mathrm{h}\!:\!\mathrm{g}
& + & \mathrm{i}\!:\!\mathrm{e}
& + & \mathrm{j}\!:\!\mathrm{f}
\\[4pt]
\mathrm{j}
& = & \mathrm{e}\!:\!\mathrm{j}
& + & \mathrm{f}\!:\!\mathrm{h}
& + & \mathrm{g}\!:\!\mathrm{i}
& + & \mathrm{h}\!:\!\mathrm{f}
& + & \mathrm{i}\!:\!\mathrm{g}
& + & \mathrm{j}\!:\!\mathrm{e}
\end{array}</math>
|}

In (2) we consider the effects of each <math>x</math> in its practical bearing on contexts of the form <math>(y, \underline{~~}),</math> as <math>y</math> ranges over <math>G,</math> and the effects are such that <math>x</math> takes <math>(y, \underline{~~})</math> into <math>yx,</math> for <math>y</math> in <math>G,</math> all of which is notated as <math>x = \{ (y : yx) ~|~ y \in G \}.</math> The pairs <math>(y : yx)</math> can be found by picking an <math>x</math> on the right margin of the group operation table and considering its effects on each <math>y</math> in turn as these run along the left margin. This produces the ''regular post-representation'' of <math>S_3,</math> like so:

{| align="center" cellpadding="10" style="text-align:center"
|
<math>\begin{array}{*{13}{c}}
\mathrm{e}
& = & \mathrm{e}\!:\!\mathrm{e}
& + & \mathrm{f}\!:\!\mathrm{f}
& + & \mathrm{g}\!:\!\mathrm{g}
& + & \mathrm{h}\!:\!\mathrm{h}
& + & \mathrm{i}\!:\!\mathrm{i}
& + & \mathrm{j}\!:\!\mathrm{j}
\\[4pt]
\mathrm{f}
& = & \mathrm{e}\!:\!\mathrm{f}
& + & \mathrm{f}\!:\!\mathrm{g}
& + & \mathrm{g}\!:\!\mathrm{e}
& + & \mathrm{h}\!:\!\mathrm{i}
& + & \mathrm{i}\!:\!\mathrm{j}
& + & \mathrm{j}\!:\!\mathrm{h}
\\[4pt]
\mathrm{g}
& = & \mathrm{e}\!:\!\mathrm{g}
& + & \mathrm{f}\!:\!\mathrm{e}
& + & \mathrm{g}\!:\!\mathrm{f}
& + & \mathrm{h}\!:\!\mathrm{j}
& + & \mathrm{i}\!:\!\mathrm{h}
& + & \mathrm{j}\!:\!\mathrm{i}
\\[4pt]
\mathrm{h}
& = & \mathrm{e}\!:\!\mathrm{h}
& + & \mathrm{f}\!:\!\mathrm{j}
& + & \mathrm{g}\!:\!\mathrm{i}
& + & \mathrm{h}\!:\!\mathrm{e}
& + & \mathrm{i}\!:\!\mathrm{g}
& + & \mathrm{j}\!:\!\mathrm{f}
\\[4pt]
\mathrm{i}
& = & \mathrm{e}\!:\!\mathrm{i}
& + & \mathrm{f}\!:\!\mathrm{h}
& + & \mathrm{g}\!:\!\mathrm{j}
& + & \mathrm{h}\!:\!\mathrm{f}
& + & \mathrm{i}\!:\!\mathrm{e}
& + & \mathrm{j}\!:\!\mathrm{g}
\\[4pt]
\mathrm{j}
& = & \mathrm{e}\!:\!\mathrm{j}
& + & \mathrm{f}\!:\!\mathrm{i}
& + & \mathrm{g}\!:\!\mathrm{h}
& + & \mathrm{h}\!:\!\mathrm{g}
& + & \mathrm{i}\!:\!\mathrm{f}
& + & \mathrm{j}\!:\!\mathrm{e}
\end{array}</math>
|}

If the ante-rep looks different from the post-rep, it is just as it should be, as <math>S_3</math> is non-abelian (non-commutative), and so the two representations differ in the details of their practical effects, though, of course, being representations of the same abstract group, they must be isomorphic.

{| cellpadding="2" cellspacing="2" width="100%"
| width="60%" |  
| width="40%" |
the way of heaven and earth<br>
is to be long continued<br>
in their operation<br>
without stopping
|-
| height="50px" |  
| valign="top" | — i ching, hexagram 32
|}

The Reader may be wondering what happened to the announced subject of ''Dynamics And Logic''. What happened was a bit like this:

We made the observation that the shift operators <math>\{ \mathrm{E}_{ij} \}</math> form a transformation group that acts on the set of propositions of the form <math>f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}.</math> Group theory is a very attractive subject, but it did not draw us so far from our intended course as one might initially think. For one thing, groups, especially the groups that are named after the Norwegian mathematician [http://www-history.mcs.st-andrews.ac.uk/Biographies/Lie.html Marius Sophus Lie (1842–1899)], have turned out to be of critical utility in the solution of differential equations. For another thing, group operations provide us with an ample supply of triadic relations that have been extremely well-studied over the years, and thus they give us no small measure of useful guidance in the study of sign relations, another brand of 3-adic relations that have significance for logical studies, and in our acquaintance with which we have barely begun to break the ice. Finally, I couldn't resist taking up the links between group representations, amounting to the very archetypes of logical models, and the pragmatic maxim.

We've seen a couple of groups, <math>V_4</math> and <math>S_3,</math> represented in various ways, and we've seen their representations presented in a variety of different manners. Let us look at one other stylistic variant for presenting a representation that is frequently seen, the so-called ''matrix representation'' of a group.

Recalling the manner of our acquaintance with the symmetric group <math>S_3,</math> we began with the ''bigraph'' (bipartite graph) picture of its natural representation as the set of all permutations or substitutions on the set <math>X = \{ a, b, c \}.</math>

<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px; font-size:large" |
<math>\text{Permutation Substitutions in}~ \mathrm{Sym} \{ a, b, c \}</math>
|- style="background:#f0f0ff"
| width="16%" | <math>\mathrm{e}</math>
| width="16%" | <math>\mathrm{f}</math>
| width="16%" | <math>\mathrm{g}</math>
| width="16%" | <math>\mathrm{h}</math>
| width="16%" | <math>\mathrm{i}</math>
| width="16%" | <math>\mathrm{j}</math>
|-
|
<math>\begin{matrix}
a & b & c
\\[3pt]
\downarrow & \downarrow & \downarrow
\\[6pt]
a & b & c
\end{matrix}</math>
|
<math>\begin{matrix}
a & b & c
\\[3pt]
\downarrow & \downarrow & \downarrow
\\[6pt]
c & a & b
\end{matrix}</math>
|
<math>\begin{matrix}
a & b & c
\\[3pt]
\downarrow & \downarrow & \downarrow
\\[6pt]
b & c & a
\end{matrix}</math>
|
<math>\begin{matrix}
a & b & c
\\[3pt]
\downarrow & \downarrow & \downarrow
\\[6pt]
a & c & b
\end{matrix}</math>
|
<math>\begin{matrix}
a & b & c
\\[3pt]
\downarrow & \downarrow & \downarrow
\\[6pt]
c & b & a
\end{matrix}</math>
|
<math>\begin{matrix}
a & b & c
\\[3pt]
\downarrow & \downarrow & \downarrow
\\[6pt]
b & a & c
\end{matrix}</math>
|}

<br>

These permutations were then converted to relative form as logical sums of elementary relatives:

{| align="center" cellpadding="10" width="90%"
| align="center" |
<math>\begin{matrix}
\mathrm{e}
& = & a\!:\!a
& + & b\!:\!b
& + & c\!:\!c
\\[4pt]
\mathrm{f}
& = & a\!:\!c
& + & b\!:\!a
& + & c\!:\!b
\\[4pt]
\mathrm{g}
& = & a\!:\!b
& + & b\!:\!c
& + & c\!:\!a
\\[4pt]
\mathrm{h}
& = & a\!:\!a
& + & b\!:\!c
& + & c\!:\!b
\\[4pt]
\mathrm{i}
& = & a\!:\!c
& + & b\!:\!b
& + & c\!:\!a
\\[4pt]
\mathrm{j}
& = & a\!:\!b
& + & b\!:\!a
& + & c\!:\!c
\end{matrix}</math>
|}

From the relational representation of <math>\mathrm{Sym} \{ a, b, c \} \cong S_3,</math> one easily derives a ''linear representation'' of the group by viewing each permutation as a linear transformation that maps the elements of a suitable vector space onto each other. Each of these linear transformations is in turn represented by a 2-dimensional array of coefficients in <math>\mathbb{B},</math> resulting in the following set of matrices for the group:

<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px; font-size:large" |
<math>\text{Matrix Representations of Permutations in}~ \mathrm{Sym}(3)</math>
|- style="background:#f0f0ff"
| width="16%" | <math>\mathrm{e}</math>
| width="16%" | <math>\mathrm{f}</math>
| width="16%" | <math>\mathrm{g}</math>
| width="16%" | <math>\mathrm{h}</math>
| width="16%" | <math>\mathrm{i}</math>
| width="16%" | <math>\mathrm{j}</math>
|-
|
<math>\begin{matrix}
1 & 0 & 0
\\
0 & 1 & 0
\\
0 & 0 & 1
\end{matrix}</math>
|
<math>\begin{matrix}
0 & 0 & 1
\\
1 & 0 & 0
\\
0 & 1 & 0
\end{matrix}</math>
|
<math>\begin{matrix}
0 & 1 & 0
\\
0 & 0 & 1
\\
1 & 0 & 0
\end{matrix}</math>
|
<math>\begin{matrix}
1 & 0 & 0
\\
0 & 0 & 1
\\
0 & 1 & 0
\end{matrix}</math>
|
<math>\begin{matrix}
0 & 0 & 1
\\
0 & 1 & 0
\\
1 & 0 & 0
\end{matrix}</math>
|
<math>\begin{matrix}
0 & 1 & 0
\\
1 & 0 & 0
\\
0 & 0 & 1
\end{matrix}</math>
|}

<br>

The key to the mysteries of these matrices is revealed by observing that their coefficient entries are arrayed and overlaid on a place-mat marked like so:

{| align="center" cellpadding="6" width="90%"
| align="center" |
<math>\begin{bmatrix}
a\!:\!a &
a\!:\!b &
a\!:\!c
\\
b\!:\!a &
b\!:\!b &
b\!:\!c
\\
c\!:\!a &
c\!:\!b &
c\!:\!c
\end{bmatrix}</math>
|}

==Work Area==

===Conjunction===

<br>

<table align="center" cellpadding="8px" cellspacing="0" style="border:1px solid black; font-size:larger; text-align:center; width:40%">

<caption style="height:30px"><math>\text{Logical Conjunction}~ f_{8}(p, q) = pq</math></caption>

<tr style="height:40px; background:ghostwhite">
<th style="border-bottom:1px solid black; width:33%"><math>p</math></th>
<th style="border-bottom:1px solid black; width:33%"><math>q</math></th>
<th style="border-bottom:1px solid black; width:34%"><math>pq</math></th>
</tr>

<tr><td><math>0</math></td><td><math>0</math></td><td><math>0</math></td></tr>
<tr><td><math>0</math></td><td><math>1</math></td><td><math>0</math></td></tr>
<tr><td><math>1</math></td><td><math>0</math></td><td><math>0</math></td></tr>
<tr><td><math>1</math></td><td><math>1</math></td><td><math>1</math></td></tr>

</table>

<br>

===Enlargement of Conjunction===

<br>

<table align="center" cellpadding="1px" cellspacing="0" style="border:1px solid black; font-size:larger; text-align:center; width:40%">

<caption style="height:60px"><math>\begin{matrix}
\text{Enlargement of Conjunction} \\
\mathrm{E}f_{8}(p, q, \mathrm{d}p, \mathrm{d}q) =
\texttt{(} p \texttt{,} \mathrm{d}p \texttt{)(} q \texttt{,} \mathrm{d}q \texttt{)}
\end{matrix}</math></caption>

<tr style="height:40px; background:ghostwhite">
<th style="border-bottom:1px solid black; width:20%"><math>p</math></th>
<th style="border-bottom:1px solid black; width:20%"><math>q</math></th>
<th style="border-bottom:1px solid black; width:20%"><math>\mathrm{d}p</math></th>
<th style="border-bottom:1px solid black; width:20%"><math>\mathrm{d}q</math></th>
<th style="border-bottom:1px solid black; border-left:1px solid black; width:20%">
<math>\mathrm{E}f_{8}</math></th>
</tr>

<tr>
<td><math>0</math></td>
<td><math>0</math></td>
<td><math>0</math></td>
<td><math>0</math></td>
<td style="border-left:1px solid black"><math>0</math></td>
</tr>

<tr>
<td> </td>
<td> </td>
<td><math>0</math></td>
<td><math>1</math></td>
<td style="border-left:1px solid black"><math>0</math></td>
</tr>

<tr>
<td> </td>
<td> </td>
<td><math>1</math></td>
<td><math>0</math></td>
<td style="border-left:1px solid black"><math>0</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>1</math></td>
<td style="border-bottom:1px solid black"><math>1</math></td>
<td style="border-bottom:1px solid black; border-left:1px solid black"><math>1</math></td>
</tr>

<tr>
<td><math>0</math></td>
<td><math>1</math></td>
<td><math>0</math></td>
<td><math>0</math></td>
<td style="border-left:1px solid black"><math>0</math></td>
</tr>

<tr>
<td> </td>
<td> </td>
<td><math>0</math></td>
<td><math>1</math></td>
<td style="border-left:1px solid black"><math>0</math></td>
</tr>

<tr>
<td> </td>
<td> </td>
<td><math>1</math></td>
<td><math>0</math></td>
<td style="border-left:1px solid black"><math>1</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>1</math></td>
<td style="border-bottom:1px solid black"><math>1</math></td>
<td style="border-bottom:1px solid black; border-left:1px solid black"><math>0</math></td>
</tr>

<tr>
<td><math>1</math></td>
<td><math>0</math></td>
<td><math>0</math></td>
<td><math>0</math></td>
<td style="border-left:1px solid black"><math>0</math></td>
</tr>

<tr>
<td> </td>
<td> </td>
<td><math>0</math></td>
<td><math>1</math></td>
<td style="border-left:1px solid black"><math>1</math></td>
</tr>

<tr>
<td> </td>
<td> </td>
<td><math>1</math></td>
<td><math>0</math></td>
<td style="border-left:1px solid black"><math>0</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>1</math></td>
<td style="border-bottom:1px solid black"><math>1</math></td>
<td style="border-bottom:1px solid black; border-left:1px solid black"><math>0</math></td>
</tr>

<tr>
<td><math>1</math></td>
<td><math>1</math></td>
<td><math>0</math></td>
<td><math>0</math></td>
<td style="border-left:1px solid black"><math>1</math></td>
</tr>

<tr>
<td> </td>
<td> </td>
<td><math>0</math></td>
<td><math>1</math></td>
<td style="border-left:1px solid black"><math>0</math></td>
</tr>

<tr>
<td> </td>
<td> </td>
<td><math>1</math></td>
<td><math>0</math></td>
<td style="border-left:1px solid black"><math>0</math></td>
</tr>

<tr>
<td> </td>
<td> </td>
<td><math>1</math></td>
<td><math>1</math></td>
<td style="border-left:1px solid black"><math>0</math></td>
</tr>

</table>

<br>

{| align="center" cellpadding="10" style="text-align:center"
|
<math>\begin{matrix}
\mathrm{E}f_{8}
& = & pq \cdot \mathrm{E}f_{pq}
& + & p(q) \cdot \mathrm{E}f_{p(q)}
& + & (p)q \cdot \mathrm{E}f_{(p)q}
& + & (p)(q) \cdot \mathrm{E}f_{(p)(q)}
\end{matrix}</math>
|}

====Panoptic View • Enlargement Maps====

Given the data that develops in this form of analysis, the disjoined ingredients can now be folded back into a boolean expansion or a disjunctive normal form (DNF) that is equivalent to the enlarged proposition <math>\mathrm{E}f.</math>

{| align="center" cellpadding="10" style="text-align:center"
|
<math>\begin{matrix}
\mathrm{E}f
& = &
pq \cdot \mathrm{E}f_{pq}
& + &
p(q) \cdot \mathrm{E}f_{p(q)}
& + &
(p)q \cdot \mathrm{E}f_{(p)q}
& + &
(p)(q) \cdot \mathrm{E}f_{(p)(q)}
\end{matrix}</math>
|}

Here is a summary of the result, illustrated by means of a digraph picture, where the “no change” element <math>(\mathrm{d}p)(\mathrm{d}q)</math> is drawn as a loop at the point <math>p~q.</math>

{| align="center" cellpadding="10" style="text-align:center"
| [[File:Directed Graph Enlargement pq.jpg|500px]]
|}

{| align="center" cellpadding="10" style="text-align:center"
|
<math>\begin{array}{rcccc}
f & = & p ~ ~ ~ q
\\[4pt]
\mathrm{E}f & = & p ~ ~ ~ q
& \cdot & \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)}
\\[4pt]
& + & ~~ p ~~ \texttt{(} q \texttt{)}
& \cdot & \texttt{(} \mathrm{d}p \texttt{)} ~~ \mathrm{d}q ~~
\\[4pt]
& + & \texttt{(} p \texttt{)} ~~ q ~~
& \cdot & ~~ \mathrm{d}p ~~ \texttt{(} \mathrm{d}q \texttt{)}
\\[4pt]
& + & \texttt{(} p \texttt{)(} q \texttt{)}
& \cdot & \mathrm{d}p ~ ~ ~ \mathrm{d}q
\end{array}</math>
|}

We may understand the enlarged proposition <math>\mathrm{E}f</math> as telling us all the ways of reaching a satisfying interpretation or “model” of the proposition <math>f</math> from each point of the universe <math>X.</math>

====Ef Expanded over Ordinary Features====

<br>

{| align="center" cellpadding="4" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; font-size:larger; text-align:center; width:80%"
|+ style="height:30px; font-size:large" | <math>\mathrm{E}f ~\text{Expanded over Ordinary Features}~ \{ p, q \}</math>
|- style="background:ghostwhite"
| style="width:10%; border-bottom:1px solid black" |  
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f</math>
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |
<math>\mathrm{E}f|_{pq}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{E}f|_{p \texttt{(} q \texttt{)}}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{E}f|_{\texttt{(} p \texttt{)} q}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{E}f|_{\texttt{(} p \texttt{)(} q \texttt{)}}</math>
|-
| style="border-top:1px solid black" | <math>f_{8}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>p ~ ~ ~ q</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\texttt{(} \mathrm{d}p \texttt{)} ~~ \mathrm{d}q ~~</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>~~ \mathrm{d}p ~~ \texttt{(} \mathrm{d}q \texttt{)}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\mathrm{d}p ~ ~ ~ \mathrm{d}q</math>
|}

<br>

Here is a summary of the result, illustrated by means of a digraph picture, where the “no change” element <math>(\mathrm{d}p)(\mathrm{d}q)</math> is drawn as a loop at the point <math>p~q.</math>

{| align="center" cellpadding="10" style="text-align:center"
| [[File:Directed Graph Enlargement pq.jpg|500px]]
|}

<br>

====Ef Expanded over Differential Features====

<br>

{| align="center" cellpadding="4" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; font-size:larger; text-align:center; width:80%"
|+ style="height:30px; font-size:large" | <math>\mathrm{E}f ~\text{Expanded over Differential Features}~ \{ \mathrm{d}p, \mathrm{d}q \}</math>
|- style="background:ghostwhite"
| style="width:10%; border-bottom:1px solid black" |  
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f</math>
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |
<math>\mathrm{E}f|_{\mathrm{d}p ~ \mathrm{d}q}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{E}f|_{\mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)}}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{E}f|_{\texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{E}f|_{\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)}}</math>
|-
| style="border-top:1px solid black" | <math>f_{8}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>p ~ ~ ~ q</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\texttt{(} p \texttt{)(} q \texttt{)}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\texttt{(} p \texttt{)} ~~ q ~~</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>~~ p ~~ \texttt{(} q \texttt{)}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>p ~ ~ ~ q</math>
|}

<br>

User:Jon Awbrey/Figures and Tables 52

2026-03-12T13:32:10Z

Jon Awbrey: + User:Jon Awbrey/Figures and Tables 52

==Cactus Language Stretching Exercises Display 1==

===PNG===

{| align="center" cellpadding="8"
| [[File:Cactus Language Stretching Exercises Display 1.png|300px]]
|}

===LaTeX===

{| align="center" cellpadding="8"
| <math>F(x, y) ~=~ F_{6}^{(2)} (x, y) ~=~ \texttt{(} ~x~ \texttt{,} ~y~ \texttt{)}</math>
|}

==Cactus Language Stretching Exercises Display 2==

===PNG===

{| align="center" cellpadding="8"
| [[File:Cactus Language Stretching Exercises Display 2.png|400px]]
|}

===LaTeX===

{| align="center" cellpadding="8"
|
<math>\begin{array}{lll}
[| \downharpoonleft s \downharpoonright |]
& = & [| F |]
\\[4pt]
& = & F^{-1} (1)
\\[4pt]
& = & \{~ (x, y) \in \mathbb{B}^2 ~:~ s ~\}
\\[4pt]
& = & \{~ (x, y) \in \mathbb{B}^2 ~:~ F(x, y) = 1 ~\}
\\[4pt]
& = & \{~ (x, y) \in \mathbb{B}^2 ~:~ F(x, y) ~\}
\\[4pt]
& = & \{~ (x, y) \in \mathbb{B}^2 ~:~ \texttt{(} ~x~ \texttt{,} ~y~ \texttt{)} = 1 ~\}
\\[4pt]
& = & \{~ (x, y) \in \mathbb{B}^2 ~:~ \texttt{(} ~x~ \texttt{,} ~y~ \texttt{)} ~\}
\\[4pt]
& = & \{~ (x, y) \in \mathbb{B}^2 ~:~ x ~\mathrm{exclusive~or}~ y ~\}
\\[4pt]
& = & \{~ (x, y) \in \mathbb{B}^2 ~:~ \mathrm{just~one~true~of}~ x, y ~\}
\\[4pt]
& = & \{~ (x, y) \in \mathbb{B}^2 ~:~ x ~\mathrm{not~equal~to}~ y ~\}
\\[4pt]
& = & \{~ (x, y) \in \mathbb{B}^2 ~:~ x \nLeftrightarrow y ~\}
\\[4pt]
& = & \{~ (x, y) \in \mathbb{B}^2 ~:~ x \neq y ~\}
\\[4pt]
& = & \{~ (x, y) \in \mathbb{B}^2 ~:~ x + y ~\}.
\end{array}</math>
|}

==Cactus Language Stretching Exercises Display 3==

===PNG===

===LaTeX===

{| align="center" cellpadding="8"
|
<math>\begin{matrix}
p & = & \upharpoonleft P \upharpoonright & : & X \to \mathbb{B}
\\[4pt]
q & = & \upharpoonleft Q \upharpoonright & : & X \to \mathbb{B}
\\[4pt]
(p, q) & = & (\upharpoonleft P \upharpoonright, \upharpoonleft Q \upharpoonright) & : & (X \to \mathbb{B})^2
\end{matrix}</math>
|}

==Cactus Language Stretching Exercises Display 4==

===PNG===

===LaTeX===

{| align="center" cellpadding="8"
|
<math>\begin{array}{ccccl}
F^\$ & = & \underline{(} \ldots, \ldots \underline{)}^\$ & : & (X \to \mathbb{B})^2 \to (X \to \mathbb{B})
\\[4pt]
F^\$ (p, q) & = & \underline{(}~p~,~q~\underline{)}^\$ & : & X \to \mathbb{B}
\end{array}</math>
|}

==Cactus Language Stretching Exercises Display 5==

===PNG===

===LaTeX===

{| align="center" cellpadding="8"
|
<math>\begin{matrix}
F^\$ (p, q)(x) & = & \underline{(}~p~,~q~\underline{)}^\$ (x) & \in & \mathbb{B}
\\[4pt]
\Updownarrow & & \Updownarrow
\\[4pt]
F(p(x), q(x)) & = & \underline{(}~p(x)~,~q(x)~\underline{)} & \in & \mathbb{B}
\end{matrix}</math>
|}

==Cactus Language Stretching Exercises Display 6==

===PNG===

===LaTeX===

{| align="center" cellpadding="8"
|
<math>\begin{array}{lll}
[| F^\$ (p, q) |]
& = & [| \underline{(}~p~,~q~\underline{)}^\$ |]
\\[4pt]
& = & (F^\$ (p, q))^{-1} (1)
\\[4pt]
& = & \{~ x \in X ~:~ F^\$ (p, q)(x) ~\}
\\[4pt]
& = & \{~ x \in X ~:~ \underline{(}~p~,~q~\underline{)}^\$ (x) ~\}
\\[4pt]
& = & \{~ x \in X ~:~ \underline{(}~p(x)~,~q(x)~\underline{)} ~\}
\\[4pt]
& = & \{~ x \in X ~:~ p(x) + q(x) ~\}
\\[4pt]
& = & \{~ x \in X ~:~ p(x) \neq q(x) ~\}
\\[4pt]
& = & \{~ x \in X ~:~ \upharpoonleft P \upharpoonright (x) ~\neq~ \upharpoonleft Q \upharpoonright (x) ~\}
\\[4pt]
& = & \{~ x \in X ~:~ x \in P ~\nLeftrightarrow~ x \in Q ~\}
\\[4pt]
& = & \{~ x \in X ~:~ x \in P\!-\!Q ~\mathrm{or}~ x \in Q\!-\!P ~\}
\\[4pt]
& = & \{~ x \in X ~:~ x \in P\!-\!Q ~\cup~ Q\!-\!P ~\}
\\[4pt]
& = & \{~ x \in X ~:~ x \in P + Q ~\}
\\[4pt]
& = & P + Q ~\subseteq~ X
\\[4pt]
& = & [|p|] + [|q|] ~\subseteq~ X
\end{array}</math>
|}

User:Jon Awbrey/Figures and Tables 51

2026-03-12T13:30:16Z

Jon Awbrey: + User:Jon Awbrey/Figures and Tables 51

==Format Examples==

<ul>
<li><math>\rightsquigarrow</math></li>

<li><math>\leftrightsquigarrow</math></li>

<li><math>\xrightarrow{\mathrm{Parse}}</math></li>

<li><math>\xrightarrow[\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}]{\mathrm{Parse}}</math></li>
</ul>

==Font Size Examples==

{| align="center" cellpadding="10" cellspacing="0"
| align="left" width="20%" | Medium
| align="center" style="font-size:medium" | <math>\text{Algorithmic Translation Rules}</math>
|-
| Larger
| align="center" style="font-size:larger" | <math>\text{Algorithmic Translation Rules}</math>
|-
| 125%
| align="center" style="font-size:125%" | <math>\text{Algorithmic Translation Rules}</math>
|-
| Large
| align="center" style="font-size:large" | <math>\text{Algorithmic Translation Rules}</math>
|-
| 150%
| align="center" style="font-size:150%" | <math>\text{Algorithmic Translation Rules}</math>
|}

==Algorithmic Translation Rules==

===PNG===

{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"
| style="height:25px; font-size:large" | <math>\text{Algorithmic Translation Rules}</math>
|-
| [[File:Cactus Language Algorithmic Translation Rules.png|600px]]
|}

===LaTeX===

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%"
|+ style="height:30px; font-size:large" | <math>\text{Algorithmic Translation Rules}</math>
|- style="height:40px; background:ghostwhite"
|
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:ghostwhite; text-align:center; width:100%"
| width="33%" | <math>\text{Sentence in PARCE}</math>
| width="33%" | <math>\xrightarrow{\mathrm{Parse}}</math>
| width="33%" | <math>\text{Graph in PARC}</math>
|}
|-
|
{| align="center" border="0" cellpadding="8" cellspacing="0" style="text-align:center; width:100%"
| width="33%" | <math>\mathrm{Conc}^0</math>
| width="33%" | <math>\xrightarrow{\mathrm{Parse}}</math>
| width="33%" | <math>\mathrm{Node}^0</math>
|-
| width="33%" | <math>\mathrm{Conc}_{j=1}^k s_j</math>
| width="33%" | <math>\xrightarrow{\mathrm{Parse}}</math>
| width="33%" | <math>\mathrm{Node}_{j=1}^k \mathrm{Parse} (s_j)</math>
|}
|-
|
{| align="center" border="0" cellpadding="8" cellspacing="0" style="text-align:center; width:100%"
| width="33%" | <math>\mathrm{Surc}^0</math>
| width="33%" | <math>\xrightarrow{\mathrm{Parse}}</math>
| width="33%" | <math>\mathrm{Lobe}^0</math>
|-
| width="33%" | <math>\mathrm{Surc}_{j=1}^k s_j</math>
| width="33%" | <math>\xrightarrow{\mathrm{Parse}}</math>
| width="33%" | <math>\mathrm{Lobe}_{j=1}^k \mathrm{Parse} (s_j)</math>
|}
|}

<br>

==Semantic Translation • Functional Form==

===PNG===

{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"
| style="height:25px; font-size:large" | <math>\text{Semantic Translation}</math> • <math>\text{Functional Form}</math>
|-
| [[File:Cactus Language Semantic Translation Functional Form.png|600px]]
|}

===LaTeX===

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%"
|+ style="height:30px; font-size:large" | <math>\text{Semantic Translation}</math> • <math>\text{Functional Form}</math>
|- style="height:40px; background:ghostwhite"
|
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:ghostwhite; width:100%"
| width="20%" | <math>\mathrm{Sentence}</math>
| width="20%" | <math>\xrightarrow[\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}]{\mathrm{Parse}}</math>
| width="20%" | <math>\mathrm{Graph}</math>
| width="20%" | <math>\xrightarrow[\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}]{\mathrm{Denotation}}</math>
| width="20%" | <math>\mathrm{Proposition}</math>
|}
|-
|
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
| width="20%" | <math>s_j</math>
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
| width="20%" | <math>C_j</math>
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
| width="20%" | <math>q_j</math>
|}
|-
|
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
| width="20%" | <math>\mathrm{Conc}^0</math>
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
| width="20%" | <math>\mathrm{Node}^0</math>
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
| width="20%" | <math>1</math>
|-
| width="20%" | <math>\mathrm{Conc}^k_j s_j</math>
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
| width="20%" | <math>\mathrm{Node}^k_j C_j</math>
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
| width="20%" | <math>\mathrm{Conj}^k_j q_j</math>
|}
|-
|
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
| width="20%" | <math>\mathrm{Surc}^0</math>
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
| width="20%" | <math>\mathrm{Lobe}^0</math>
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
| width="20%" | <math>0</math>
|-
| width="20%" | <math>\mathrm{Surc}^k_j s_j</math>
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
| width="20%" | <math>\mathrm{Lobe}^k_j C_j</math>
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
| width="20%" | <math>\mathrm{Surj}^k_j q_j</math>
|}
|}

<br>

==Semantic Translation • Equational Form==

===PNG===

{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"
| style="height:25px; font-size:large" | <math>\text{Semantic Translation}</math> • <math>\text{Equational Form}</math>
|-
| [[File:Cactus Language Semantic Translation Equational Form.png|600px]]
|}

===LaTeX===

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%"
|+ style="height:30px; font-size:large" | <math>\text{Semantic Translation}</math> • <math>\text{Equational Form}</math>
|- style="height:40px; background:ghostwhite"
|
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:ghostwhite; width:100%"
| width="20%" | <math>\downharpoonleft \mathrm{Sentence} \downharpoonright</math>
| width="20%" | <math>\stackrel{\mathrm{Parse}}{=}</math>
| width="20%" | <math>\downharpoonleft \mathrm{Graph} \downharpoonright</math>
| width="20%" | <math>\stackrel{\mathrm{Denotation}}{=}</math>
| width="20%" | <math>\mathrm{Proposition}</math>
|}
|-
|
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
| width="20%" | <math>\downharpoonleft s_j \downharpoonright</math>
| width="20%" | <math>=</math>
| width="20%" | <math>\downharpoonleft C_j \downharpoonright</math>
| width="20%" | <math>=</math>
| width="20%" | <math>q_j</math>
|}
|-
|
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
| width="20%" | <math>\downharpoonleft \mathrm{Conc}^0 \downharpoonright</math>
| width="20%" | <math>=</math>
| width="20%" | <math>\downharpoonleft \mathrm{Node}^0 \downharpoonright</math>
| width="20%" | <math>=</math>
| width="20%" | <math>1</math>
|-
| width="20%" | <math>\downharpoonleft \mathrm{Conc}^k_j s_j \downharpoonright</math>
| width="20%" | <math>=</math>
| width="20%" | <math>\downharpoonleft \mathrm{Node}^k_j C_j \downharpoonright</math>
| width="20%" | <math>=</math>
| width="20%" | <math>\mathrm{Conj}^k_j q_j</math>
|}
|-
|
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
| width="20%" | <math>\downharpoonleft \mathrm{Surc}^0 \downharpoonright</math>
| width="20%" | <math>=</math>
| width="20%" | <math>\downharpoonleft \mathrm{Lobe}^0 \downharpoonright</math>
| width="20%" | <math>=</math>
| width="20%" | <math>0</math>
|-
| width="20%" | <math>\downharpoonleft \mathrm{Surc}^k_j s_j \downharpoonright</math>
| width="20%" | <math>=</math>
| width="20%" | <math>\downharpoonleft \mathrm{Lobe}^k_j C_j \downharpoonright</math>
| width="20%" | <math>=</math>
| width="20%" | <math>\mathrm{Surj}^k_j q_j</math>
|}
|}

<br>

==Boolean Functions on Zero Variables==

===PNG===

{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"
|+ style="height:25px; font-size:large" | <math>\text{Boolean Functions on Zero Variables}</math>
|-
| [[File:Boolean Functions on Zero Variables • Truth Table.png|600px]]
|}

===LaTeX===

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%"
|+ style="height:30px; font-size:large" | <math>\text{Boolean Functions on Zero Variables}</math>
|- style="height:40px; background:ghostwhite"
| width="14%" | <math>F</math>
| width="14%" | <math>F</math>
| width="48%" | <math>F()</math>
| width="24%" | <math>F</math>
|-
| <math>0</math>
| <math>F_0^{(0)}</math>
| <math>0</math>
| <math>\texttt{( )}</math>
|-
| <math>1</math>
| <math>F_1^{(0)}</math>
| <math>1</math>
| <math>\texttt{(( ))}</math>
|}

==Boolean Functions on One Variable==

===PNG===

{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"
|+ style="height:25px; font-size:large" | <math>\text{Boolean Functions on One Variable}</math>
|-
| [[File:Boolean Functions on One Variable • Truth Table.png|600px]]
|}

===LaTeX===

{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:60%"
|+ style="height:30px; font-size:large" | <math>\text{Boolean Functions on One Variable}</math>
|- style="height:40px; background:ghostwhite"
| width="14%" | <math>F</math>
| width="14%" | <math>F</math>
| colspan="2" | <math>F(x)</math>
| width="24%" | <math>F</math>
|- style="height:40px; background:ghostwhite"
| width="14%" |  
| width="14%" |  
| width="24%" | <math>F(1)</math>
| width="24%" | <math>F(0)</math>
| width="24%" |  
|-
| <math>F_0^{(1)}</math>
| <math>F_{00}^{(1)}</math>
| <math>0</math>
| <math>0</math>
| <math>\texttt{( )}</math>
|-
| <math>F_1^{(1)}</math>
| <math>F_{01}^{(1)}</math>
| <math>0</math>
| <math>1</math>
| <math>\texttt{(} x \texttt{)}</math>
|-
| <math>F_2^{(1)}</math>
| <math>F_{10}^{(1)}</math>
| <math>1</math>
| <math>0</math>
| <math>x</math>
|-
| <math>F_3^{(1)}</math>
| <math>F_{11}^{(1)}</math>
| <math>1</math>
| <math>1</math>
| <math>\texttt{(( ))}</math>
|}

<br>

==Boolean Functions on Two Variables==

===PNG===

{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"
|+ style="height:25px; font-size:large" | <math>\text{Boolean Functions on Two Variables}</math>
|-
| [[File:Boolean Functions on Two Variables • Truth Table.png|600px]]
|}

===LaTeX===

{| align="center" border="1" cellpadding="4" cellspacing="0" style="text-align:center; width:60%"
|+ style="height:30px; font-size:large" | <math>\text{Boolean Functions on Two Variables}</math>
|- style="height:40px; background:ghostwhite"
| width="14%" | <math>F</math>
| width="14%" | <math>F</math>
| colspan="4" | <math>F(x, y)</math>
| width="24%" | <math>F</math>
|- style="height:40px; background:ghostwhite"
| width="14%" |  
| width="14%" |  
| width="12%" | <math>F(1, 1)</math>
| width="12%" | <math>F(1, 0)</math>
| width="12%" | <math>F(0, 1)</math>
| width="12%" | <math>F(0, 0)</math>
| width="24%" |  
|-
| <math>F_{0}^{(2)}</math>
| <math>F_{0000}^{(2)}</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>\texttt{( )}</math>
|-
| <math>F_{1}^{(2)}</math>
| <math>F_{0001}^{(2)}</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>1</math>
| <math>\texttt{(} x \texttt{)(} y \texttt{)}</math>
|-
| <math>F_{2}^{(2)}</math>
| <math>F_{0010}^{(2)}</math>
| <math>0</math>
| <math>0</math>
| <math>1</math>
| <math>0</math>
| <math>\texttt{(} x \texttt{)} y</math>
|-
| <math>F_{3}^{(2)}</math>
| <math>F_{0011}^{(2)}</math>
| <math>0</math>
| <math>0</math>
| <math>1</math>
| <math>1</math>
| <math>\texttt{(} x \texttt{)}</math>
|-
| <math>F_{4}^{(2)}</math>
| <math>F_{0100}^{(2)}</math>
| <math>0</math>
| <math>1</math>
| <math>0</math>
| <math>0</math>
| <math>x \texttt{(} y \texttt{)}</math>
|-
| <math>F_{5}^{(2)}</math>
| <math>F_{0101}^{(2)}</math>
| <math>0</math>
| <math>1</math>
| <math>0</math>
| <math>1</math>
| <math>\texttt{(} y \texttt{)}</math>
|-
| <math>F_{6}^{(2)}</math>
| <math>F_{0110}^{(2)}</math>
| <math>0</math>
| <math>1</math>
| <math>1</math>
| <math>0</math>
| <math>\texttt{(} x \texttt{,} y \texttt{)}</math>
|-
| <math>F_{7}^{(2)}</math>
| <math>F_{0111}^{(2)}</math>
| <math>0</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>\texttt{(} x y \texttt{)}</math>
|-
| <math>F_{8}^{(2)}</math>
| <math>F_{1000}^{(2)}</math>
| <math>1</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>x y</math>
|-
| <math>F_{9}^{(2)}</math>
| <math>F_{1001}^{(2)}</math>
| <math>1</math>
| <math>0</math>
| <math>0</math>
| <math>1</math>
| <math>\texttt{((} x \texttt{,} y \texttt{))}</math>
|-
| <math>F_{10}^{(2)}</math>
| <math>F_{1010}^{(2)}</math>
| <math>1</math>
| <math>0</math>
| <math>1</math>
| <math>0</math>
| <math>y</math>
|-
| <math>F_{11}^{(2)}</math>
| <math>F_{1011}^{(2)}</math>
| <math>1</math>
| <math>0</math>
| <math>1</math>
| <math>1</math>
| <math>\texttt{(} x \texttt{(} y \texttt{))}</math>
|-
| <math>F_{12}^{(2)}</math>
| <math>F_{1100}^{(2)}</math>
| <math>1</math>
| <math>1</math>
| <math>0</math>
| <math>0</math>
| <math>x</math>
|-
| <math>F_{13}^{(2)}</math>
| <math>F_{1101}^{(2)}</math>
| <math>1</math>
| <math>1</math>
| <math>0</math>
| <math>1</math>
| <math>\texttt{((} x \texttt{)} y \texttt{)}</math>
|-
| <math>F_{14}^{(2)}</math>
| <math>F_{1110}^{(2)}</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>0</math>
| <math>\texttt{((} x \texttt{)(} y \texttt{))}</math>
|-
| <math>F_{15}^{(2)}</math>
| <math>F_{1111}^{(2)}</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>\texttt{(( ))}</math>
|}

<br>

Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 5

2026-03-02T17:00:16Z

Jon Awbrey: /* 5.3.4. Points Forward */ fix subheads

{{DISPLAYTITLE:Inquiry Driven Systems : Part 5}}
----
<div align="center">
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems|Contents]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 1|Part 1]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 2|Part 2]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 3|Part 3]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 4|Part 4]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 5|Part 5]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6|Part 6]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 7|Part 7]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 8|Part 8]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Appendices|Appendices]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : References|References]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Document History|Document History]]
•
</div>
----

<div class="nonumtoc">__TOC__</div>

==5. Interlude : The Medium and Its Message==

===5.1. Reflective Expression===

====5.1.1. Casual Reflection====

Recall that an ''ostensibly recursive text'' (ORT), already encountered a bit less formally in discussing the issue of the informal context, is a text that cites itself by title at some site within its body.

Consider a ''text in progress'' (TIP) at its growing edge, anywhere that it joins new text to a body of work already established, anywhere that it sends out buds and shoots from a secular truncation of its author's intention. Here is where a growing text advances through its media of language and communication, the potentially nurturing environments and the invariably constraining surroundings that make a definite array of resources available to a text, grant it certain options for continuing its development, and limit the effects of meaning that it can achieve. A text that can form in such a medium is part of what one has in mind whenever one opens a sentence with a phrase like “A text that can ___” and continues by elaborating on a text's ''abilities'', ''capacities'', or ''intentions'' as if this text is not a fixed or a static entity but one whose free selection and future development are still open to question.

A text that can cite itself by title is a limiting case of a text that can cite itself by chapter and verse, in other words, a text with a sufficient degree of articulation that it can make appropriate references to its own parts and sections, and can thus invoke the objects, the functions, and the structures that are represented in them. This should go to explain the interest I am taking in ORTs, their kin, and their generalizations. These kinds of texts exhibit an aspect of self reference that is usually taken for granted, to the point that it is hardly recognized as such, but one that is implied in all attempts “to make infinite use of finite means”. A program is generally a text of this sort. A non trivial program, one that wraps an infinite object in a finite sign, whether it numbers its lines and directs its execution by means of instructions that have its interpreter ''go to'' this or that place in its own text, or whether its modules call on each other by name, is always ''recursive'' in this sense.

The deeper that one looks into a species of text, the further that one's interest tends to shift from the distinctive features of individual texts to the properties of the medium that supports their growth. Although a medium is initially conceived to be a source of texts or a constraint on their production, that is, as a generative facility or a generated space, it is often possible to formalize it as the full grammar of a discursive language, in other words, as a comprehensive theory for that species of texts, accounting for their syntactic, semantic, and pragmatic aspects.

Still, what is the status in reality of these conceptual constructions: a medium, a grammar, or a theory for a species of texts? They have no meaning apart from the texts that they admit to exist. They are only known by means of the texts that they allow to subsist in them, that they enable to live and to grow, and everything that is learned about them ultimately needs to be expressed in a text, or something like it, even if not always a text of the very same order or species.

=====5.1.1.1. Ostensibly Recursive Texts=====

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | Bein' on the twenty third of June,
|-
| width="5%" |   || As I sat weaving all at my loom,
|-
| colspan="2" | Bein' on the twenty third of June,
|-
| width="5%" |   || As I sat weaving all at my loom,
|-
| colspan="2" | I heard a thrush, singing on yon bush,
|-
| width="5%" |   || And the song she sang was ''The Jug of Punch''.
|}

Is there a true poem, a verse with the echo of divine inspiration, that resounds in the memories of this minor ditty, this song of the pubs and this ballad of the streets? Does ambrosia yet flow in the veins of the one who sings this ode on an amphoric urn? If there is, and if it does, then it is likely to involve a wholly different order of interpretation, one where the reference to a jug of punch, ostensibly denoting a simple demijohn, is sure to denote an object of much greater significance than its literal denotation can convey, and one where the cryptic imports of its invocations are intended to descant a more figurative, metaphorical, and transcendental sense.

For the sake of shortening future references to the epigraph at the top of this subsection, let the acronym “JOP” be taken as equivalent to the phrase “jug of punch”, and let the italicized tag ''“TJOP”'' be taken as tantamount to the title ''“The Jug of Punch”''.

There are features of this style of abbreviation that need to be noted. Instead of letting the acronym “JOP” denote the phrase “jug of punch”, and rather than letting the italicized acronym ''“TJOP”'' denote the title ''“The Jug of Punch”'', I am merely asking for the reader to take part in forming an augmented scheme of interpretation, one that adds new signs to a formerly established sign relation, but does it in a way that does nothing of serious consequence to its underlying semantic properties. This involves the construction of a ''semiotic partition'' (SEP), along with its corresponding ''semiotic equivalence relation'' (SER), in which the associated set of ''semiotic equivalence classes'' (SECs) serves to stake out a number of parts. In the present illustration there are two sets of synonyms, constellating a pair of mutually exclusive classes of signs that denote their respective objects in parallel, as in Table 17.

<pre>
Table 17. Semiotic Partition Implied by the ACE of J
Object SEC
jug of punch {"jug of punch", "JOP"}
The Jug of Punch {"The Jug of Punch", "TJOP"}
</pre>

In each case, the abbreviated form and its expansion are set to connote each other all within a single level of signs, while both signs are set to denote their common object in a parallel fashion. This strategy for annexing compressed references to a sign relation can be referred to as an ''acronymically connotative extension'' (ACE) of that sign relation.

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | What more pleasure can a boy desire,
|-
| width="5%" |   || Than sitting down beside the fire?
|-
| colspan="2" | What more pleasure can a boy desire,
|-
| width="5%" |   || Than sitting down beside the fire?
|-
| colspan="2" | And in his hand a jug of punch,
|-
| width="5%" |   || And on his knee a tidy wench.
|}

Let <math>J\!</math> be a hypothetical sign relation that is adequate to interpret the ''The Jug of Punch''. Table 18 gives a bit of <math>J\!</math> that is called for in order to accomplish this interpretation. Table 19 shows an ACE of this bit. Table 20 gives a bit of this ACE that suffices to get the gist of it.

<pre>
Table 18. Bit of the Sign Relation J
Object Sign Interpretant
jug of punch "jug of punch" "jug of punch"
The Jug of Punch "The Jug of Punch" "The Jug of Punch"
</pre>

<pre>
Table 19. ACE of a Bit of J
Object Sign Interpretant
JOP "JOP" "JOP"
JOP "JOP" "jug of punch"
JOP "jug of punch" "JOP"
JOP "jug of punch" "jug of punch"
TJOP "TJOP" "TJOP"
TJOP "TJOP" "The Jug of Punch"
TJOP "The Jug of Punch" "TJOP"
TJOP "The Jug of Punch" "The Jug of Punch"
</pre>

<pre>
Table 20. Bit of an ACE of a Bit of J
Object Sign Interpretant
JOP "jug of punch" "JOP"
TJOP "The Jug of Punch" "TJOP"
</pre>

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | When I am dead and left in my mould,
|-
| width="5%" |   || At my head and feet place a flowing bowl,
|-
| colspan="2" | When I am dead and left in my mould,
|-
| width="5%" |   || At my head and feet place a flowing bowl,
|-
| colspan="2" | And every young man that passes by,
|-
| width="5%" |   || He can have a drink and remember I.
|}

Table 21

<pre>
Table 21. Budget of a Sign Relation: The Bottom of the Bit
Object Sign Interpretant
JOP "JOP" "jug of punch"
TJOP "TJOP" "The Jug of Punch"
... TJOP "Bein' ... I."
"Being on the "Being on the
twenty third of June, twenty third of June,
as I sat weaving as I sat weaving
all at my loom, all at my loom,
I heard a thrush, I heard a thrush,
singing on yon bush, singing on yon bush,
and the song she sang and the song she sang
was The Jug of Punch." was just this song."
"What more pleasure "No more pleasure
can a boy desire, can a boy desire,
than sitting down than sitting down
beside the fire, beside the fire,
and in his hand and in his hand
a jug of punch, a jug of punch,
and on his knee and on his knee
a tidy wench?" a tidy wench."
"When I am dead "When I am dead
and left in my mould, and left in my mould,
at my head and feet at my head and feet
place a flowing bowl, place a flowing bowl,
and every young man and every young man
that passes by, that passes by,
he can have a drink he can have a drink
and remember I." and remember me."
</pre>

Needless to say, acute enough hearers of this humble hymn or insightful enough interpreters of its lilting lyrics would no doubt find traces of Arthurian legend and clues to the Grail mythology woven all through its homespun text, but that is not my present task, to guess at its deepest drafts of meaning. Although there are features of this particular text that possess an incidental interest here, the reason for introducing it into the current context is more to illustrate the kinds of issues that are involved in a complex text, one with recursions and multiple levels of interpretation.

=====5.1.1.2. Analogical Recursion=====

With these preparations it is possible to return to the problems of analogical recursion, as illustrated by the poem ''The Lady of Shalott''. In the body of the poem, the italicized phrase ''“The Lady of Shalott”'' is triply ambiguous, being amenable to any one of the following readings:

# It can be the title of a person, italicized for emphasis.
# It can be the name of a vehicle, for instance, a boat or a ship that is named after a person.
# It can be the title of a text, for instance, a poem named after its principal subject or a story named after its chief character.

The first two readings are available on a literal interpretation and can be distinguished if the difference of emphasis is detected by the reader. The third reading is subtler, requiring both a figurative interpretation and a reason to suspect that some sort of subtext is possibly in force. How is the reader supposed to deal with this three headed equivocation? Is it a deliberate ambiguity on the author's part, one whose design is plotted with the aim of conveying a meaning? Does this question really matter, or does the syntactic structure of text still betray a form of intention, whether or not a conscious one?

If it is frequently necessary to distinguish the equally likely readings of equivocal signs, and if the design of the language in use is a topic open to discussion, then it is possible to bring in a requisite array of typographical conventions and a suitable set of type marking devices to indicate more explicitly the types of objects that are being denoted or the senses of signs that are being intended. But this kind of strategy only puts off the day when a capacity for intelligent interpretation is called on to resolve the ambiguities and the uncertainties that remain to all orders of finite signs. Keeping the inevitability of this outcome in mind, it is probably a good idea to spend a reasonable proportion of the meantime thinking of ways to build a capacity for flexible interpretation into a language from its very conception, or at least to leave room for its growth, and thus to facilitate an aptitude for interpretation under conditions of uncertainty throughout the entire course of development of a sign using capacity.

An intentional ambiguity in the reference of a sign is a primitive way of suggesting that there is an aspect of analogy or equality among the objects denoted, in other words, that there is a respect in which they are similar or a feature they have in common. In short, equivocation is akin to equation, becoming more pertinent the more persistent it is, and ambiguities that are systematic enough can amount to valid abstractions.

In the present case, one can observe the possibility that the author is suggesting the following analogies:

<ol style="list-style-type:decimal">

<li><p>One analogy says that authoring a text is like piloting a vehicle. This can be written in either one of two ways.</p></li>

<ol style="list-style-type:lower-alpha">

<li><p>Poet / Poem = Pilot / Boat.</p></li>
<li><p>Poet / Pilot = Poem / Boat.</p></li>
<li><p>Pilot / Poet = Boat / Poem.</p></li>

</ol>

<li>…</li>

</ol>

The arrangements of SECs, SEPs, SEQs, and SERs have to do with the analogies that can be discovered and the equalities that can be created among signs, but a hint of the relevant similarities can be found in the ''categorical analogies'' (CANs) or the ''categorical equations'' (CEQs) that it is frequently possible to recognize among general terms, namely, the class names that apply to the corresponding categories of objects.

Tables 22 and 23 show two ways of expressing these general kinds of relationship, as they apply to the present example.

<pre>
Table 22. A Categorical Analogy
Pilot / Poet
=
Boat / Poem
</pre>

<pre>
Table 23. A Categorical Equation
Pilot = Poet
<=>
Boat = Poem
</pre>

Thus, a rough outline of the ARK that transports ''The Lady of Shalott'' from a PORT or a QORT to an ORT is provided by the following example of a "categorical equation" (CEQ).

: Pilot = Poet <=> Boat = Poem.

This means that the reader can get a clue as to how the author relates to his text by reading, in a metaphorical way, the statements as to how the pilot or the passenger (the Lady of Shalott) relates to her vehicle (''The Lady of Shalott''). What one sees illustrated here is a particular form of literary device, one that I refer to as ''analogical recursion''. Given the intricacy of this form, it is probably useful to analyze its workings into several steps.

For the sake of shortening future references to the epitext at the top of the last subsection, the sequence of epigraphs that lace its prose discussion, let the acronym “TLOS” stand for “The Lady of Shalott”, the unofficial title of a legendary person, and let the italicized acronym ''“TLOS”'' be taken in token for ''“The Lady of Shalott”'', the title of a poem.

<pre>
Table 24. Semiotic Partition Implied by the ACE of L
Object SEC
The Lady of Shalott {"The Lady of Shalott", "TLOS"}
The Lady of Shalott {"The Lady of Shalott", "TLOS"}
</pre>

<pre>
Table 25. Bit of an ACE of a Bit of L
Object Sign Interpretant
TLOS "The Lady of Shalott" "TLOS"
TLOS "The Lady of Shalott" "TLOS"
</pre>

====5.1.2. Conscious Reflection====

In this Section I examine how the intellectual process of reflection is expressed in reflective writing. Not so much as materials for analysis, since the power of analysis present in this work remains in a primitive state, but more to provide a constant reminder of what a reflective text is like, I am taking my epitext from the lifelong work of a single author, who made the aim of reflection an integral part of a whole life's work.

The thesis that is developing here is this: Language users have an innate knowledge of the ''situation of communication'' (SOC), a knowledge that is built into the language itself and gives its users an inkling of the social setting of communication that the language is meant to serve. Such a knowledge is tantamount to a science of communication that its users develop from its initial state by dint or by virtue of using it. Language users possess an intuitive, if imperfect, appreciation of the forms that are inherent in the social ''task of communication'' (TOC), and they exercise an implicit, if incipient, understanding of the practical roles that are constrained by the social ''hold of communication'' (HOC). Although every language user actualizes these roles with more or less competence and participates in the requisite forms of relationship with more or less cognizance, it is poets, playwrights, programmers, policy analysts, and other sorts of reflective writers that are especially charged to articulate these forms in a relatively explicit fashion.

The fact that reflective writers are driven to comment on the SOC itself, even if it is not always desirable or feasible to do so straightforwardly, and the fact that perceptive writers are able to find symbols of the SOC in the most inobvious places, reflected in the most refractory settings, and even when its likenesses are cast into the most unlikely images — these are two of the factors that combine to give creative writing its notably recursive and often cryptic character.

A reflective writer converts a ''situation of communication'' (SOC), a type of object, into a ''communication of situation'' (COS), a type of sign, and links the succession of reflective signs into the ongoing reflective text. But what are the forces that force a text, developing freely in a medium of communication, furnishing the vehicle of an observation, and bearing the impression of an object that occasions it, to double back on its writer and itself, to turn back through the medium of communication, all to form a sign for itself and to make a name for its author?

The relevance of reflective writing to the inquiry into inquiry can be seen in the following way. Let one examine a reflective text, a sample from the work of a suitably reflective writer, and one often discovers, besides the interpretation that bears on the obvious subject and serves to carry the ostensible theme, that there is coded or woven through the covering text a comment on the SOC itself, that is, a reflection on the writer, the reader, and the text itself. This reflexive interpretation reveals the writer's impressions about the process of writing, the very process that led to the text as its end result.

By tracing the analogies that exist between reflective writing and the inquiry into inquiry it is possible to gain a measure of insight into the character of the latter task. It is not a strange circumstance for the life and work of a writer to be represented again in that work, indeed, to be critically reflected there. And there is no question that a text can be used by a writer to talk about itself and its author, in a way that conceivably makes sense of them both — the question is whether what a reflexive text says can be interpreted in the same way as a text about external objects, or has to be taken with a distinct grain of salt.

Reflective writing arises from reflection on life and conduct and issues in a description of what goes on in the scene surrounding the reflective writer's ''point of view'' (POV). Of course, among the forms of conduct that are subject to inspection, a piece of reflective writing can also reflect on the process of writing itself, detailing the conditions that affect its intentions and its outcome, and thus taking on a “reflexive” character, though it is customary to express these narrower reflections in any number of less direct manners. With regard to the scene about a POV, a piece of reflective writing can take any stance from admiring, to amused, to bemused, to critical, to simply trying to puzzle out a fraction of what is going on. With respect to the process of writing and the development of a writer, a piece of reflective writing and what it articulates can be a crucial part of changing or preserving a POV. From this description, it ought to be clear that reflective writing is a naturally occurring species of inquiry into inquiry. That is to say, in analyzing the varieties of syntactic, semantic, and pragmatic phenomena that occur in reflective writing, one is performing a task that parallels the inquiry into inquiry.

Notions of recursion, informally taken, arise in this discussion for several reasons. First, there is the appearance of self application that is involved in an inquiry into inquiry, in the idea that an instrumental activity called “inquiry” can have application to an objective argument called “inquiry” and yield a meaningful result. Second, there is the appearance of self reference that is involved in an inquiry into inquiry, in the fact that a textual record of a self described inquiry needs to refer to itself as falling under the general topic of inquiry.

Strictly speaking, the themes of self application and self reference are less properly described as ''recursive'' than ''reflective'' or ''reflexive'', but it is easy to see how these issues arise in the process of carrying out a genuinely recursive project in an effectively pragmatic context.

In order to do this, I need to give a rough description of these two ideas, that of a recursive project and that of a pragmatic context.

# In a recursive project, one attempts to clarify a complex concept in terms of simpler concepts. For instance, an important special case occurs when one tries to analyze a complex process in terms of simpler processes. A recursive project recurs upon a type of situation where the ''same'' concept is applied to simpler objects, in particular, where the same process, procedure, or function is applied to simpler arguments, proceeding to increasingly simpler arguments until the simplest arguments are reached. A recursive project is sound if there is a bound that can be wound around it, and it redounds to good effect if there is a ground that can be found to found it.
# In a pragmatic context, the canonical way to clarify any concept is to give it an effective representation or an operational definition, that is, to detail the effects that the object of the concept is conceived to have when it is applied to the objects available in a specifiable variety of practical situations. An interpretive agent that follows this pragmatic prescription for clarifying concepts, persisting at it long enough and pursuing it through an adequate array of applications, can convert each concept analyzed into its corresponding "active formula". This is a form of expression that is logically equivalent, or as nearly as possible, to the intended concept, but suitable for immediate application to the contemplated domain of objects.

Now, consider the concept of inquiry as a candidate for clarification. In the case of a concept like inquiry, the object of the concept in question is an activity that applies to the broadest conceivable variety of objects, one of these arguments being the topic of inquiry itself. As a result, if one approaches a definition of inquiry by way of the pragmatic prescription for clarifying concepts, one quickly discovers that an important ingredient in the active formula for inquiry is a component that characterizes the concept ''inquiry'' in terms of its action on itself.

What does the object denoted by “inquiry” have to do with respect to the object denoted by “inquiry”, in the first place, to qualify as a genuine inquiry, in the end, to succeed as an inquiry into inquiry? Evidently, something about the sign, the object, or what transpires between the sign and the object is conducive to attaining a better description of inquiry than that given by the mere name “inquiry”. The word “inquiry” and the symbol <math>y\!</math> are like a host of difficult signs, starting with “I” and “you”, where knowing the sign does not mean knowing the object perfectly, although it can lead to a knowledge of it. Simply knowing the word “I” and being able to use it adequately does not mean that I know myself perfectly, or that I can articulate my own nature. At best, these signs can serve to indicate the direction of the object pointed out or help to remind an agent of the action that is called for to be carried out.

Can the senses of a sign be so confused, or the sense of an interpreter be so confounded by it, that between the two it is difficult to know if the sign refers to something inside the self that is in the world or to something in the world that is outside the self?

Given the description of a question as an unclear sign, it might be thought that the sole purpose of an inquiry is to clarify a question until it acquires the status of an answer, and thus to operate wholly within the syntactic realm of signs and ideas. But this would ignore many cases of experimental inquiry and active problem solving, natural to include among inquiries in general, that involve the manipulation of external objects and the alteration of objective states as they occur in the objective world. With these things in mind, it is best to define an inquiry as a ''sign relation transformation'' (SRT), in other words, and a bit more pronounceably, as a ''transformation of sign relations'' (TOSR). This is conceived to be an operation that acts on whole sign relations, changing one into another, typically by acting in specified ways on the ''elementary sign relations''(ESRs), or on the ordered triples <math>(o, s, i).\!</math> An inquiry, regarded as a TOSR, can be treated as a generalized form of ''sign process''. Whereas a sign process is restricted to acting within a single sign relation and can only change signs into their interpretants, a TOSR can subject elements of an object domain to experimental actions and induce objective states to undergo a variety of intentional changes.

From an abstract relational point of view, it is not too far from grasping the concept of a sign relation to seeing that transformations, operations, and other sorts of relations that are possible to define on sign relations are bound to become of significant interest. But the present concern is to decide whether the identification of inquiries with TOSRs constitutes a good definition in practice. A good definition in practice, aside from capturing the necessary and sufficient properties of its subject, is one that facilitates the generation of fruitful, incisive, material, pertinent, and relevant ideas about it. To some extent this depends on the context of practices and the specific purposes that a particular interpreter has in mind. Still, some definitions are more generally useful than others. Accordingly, the task I need to take up next is to examine the abstract concept of a TOSR with regard to its utility in practice, in other words, to determine its practical bearing on a concrete conception of inquiry, as it is topically understood.

If the essence of inquiry, or any aspect of what an inquiry can be, is captured by the concept of a TOSR, then a lot can be learned about the nature of inquiry by studying the manifest varieties and the internal structures of TOSRs.

Expressing this in terms of a prospective calculus, the present inquiry, <math>y_0,\!</math> constituted as an inquiry into inquiry, <math>y \cdot y,\!</math> ...

As the clarification of a concept is pursued to the limit, it approaches the status of a definition. And so one finds oneself contemplating a definition of inquiry that defines it at least partially in terms of itself. But an attempt to define a concept in terms of itself is ordinarily considered to be a bad thing, leading to the sort of circular definition that vitiates the utility of the whole effort toward clarity.

In summary, because the subject of inquiry is something that one can reasonably be in question about, and because the topic of inquiry is something that one can sensibly inquire into, the chances that one can make sense of an inquiry into inquiry is not merely an interesting and diverting possibility but a necessary part of the meaning of inquiry.

Definitions are limiting cases of clarifications, since a process of clarification pursued far enough approaches a formulation of a concept that is tantamount to its definition.

Before an inquiry can proceed very far, it needs to develop a map or a plan of the territory that the agent of inquiry intends to investigate. This task involves the drawing of distinctions, the finding of natural differences and the making of useful separations, among the objects of inquiry.

Let me call attention to a compound form of existence, the kind that is composed of a sign and the interpreter that authors it, and describe it more briefly as a ''sign and issuer'' (SAI) or a ''text and writer'' (TAW). As long as one moves through a casual context it is convenient to carry along these portmanteau words, precisely because their two components are confounded so consistently in informal speech, where it is hardly polite to keep on objecting to their ambiguities and anthropomorphisms. When I say that a SAI does this or that, it is up to the good sense of a charitable interpreter to decide whether this or that is something that a sign or rather its issuer is supposed to be able to do. In these terms, a SAI that speaks of and to itself and addresses its own composition or a TAW that talks about itself in either sense are not likely to have the same interest for others as they do in themselves.

It is ordinarily thought to be a good thing for a SAI or a TAW to be able to reflect on itself, but one whose subject is solely oneself is not ordinarily thought to be of interest to others.

In this work, I am interested in SAIs and TAWs that survive the onset of recursion while avoiding the snares of sheer self reference, that pose patterns of self reference but only in the service of a greater subject, and that slip the snarly bonds of narcissism frequently enough to say something significant about something else.

It is a form of narcissism to think that others are necessarily as interested in every detail of one's existence as one is oneself. But narcissism is an unnatural condition that has to be distinguished from one's more commonly understandable interest in oneself. In its extreme forms, a full blown narcissism is not the natural flourishing of a healthy self interest but the outgrowth of deep and typically early disturbances in the systematic structure of the self.

In order to understand how a sign functions as a sign it is necessary to understand the interpreter for whom it actually functions as a sign. The ways that a sign denotes its objects and connotes its interpretants say a lot about the interpreter for whom it denotes its objects and for whom it connotes its interpretants, where the antecedents of all these occurrences of “its” can be either the sign or its interpreter. To the extent that all knowledge is expressed in signs, to know anything at all is to know an aspect of oneself, however unwittingly. In this way, one can arrive at the epigrammatic formulas that “all knowledge is self knowledge” and that “every inquiry is an inquiry into inquiry”.

A ''symbol'' is a type of sign whose relation to its object is constituted solely by the fact that an interpreter employs it to denote that object, in other words, that an interpretant connects it with that object in an ''elementary sign relation'', or an ordered triple of the form <math>o, s, i.\!</math> This means that the nature and the character of an interpreter can be studied especially well as reflected in the symbols that it employs.

Unlike icons and indices, which have rationales for their denotations in the properties and instances, respectively, which are common to objects and their signs, ...

: term/premiss/argument: symbols with internal or instructive hints?

An ''argument'' is a type of symbol that incorporates among its syntactic provisions an independent indication of the method that is intended for its interpretation, that is, it embodies a series of hints about the ways and the means that its issuer intends its prospective interpreter to use in order to achieve its interpretant, in short, to reach its conclusion.

=====5.1.2.1. The Signal Moment=====

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | One night as I did wander,
|-
| width="5%" |   || When corn begins to shoot,
|-
| colspan="2" | I sat me down to ponder
|-
| width="5%" |   || Upon on auld tree root.
|-
| colspan="2" | Auld Ayr ran by before me,
|-
| width="5%" |   || And bicker'd to the seas;
|-
| colspan="2" | A cushat crooded o'er me,
|-
| width="5%" |   || That echoed through the trees.
|-
| colspan="2" align="right" | — Robert Burns, ''One Night As I Did Wander'', [CPW, 48]
|}

There is a thought that forms the theme of the present inquiry, indeed, as a chorus to a lyric are its evocations to the text that records this inquiry, and I find myself returning to its expressions on a constantly recurring basis, however much I strive to introduce variations for the sake of developing its implications and reflecting on its meanings from a fresh angle. So let me give the current rendition:

The present inquiry, <math>y_0,\!</math> portraying itself as an inquiry into inquiry, <math>y \cdot y,\!</math> proceeds on the premiss that a generic inquiry, <math>y,\!</math> can generally inquire into a generic inquiry, <math>y,\!</math> thereby achieving a settled result, one that awaits a mere determination to be signified by the name <math>{}^{\backprime\backprime} y \cdot y {}^{\prime\prime}.\!</math> Thus the present inquiry, acting on the pretext of a ''formal posability'', that is, a poetic license, a verbal permission, or a written suggestion, being motivated and justified by no more authority than these connote, is led to define itself in terms that appose its own term to its own term, and so it is led to take on a recursive, a reflective, or a reflexive cast.

The terms of this description need to be inquired into, and their implications pursued in greater detail.

The present inquiry, <math>y_0,\!</math> portraying itself as an inquiry into inquiry, <math>y \cdot y,\!</math> proceeds on the premiss that a generic inquiry, <math>y,\!</math> can generally inquire into a generic inquiry, <math>y,\!</math> and thereby achieve a settled result, and that this result awaits nothing other than its determination by the present inquirer to confer an objective significance on the name <math>{}^{\backprime\backprime} y \cdot y {}^{\prime\prime}.\!</math> All of this is summed up in the formula: <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}.\!</math>

Thus the present inquiry, acting on the pretext of a ''formal posability'', namely, the circumstance that the rules of a prospective formal grammar allow one to write the expression <math>{}^{\backprime\backprime} y \cdot y {}^{\prime\prime}\!</math> and to inquire after its meaning, is led to define itself in terms that apply to its own case as argument, since the present inquiry, <math>y_0,\!</math> must be an example of whatever genus, <math>Y,\!</math>, that a generic inquiry, <math>y,\!</math> is selected to represent. As a consequence, the present inquiry is forced to pursue the development of its own case in terms that appose its own actions to its own motives, and so is led to take on a recursive, a reflective, or a reflexive cast.

=====5.1.2.2. The Symbolic Object=====

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | I dream'd I lay where flowers were springing
|-
| width="5%" |   || Gaily in the sunny beam,
|-
| colspan="2" | List'ning to the wild birds singing,
|-
| width="5%" |   || By a falling crystal stream;
|-
| colspan="2" | Straight the sky grew black and daring,
|-
| width="5%" |   || Thro the woods the whirlwinds rave,
|-
| colspan="2" | Trees with aged arms were warring
|-
| width="5%" |   || O'er the swelling, drumlie wave.
|-
| colspan="2" align="right" | — Robert Burns, ''I Dream'd I Lay'', [CPW, 45]
|}

To paraphrase, the present inquiry acts on the pretense that an inquiry can inquire into other inquiries, perhaps even those that are presently ongoing, and even inquire into itself, in sum, being entitled to inquire into the full genus of inquiry, <math>Y,\!</math> a class that includes <math>y_0\!</math> as a member. But these representations, under cross examination, lead to a number of unanswered questions, like: Just what is a “generic inquiry”, anyway? Even more critically, their close and repeated examination leads to a host of “unquestioned answers”, answers already accepted as adequate, but whose appearances as answers need to be questioned again.

The ''formal posability'' of a self application, for example, as expressed by the term <math>{}^{\backprime\backprime} y \cdot y {}^{\prime\prime},\!</math> especially when the formal calculus that is called on to make sense of these applications is still merely prospective and still highly speculative, ought to arouse a lot of suspicion from the purely formal point of view. Indeed, I cannot justify this way of proceeding, beginning in the middle of things and without stopping to establish a well defined formal system ahead of time, except to say that something very like it is unavoidable in a large number of natural circumstances, and so one ought to find a way of getting used to it. A way of getting used to the natural situation of inquiry is one of the things that the present inquiry hopes to find.

If it appears that this allows the present inquiry an unlimited scope or an excessive freedom, it has to be remembered that a ''formal posability'' is barely enough of a formal subsistence to begin an inquiry, but rarely enough to finish it. It can be invaluable as the provisional ''grubstake'' for a prospecting expedition, supplying the initial overhead it takes to ''prime the pump'' of subsequent exploration, but it is not sufficient to continue very far with an investigation. In essence, it is nothing more substantial than a grammatical allowance or a syntactic hypothesis, in effect, a poetic license, a verbal permission, or a written suggestion. Taking all of these cautions into account, it leaves the present inquiry motivated and justified by no more authority than their titles connote, and it obliges the precocity of what is written to be atoned for with all the critical benevolence of afterthought that can be mustered after the fact, to wit, through the diligent application of that turn of mind that allows one to write first and only later to think on the meaning.

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | Such was my life's deceitful morning,
|-
| width="5%" |   || Such the pleasures I enjoy'd!
|-
| colspan="2" | But lang or noon, loud tempests storming,
|-
| width="5%" |   || A' my flowery bliss destroy'd.
|-
| colspan="2" | Tho fickle Fortune has deceiv'd me
|-
| width="5%" |   || (She promis'd fair, and perform'd but ill),
|-
| colspan="2" | Of monie a joy and hope bereav'd me,
|-
| width="5%" |   || I bear a heart shall support me still.
|-
| colspan="2" align="right" | — Robert Burns, ''I Dream'd I Lay'', [CPW, 45]
|}

The present inquiry acts on the purely formal suggestion that a generic inquiry can inquire into other inquiries, perhaps even those that remain ongoing, moreover, that a particular inquiry can even inquire into itself. Interpolating the appropriate symbols, the present inquiry, referring to itself as <math>{}^{\backprime\backprime} y_0 {}^{\prime\prime},\!</math> acts on the instance of a purely formal possibility, one that it expresses as a premiss in the formula <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime},\!</math> intending this to be interpreted to the effect that an inquiry can inquire into a class of inquiries that includes itself as a member, and this is a hypothesis that is based on little more authority than the fact of its expression a prospective formal language, in other words, one whose interpretation is still a largely prospective matter.

Stepping back and reflecting on the situation, one needs to ask how in general and how in particular does one fall so blithely into these forms and into these manners of representation. Once that process is better understood then it becomes possible to evaluate in a fairer way whether this direction of fall is tantamount to a happy accident of the natural intuition or whether it constellates a disastrous catastrophe that needs to be remedied through the application of a severer style of reasoning. Generally speaking, the point at which intellectual developments like these begin to take on an automatic character is when the intention is formed of devising a formal calculus, in the present case, a prospective calculus of ''applications'' or ''appositions'' of the form <math>f \cdot g,\!</math> the terms of which are intended to be capable of referring to processes potentially as complex as inquiries. The project of an ''appositional calculus'' (AC) is what formalizes the intuitive possibility of an inquiry into inquiry and continues to suggest the formal possibility that any inquiry can be applied to itself, at least, any inquiry that can be symbolized in this calculus.

But not every form of words that can be formed within the permissions of a formal language does in fact point to a form of objective reality. Whether an inquiry into inquiry is a real possibility, how its possibility is to be actualized if it is indeed real in fact, and why it is necessary for an individual species of agents to bother with the actualization of this possibility — these are just some of the questions that demand to be addressed at this point, no matter how gingerly and how tentatively it is presently conceivable to respond to them, and they are just a few of the issues the distribution of whose partial solutions are found to occupy the greater body of this work.

=====5.1.2.3. The Endeavor to Communicate=====

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | When o'er the hill the eastern star
|-
| width="5%" |   || Tells bughtin time is near, my jo,
|-
| colspan="2" | And owsen frae the furrow'd field
|-
| width="5%" |   || Return sae dowf and weary, O,
|-
| colspan="2" | Down by the burn, where scented birks
|-
| width="5%" |   || Wi dew are hangin clear, my jo,
|-
| colspan="2" | I'll meet thee on the lea-rig,
|-
| width="5%" |   || My ain kind dearie, O!
|-
| colspan="2" align="right" | — Robert Burns, ''The Lea-Rig'', [CPW, 474]
|}

An agent involved in an ''effort to communicate'', no matter how various the signs and the media that make its conveyance conceivable, and no matter how articulately the character of its endeavor is styled, whether it is pointed and straightforward, or allusive and recursive, whether it is elliptic, hyperbolic, parabolic, or otherwise conically sectioned, or whether it is much less smoothly sliced into its initial approximations and final truncations, there are only so many ways that a ''finitely informed creature'' can find to figure out what meaning the world has and to formulate what sense a life's work can add to it.

The present situation, as far as it goes, is a suitable subject for being investigated along the lines of the pragmatic theory of sign relations.

Since <math>{}^{\backprime\backprime} x {}^{\prime\prime}\!</math> is a sign, it has the potential to denote an object <math>x,\!</math> if and when there is determined to be a signified object, and one with a power to impress itself on the mind of the operative interpreter of that sign. Likewise, since <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}\!</math> is a sign, it has the potential to denote an object, one that syntactic compunctions stop me from saying is <math>y_0 = y \cdot y,\!</math> that is, if I want to avoid a definite risk of failing to be understood. But what is this object, if it exists? At any rate, what sort of object is the receiver of the sign thereby entitled to expect it to be, whether or not the object that it foreshadows ever does come to be actualized?

In order to have a variety of more convenient names for referring to the object potentially denoted by the sign <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime},\!</math> I refer to the expression <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}\!</math> as ''“The Initial Equation”'', or as ''“TIE”'', for short. Although it is not strictly necessary for such a small piece of text as <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime},\!</math> I here obey the rule that the titles of texts are italicized. Furthermore, the object, situation, or state that satisfies ''TIE'', to the effect that <math>y_0 = y \cdot y,\!</math> and is therefore potentially denoted by ''TIE'', can also be referred to as “the intended state”, or as “TIS”, for short.

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | At midnight hour in mirkest glen,
|-
| width="5%" |   || I'd rove, and ne'er be eerie, O,
|-
| colspan="2" | If thro that glen I gaed to thee,
|-
| width="5%" |   || My ain kind dearie, O!
|-
| colspan="2" | Altho, the night were ne'er sae wild,
|-
| width="5%" |   || And I were ne'er sae weary, O,
|-
| colspan="2" | I'll meet thee on the lea-rig,
|-
| width="5%" |   || My ain kind dearie, O!
|-
| colspan="2" align="right" | — Robert Burns, ''The Lea-Rig'', [CPW, 474]
|}

For the sake of shortening future references to the chief object of the present inquiry and the initial sign of its potential existence, let the acronym “TIS” be equiferent to the phrase “the intended state”, and let the italic tag ''“TIE”'' be equiferent to the title ''“The Initial Equation”''. Further, let the connotations be so arranged that “TIS” is semiotically equivalent to “the intended state” and ''“TIE”'' is semiotically equivalent to ''“The Initial Equation”''. It is important to note that a set of signs can be equiferent among themselves in the wholly vacuous sense that all of them have no objective reference, and, strictly speaking of what they denote, that all of them refer to nothing at all, whereas a set of signs that are equivalent in the properly semiotic sense still have each other as their connotations.

There is a feature of this style of abbreviation to which it is useful to call attention. Rather than letting the acronym “TIS” strictly denote the phrase “the intended state” and instead of letting the tag ''“TIE”'' strictly denote the title ''“The Initial Equation”'', I am merely asking the reader to arrange in behalf of the interpretation a ''semiotic partition'' (SEP), along with its corresponding ''semiotic equivalence relation'' (SER), in which a particular pair of ''semiotic equivalence classes'' (SECs) serve to stake out a couple of parts, that is, to represent mutually exclusive classes of signs that denote their respective objects in parallel. This situation is depicted in Table 26.

<pre>
Table 26. Semiotic Partition Implied by the ACE of Q
Object SEC
the intended state {"the intended state", "TIS"}
The Initial Equation {"The Initial Equation", "TIE"}
</pre>

In each case, the abbreviated form and its expansion are set to connote each other all within a single level of signs, while both signs are set to denote their common object in a parallel fashion. This strategy for annexing compressed references to a sign relation can be referred to as an ''acronymically connotative extension'' (ACE) of that sign relation.

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | The hunter lo'es the morning sun,
|-
| width="5%" |   || To rouse the mountain deer, my jo,
|-
| colspan="2" | At noon the fisher takes the glen
|-
| width="5%" |   || Adown the burn to steer, my jo:
|-
| colspan="2" | Gie me the hour o gloamin grey
|-
| width="5%" |   || It maks my heart sae cheery, O,
|-
| colspan="2" | To meet thee on the lea-rig,
|-
| width="5%" |   || My ain kind dearie, O!
|-
| colspan="2" align="right" | — Robert Burns, ''The Lea-Rig'', [CPW, 474]
|}

=====5.1.2.4. The Medium of Communication=====

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| The gloomy night is gath'ring fast,
|-
| Loud roars the wild inconstant blast;
|-
| Yon murky cloud is filled with rain,
|-
| I see it driving o'er the plain;
|-
| The hunter now has left the moor,
|-
| The scatt'red coveys meet secure;
|-
| While here I wander, prest with care,
|-
| Along the lonely banks of Ayr.
|-
| align="right" | — Robert Burns, ''The Gloomy Night is Gath'ring Fast'', [CPW, 250]
|}

Before I can advance this discussion to a higher level of reflection on my own and other texts, in other words, to augment its participation in syntactic and textual processes beyond the level of rote recitation and routine replication of whatever text comes to mind, I need to introduce a minimal set of syntactic devices and textual mechanisms for adverting to and reflecting on syntactic entities and textual objects, in other words, for generating and interpreting, or else recognizing and elaborating, a level of references to objects that are themselves composed of signs and that therefore have the characters of complex signs in their own rights.

In general, signs that denote signs are called ''higher order'' (HO) signs, leaving the signs denoted to be referred to as ''lower order'' (LO) signs. These form the subject of detailed discussions later on in this work, but the critical need for now is merely to make available an informal set of plausible devices for availing the discussion of names for pieces of text. Thus, the tools that are required can be sufficiently well illustrated in their immediate applications to the present materials.

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| The Autumn mourns her rip'ning corn
|-
| By early Winter's ravage torn;
|-
| Across her placid, azure sky,
|-
| She sees the scowling tempest fly;
|-
| Chill runs my blood to hear it rave;
|-
| I think upon the stormy wave,
|-
| Where many a danger I must dare,
|-
| Far from the bonie banks of Ayr.
|-
| align="right" | — Robert Burns, ''The Gloomy Night is Gath'ring Fast'', [CPW, 250]
|}

Depending on whether it is possible to adapt, appropriate, and otherwise make use of what HO signs are already extant in a field of discussion or whether it is necessary to create, invent, or otherwise make up further HO signs to denote the signs and the texts that one notices in an area, one finds that the sorts of syntactic devices, textual mechanisms, and reflective operators that one needs are divided into two broad camps:

# There are the anaclitic, ancillary, or auxiliary devices that an agent uses to imp out each HO connotative plane by extrapolating its indirections in novel directions, to allude in a connotative fashion to the current signs of signs and the established titles of texts, to sharpen up the reflective references already extant, and to take full advantage of the ancient orders of associations and the antecedent layers of citations that are already in place.
# There are the creative, generative, or productive devices that an agent uses to eke out each HO denotative plane in the first place, to adduce the initial signs in that order, to create new HO signs, to refer in a denotative fashion to what thereby becomes an order of comparatively LO signs, and to issue HO citations of LO texts.

The connotative mechanism, relying on prior quotations and established titles, ...

'''Acronyms.'''

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| 'Tis not the surging billows' roar,
|-
| 'Tis not that fatal, deadly shore;
|-
| Tho death in ev'ry shape appear,
|-
| The wretched have no more to fear;
|-
| But round my heart the ties are bound,
|-
| That heart transpierc'd with many a wound;
|-
| These bleed afresh, those ties I tear,
|-
| To leave the bonie banks of Ayr.
|-
| align="right" | — Robert Burns, ''The Gloomy Night is Gath'ring Fast'', [CPW, 250]
|}

The denotative mechanism, ...

These are devices whose function it is to operate on signs, including all sorts of characters, expressions, phrases, and texts, and whose result it is to generate signs that refer to their respective arguments as objects.

'''Quotation marks.''' Ordinary quotation marks (“ ”) can be used in the customary ways to create names for signs, concatenated signs, or pieces of text that they enclose. Unfortunately, for formal purposes, ordinary quotation marks have the disadvantage of being used for several other functions besides that of creating names for enclosed signs and texts. In particular, the same marks are frequently used for a motley crew of ''emphatic functions'' or ''monitory purposes'', that is, simply to call an extra measure of attention to the sign or the text enclosed, but without necessarily intending to interrupt its significance or to interfere with the corresponding process of denotation.

'''Arch quotations.''' An alternative form of quotation is provided through the employment of ''raised angle brackets'' (<sup><</sup> <sup>></sup>), also called ''arches'' or ''supercilia''. These marks are reserved to the sole purpose of creating signs for signs and generating names for pieces of text, in keeping with the ''nominal intention'' and the ''normal use'' of quotation marks.

'''Titles and headings.''' An arbitrary title for a syntactic object or a textual segment is created simply by designating anything whatsoever to a service in that role. Whatever it is before being dubbed as the title of the material in question, it becomes a pointer to its appointed object simply by virtue of being so dubbed, if nothing else, at least as regarded by a single interpreter that is duly appointed to appoint things so, if only for the sake of a purely personal recognizance.

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| Farewell, old Coila's hills and dales,
|-
| Her heathy moors and winding vales;
|-
| The scenes where wretched Fancy roves,
|-
| Pursuing past unhappy loves!
|-
| Farewell my friends! farewell my foes!
|-
| My peace with these, my love with those —
|-
| The bursting tears my heart declare,
|-
| Farewell, my bonie banks of Ayr!
|-
| align="right" | — Robert Burns, ''The Gloomy Night is Gath'ring Fast,'', [CPW, 251]
|}

The highest order of generality among titles is not absolutely necessary in the present context. More commonly, a title is a pre-arranged sign, a pre-established mark, a prefixed epithet, or a pre-ordained piece of text that gets re used, perhaps subject to a conventional modification or a special inflection, to serve as a sign or a name for what is customarily a disjoint sign or a distinct piece of text. Under typical circumstances, although not universal, the syntactic entity or the textual object to which a title refers is a much longer text, and thus one that occasions the practical need among its interpreters of having a briefer alias or a compressed designation for it. In short, a title is intended to serve a purpose that is similar to one of the roles of ordinary quotation, but subject to orders of pragmatic constraints that quotation marks, when literally taken and expressly used, are clearly not able to satisfy. Putting aside for the time being the issues that are raised by this general discussion, I revert to the ordinary use of quoted expressions and italicized phrases as the titles of texts.

=====5.1.2.5. The Ark of Types : The Order of Things to Come=====

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | Now westlin winds and slaught'ring guns
|-
| width="5%" |   || Bring Autumn's pleasant weather;
|-
| colspan="2" | The moorcock springs on whirring wings
|-
| width="5%" |   || Amang the blooming heather:
|-
| colspan="2" | Now waving grain, wide o'er the plain,
|-
| width="5%" |   || Delights the weary farmer;
|-
| colspan="2" | And the moon shines bright, as I rove by night,
|-
| width="5%" |   || To muse upon my charmer.
|-
| colspan="2" align="right" | — Robert Burns, ''Now Westlin Winds'', [CPW, 44]
|}

<pre>
The present situation, as far as it goes, is a suitable subject for being investigated along the lines of the pragmatic theory of sign relations. The state of the resulting examination, as it stands at the current stage of analysis, is summarized in Table 27, indicating little more than this hypothetical circumstance: That a couple of terms of a formal language, a prospective calculus of "applications" or "appositions" of the form f.g, are intended to be identified in all of their current objective references. Thus, the terms "y0" and "y.y", formed in accord with the still inchoate and yet developing grammar of the intended "appositional calculus" (AC), are set to denote the very same object or objects, all the while that the precise nature of what these signs actually denote is still up for grabs, and in spite of the circumstance that the bare consistency of its logical possibility remains unknown, for all the plausibility of the posability.

Recalling that a "bit" of a sign relation is any subset of its extension, that is, an arbitrary selection of its ordered triples, Table 27 presents a bit of a sign relation that is needed to interpret "The Initial Equation", also known as "TIE".

Table 27. Bit of a Sign Relation for TIE
Object Sign Interpretant
Y "Y" "Y"
y "y" "y"
y0 "y0" "y0"
y0 "y0" "y.y"
y0 "y.y" "y0"
y0 "y.y" "y.y"
</pre>

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | The paitrick lo'es the fruitfu fells,
|-
| width="5%" |   || The plover lo'es the mountains;
|-
| colspan="2" | The woodcock haunts the lonely dells,
|-
| width="5%" |   || The soaring hern the fountains:
|-
| colspan="2" | Thro lofty groves the cushat roves,
|-
| width="5%" |   || The path o man to shun it;
|-
| colspan="2" | The hazel bush o'erhangs the thrush,
|-
| width="5%" |   || The spreading thorn the linnet.
|-
| colspan="2" align="right" | — Robert Burns, ''Now Westlin Winds'', [CPW, 44]
|}

<pre>
In order to refer to an object x, it is necessary to use a sign "x", or something equally good, an equiferent sign, to do so. In a similar vein, in order to refer to a sign "x", it is necessary to use a HO sign <"x">, or something equally good, an equiferent HO sign, to do so.

In referring to the signs "x" and "y0 = y.y", I am of course using a definite style of HO signs to do so, while the corresponding LO signs, the ones that are being denoted by these mechanisms, are homologous to the portions of text that appear within the bounds of these quotations. The chief exception to this rule, attaching a note of practical caution to its exercise that precludes its overly automatic use, is due to the problem already noted, that not every LO sign extracted from quotation is safe to use, semantically speaking, in every discursive context, grammatical environment, syntactic frame, or textual niche.

As long as I am referring to the signs "x" and "y0 = y.y", I can keep on using the HO signs that refer to them, all without having to employ the next layer of encapsulation in arch quotes. I am obligated to use the new order of arches only when I want to awake to, become aware of, and directly mention the order of signs that I find myself employing, in the present case, when I get a notion to critically reflect on and thus to make explicit reference to the HO signs <"x"> and <"y0 = y.y">. It is almost as if, in using an order of signs, that one takes off the wraps that one uses in order to mention them. This is generally true, but subject to exceptions at the boundary conditions, where there are no more lamina to strip away.

Table 28

The bit of a sign relation that is shown in Table 28 is an example of this type of arbit.

Table 28. Arbit of a Sign Relation for TIE
Object Sign Interpretant
y0 "y0" = "y.y"
</pre>

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | Thus ev'ry kind their pleasure find,
|-
| width="5%" |   || The savage and the tender;
|-
| colspan="2" | Some social join, and leagues combine,
|-
| width="5%" |   || Some solitary wander:
|-
| colspan="2" | Avaunt, away, the cruel sway!
|-
| width="5%" |   || Tyrannic man's dominion!
|-
| colspan="2" | The sportsman's joy, the murd'ring cry,
|-
| width="5%" |   || The flutt'ring, gory pinion!
|-
| colspan="2" align="right" | — Robert Burns, ''Now Westlin Winds'', [CPW, 44]
|}

<pre>
Table 29 shows a variety of notations that are available for the first two orders of signs above "x".

Table 29. Simple Signs & Their Higher Order Signs
Object Sign Interpretant

x "x" = <x>

"x" <"x"> = <<x>>

"x" <"x"> = "<x>"
</pre>

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | But, Peggy dear, the ev'ning's clear,
|-
| width="5%" |   || Thick flies the skimming swallow;
|-
| colspan="2" | The sky is blue, the fields in view,
|-
| width="5%" |   || All fading green and yellow:
|-
| colspan="2" | Come let us stray our gladsome way,
|-
| width="5%" |   || And view the charms of Nature;
|-
| colspan="2" | The rustling corn, the fruited thorn,
|-
| width="5%" |   || And ilka happy creature.
|-
| colspan="2" align="right" | — Robert Burns, ''Now Westlin Winds'', [CPW, 44]
|}

<pre>
Table 30

Table 30. Complex Signs & Their Higher Order Signs
Object Sign Interpretant

TIS "y0 = y.y" = <y0 = y.y>

TIS "y0 = y.y" = <"y0" =E "y.y">

"y0 = y.y" <"y0 = y.y"> = <<y0 = y.y>>

"y0 = y.y" "TIE" = <TIE>
</pre>

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | We'll gently walk, and sweetly talk,
|-
| width="5%" |   || While the silent moon shines clearly;
|-
| colspan="2" | I'll clasp thy waist, and, fondly prest,
|-
| width="5%" |   || Swear how I lo'e thee dearly:
|-
| colspan="2" | Not vernal show'rs to budding flow'rs,
|-
| width="5%" |   || Not Autumn to the farmer,
|-
| colspan="2" | So dear can be as thou to me,
|-
| width="5%" |   || My fair, my lovely charmer!
|-
| colspan="2" align="right" | — Robert Burns, ''Now Westlin Winds'', [CPW, 44]
|}

<pre>
Table 31

Table 31. More Complex Signs & Their Higher Order Signs
Object Sign Interpretant
TIS "TIS" = "The Intended State"
TIS TIE = The Initial Equation
TIS TIE = "y0 = y.y"
"TIS" <"TIS"> = <<TIS>>
TIE "TIE" = "The Initial Equation"
TIE "TIE" = <"y0 = y.y">
TIE "TIE" = <TIE>
"TIE" <"TIE"> = <<TIE>>
</pre>

=====5.1.2.6. The Epitext=====

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | ''Green grow the rashes, O;''
|-
| width="5%" |   || ''Green grow the rashes, O;''
|-
| colspan="2" | ''The sweetest hours that e'er I spend,''
|-
| width="5%" |   || ''Are spent among the lasses, O.''
|-
| colspan="2" align="right" | — Robert Burns, ''Green Grow the Rashes, O'', [CPW, 81]
|}

It is time to make explicit mention of a certain wrinkle in this text, even at the risk of warping the record that records this reconnaissance and rendering all the rest of its traces an unceasing problem to itself.

Throughout the course of this ongoing discussion is threaded what I refer to as an “epitext”, a linked succession of epigraphs, that, aside from the incidental interest that their contents may hold, are designed to keep before the mind what a real text is like, with all the potential for meaning and all the problems of interpretation that genuine symbols, complex imagery, and transcendental allusions can present. I cannot be expected to comment on, much less to clarify, all of the relevant aspects and all of the problematic features that are likely to be obvious to even the most casual reader of this epitext, but I think it is important to keep in mind a sense of the distance that is yet to be covered before any theory can claim a true comprehension of real language use.

Why are poetic texts and lyrical materials relevant to the aims of the present project? It is because they “face the music” as soon as they can speak and continue to address it for the rest of their developments. In other words, they speak from the very beginning of their invocations to the most pressing issues of communication and they attempt to tackle in their informal ways the most difficult problems of interpretation, those that formal languages and formal logics often put off till the end of their days, if they ever face up to them at all. Of course, if a set of troubles can be avoided then perhaps it is best to do so. Therefore, if I want to convince anybody that it is worth their bother to engage a given array of issues and problems, then I ought to supply an argument to make it plausible why the types of phenomena in question are likely to be inevitable.

One of the reasons for drawing this epitext from poetic sources is that a genuine poem, aside from its commentary on the passing show, what it seems to say about this ostensibly concrete subject or that divertingly pastoral scene, usually has something extra to say, a surplus meaning or an ulterior motive that sets its aim above and beyond the call of beauty, and this is usually something that it ventures to say about the reasons for its own existence, about the endeavor to communicate that goes into its making, and thus about its total ''context of interpretation'' (COI). In sum, a poem is often meant, at least partly, to address the implicit questions: Why am I writing this? And why am I writing this way?

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | There's nought but care on ev'ry han',
|-
| width="5%" |   || In every hour that passes, O:
|-
| colspan="2" | What signifies the life o man,
|-
| width="5%" |   || An 'twere na for the lasses, O.
|-
| colspan="2" align="right" | — Robert Burns, ''Green Grow the Rashes, O'', [CPW, 81]
|}

Aside from its apparent subject, its basic theme, and its commentary on diverse idyllic scenes, a genuine poem often has something to say about the reasons for its existence, the very idea that an author can form an intention, and the form of reception that it is aimed or averse to find. Thus its bits of reflective imagery, properly reconstituted and happily interpreted, make it tantamount to an ''implicitly recursive text'' (IRT).

What is it that forces a text to bear an immense variety of meanings, a few of them obvious, the bulk of them less so, if not the desire of its author to capture an image of a huge reality in an utterly tiny space, and to convey a fragment of a thicker truth along invisibly thin lines? A task like this can only be achieved through the use of multifaceted symbols and mirrored expressions, the results of multiple and repeated reflections. And a text like this can only be understood by means of an imaginative interpretation. Altogether, this mode of communication is comprehended by establishing a relation between writer and reader, one that is imprisioned at either end by the capacity at that terminus for imagination and reflection.

What is it that makes a text able to hold a wealth of meanings within it, if not the complementary desires of a writer and a reader to capture a huge reality between them?

The living creature, in its drive to write itself irreplacwably into the text of the universe and in its essay to render itself indispensable to the task of reading this text with any measure of understanding, ...

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | The war'ly race may riches chase,
|-
| width="5%" |   || An riches still may fly them, O;
|-
| colspan="2" | An tho at last they catch them fast,
|-
| width="5%" |   || Their hearts can ne'er enjoy them, O.
|-
| colspan="2" align="right" | — Robert Burns, ''Green Grow the Rashes, O'', [CPW, 81]
|}

The more one steps back from the objects and the phenomena that first struck one's fancy to know, the ones one really desires to understand, in order to get what one imagines will be a better, more analytic, and more conceptual view of them, the more one sees the envisioned ends of understanding receding into the distance of one's altered perspective, leaving one with respect to them barely at the beginnings of analysis. Over time, the visionary ends of inquiry appear to disappear into the mists of one's lately imposed starts, to skip over the marks of one's recently interposed stations, perhaps to await one's tardy arrival at the alpha and omega of one's upstart inquiry.

These effects, that are due to a developing perspective, especially the appearance of an expanding universe that grows out of one's increasing ability to see detail, can turn from being awe inspiring at one moment, increasing one's desire to know an object, and encouraging the work of understanding, to being disheartening at the next moment, overwhelming one's mind and senses with the power of a phenomenon, and dashing all hopes of comprehending it. As a consequence, short of renouncing the quest altogether, one is likely to restrain oneself to any fragment of the original question that seems easy enough to address in due order, and then to settle for any semblance of an answer that happens to present itself in due time.

The intervention of an epitext is designed precisely for this reason, to compensate, counteract, and remediate the more deleterious effects of an otherwise heathily growing perspective. The epitext is meant to keep the end in view, to remind the participants in a communication of the type of text that is ultimately desirable to understand, but without demanding its complete unraveling within the immediate frame of time, nor taunting each other so severely with the distances that remain to their goals that all are daunted from continuing with the ongoing task.

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | But gie me a cannie hour at e'en,
|-
| width="5%" |   || My arms about my dearie, O,
|-
| colspan="2" | An war'ly cares an war'ly men,
|-
| width="5%" |   || May a' gae tapsalteerie, O!
|-
| colspan="2" align="right" | — Robert Burns, ''Green Grow the Rashes, O'', [CPW, 81]
|}

In this way, an epitext can serve a couple of functions within a text:

<ol style="list-style-type:decimal">

<li><p>The epitext maintains an internal model of the informal context, the actual, intended, or likely "context of interpretation" (COI), or the typical "situation of communication" (SOC) that prevails in a given society of interpretive agents. It does this by preserving a constant but gentle reminder of the type of text that ultimately demands to be understood within this social context. In other words, it represents its social context in terms of its ideals, [??? the expectation that contains it dialogue between the epitext helps to provides an image of the dialogue that ???]</p></li>

<li><p>The epitext and the text are in a relation, analogous to a dialogue, that mirrors the relation of the text itself to its casual, informal, or social context. In general, the analogy can be set up in either one of two ways, and can shift its sense from moment to moment:</p></li>

<ol style="list-style-type:lower-latin">

<li><p>Epitext : Text :: Context : Text. Here, the epitext plays the part of common expectations, generic ideals, or social norms that are invoked in the process of communication.</p></li>

<li><p>Epitext : Text :: Text : Context. Here, the epitext gives vent to the individual conceits, idiosyncratic caprices, or whims of the moment that are stirred up by the process of communication.</p></li>

</ol></ol>

What type of text is best to use in an epitext, if it is going to achieve these stated objectives? A ''"paragon of writing'' is an apt title to give to the type of text at issue, since it marks the type of text that is originally desired to be understood and the type of text that ultimately demands to be understood.

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | For you sae douce, ye sneer at this;
|-
| width="5%" |   || Ye're nought but senseless asses, O:
|-
| colspan="2" | The wisest man the warl' e'er saw,
|-
| width="5%" |   || He dearly lov'd the lasses, O.
|-
| colspan="2" align="right" | — Robert Burns, ''Green Grow the Rashes, O,'', [CPW, 81]
|}

The epitext acts as a kind of tether that helps to keep the end in view. It maintains an internal model of the informal context and preserves a reminder of the kinds of texts that ultimately demand to be understood. In this way, the epitext supplies a canon, a guideline, or a standard within the text, and secrets within the main text not only an image of the object to be analyzed but also an icon of the object to be achieved. Naturally reflective texts, the kinds that occur in poems, in plays, and in other sorts of creative writing, are the best sources of stock for an epitext. These kinds of text have a double relevance to the aims of AI. As materials for analysis, they exemplify the character of real language use in natural settings. As models for methods of analysis, they are capable of addressing complex issues in a casual fashion. Because they are freely chosen from natural sources of exemplary language use, they can serve as examples for analysis that are realistic enough to exemplify the types of phenomena that one desires to understand and difficult enough to test the methods of analysis that one proposes to use. Because it is their nature in part to reflect on their own nature as texts, they often contain insightful commentaries on this very nature.

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | Auld Nature swears, the lovely dears
|-
| width="5%" |   || Her noblest work she classes, O:
|-
| colspan="2" | Her prentice han' she try'd on man,
|-
| width="5%" |   || An then she made the lasses, O.
|-
| colspan="2" align="right" | — Robert Burns, ''Green Grow the Rashes, O'', [CPW, 81]
|}

Let me then try to summarize the bearings that these poems and songs, these innocently reflective and innocuously self reverent types of texts, can be contemplated as having on the present inquiry, and again on the encompassing viability of AIR. Their relevance is twofold, bearing on the matter in question from complementary directions:

First, there is their negative or limitative bearing. As materials for analysis, poetic and lyrical texts contribute some of the hardest, most resilient, and most resistant cases that are known to confront the work of natural language understanding, and so they serve to puncture many of the rashest pretenses that this task is anywhere near to being done. Of course, if poetic and lyrical modes of production and comprehension are justly relegated to the status of mere diversions, then the task of unraveling their mysteries can be deemed a much less critical problem. However, if the resources that are elaborated and exploited in these manners of impression and expression are somehow essential to the very ideas of significant communication and meaningful interpretation, then it becomes a more crucial requirement to understand how they operate.

Second, there is their positive or productive bearing. As hints of models and heralds of methods, these types of texts hold out some hope that a text can sensibly comment on the task of achieving its own style of communication, the writing of it, the sending of it, the reading of it. More than that, their stock in variety of canonical styles helps to offer a cornucopia of constructive suggestions for ways of actually doing so. The chance of a text being able to express a perceptive reflection comes near the heart of the present matter, and the likelihood of a text having the power to embody an insightful self commentary brings the immediate discussion within a heartbeat of the inquiry into inquiry, for if a writing can teach about writing then perhaps an inquiry can learn about inquiry.

=====5.1.2.7. The Context of Interpretation=====

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | To the weaver's gin ye go, fair maids,
|-
| width="5%" |   || To the weaver's gin ye go,
|-
| colspan="2" | I rede you right, gang ne'er at night,
|-
| width="5%" |   || To the weaver's gin ye go.
|-
| colspan="2" align="right" | — Robert Burns, ''To the Weaver's Gin You Go'', [CPW, 306]
|}

As this discussion proceeds, it is designed to expand the scope of what it can analyze successfully from simple signs to complex expressions to extended texts. In this progression, the pragmatic theory of signs is used as a unifying thread to connect the different levels of complexity. Accordingly, it needs to be kept in mind throughout the discussion that references to ''signs'', unless specified otherwise, can generally be taken in a maximally inclusive sense, referring also to expressions and texts.

The more complex and more extended a sign, expression, or text becomes the more occasion it has for referring to many other objects on its way to achieving its ultimate denotation. Some of these implicit, incidental, and intermediate objects can be components, properties, and relations of the sign, expression, or text itself, perhaps amounting to its accidental connotations and its intended interpretants. When it comes to a highly involved sign, expression, or text, some of its subsidiary effects and ulterior objects can even be aspects or elements of those very agents and those very media that are actually, imagined, or intended to be involved in its production, transmission, or reception.

In many respects, these ''side effects'' are actually more important from a practical standpoint than the ''token objects'' of denotation, that is, the nominal results and the ostensible values that merely serve to mark the successful outcome of the interpretation process. Anything that an agent strives toward achieving or that a system moves toward attaining can be called its ''object'', and so there arises the possibility that a ''global object'' or a ''derivative object'', a thing constructed or reconstructed from various bits and pieces of extraneous references, is that which primarily or effectively calls or drives the greater action. If it strikes one as strange that an object construed from epiphenomenal marks and tangential signs should be the main motive and real object of the process of interpretation, it ought to be remembered that a special case of this already appears in the form of the semantic partitions that reconstruct the forms of their object domains. Accordingly, the types of global objects and derivative objects that emerge from the present considerations are just the ultimate generalizations of SECs.

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | My heart was ance as blythe and free
|-
| width="5%" |   || As simmer days were lang;
|-
| colspan="2" | But a bonie, westlin weaver lad
|-
| width="5%" |   || Has gart me change my sang.
|-
| colspan="2" align="right" | — Robert Burns, ''To the Weaver's Gin You Go'', [CPW, 306]
|}

Signs are typical of the contents of consciousness. Indeed, from the standpoint of the pragmatic theory of signs, where a maximally general definition of signs holds sway, signs are considered to be all inclusive, generically identical, or simply co extensive with the class of phenomena that are commonly known as the contents of consciousness. In this view of the matter, a complex expression is analogous to a complex concept, of an order that is typically but not exclusively constructed in deliberate and purposeful thinking processes, while an extended text is analogous to an ongoing stream of consciousness or a long drawn out process of reasoning, whatever its character may be.

This analogy or identity between signs and contents of consciousness can help to explain the pains I am taking in the present discussion to elucidate the structures of self referent signs, expressions, and texts. With this key to its interpretation, a sign that denotes itself attracts interest because it presents an icon of self awareness. In other words, a sign relation that is called on to make sense of a self denoting sign affords a particular type of formal model, one that captures a relevant aspect of the structure that is involved in a content of consciousness referring to itself. In a corresponding fashion, a text that refers to itself, in whole or in part, is analogous to a conscious process that makes reference to itself, its aspects, or its instants.

It needs to be noted what I am not saying here. I do not say that signs and texts are themselves aware, or that consciousness needs to emerge from them, however much they can serve to attract the attention of already conscious agents. Indeed, I am taking no position yet on the questions of whether or how consciousness can emerge from conceivably non conscious materials. At present, I am only interested in describing the formal relations or the structural relationships that can be noted to exist among contents of consciousness, as noted, and not to explain the bare facts of these contents, much less to explain the circumstance of consciousness itself. With regard to this purely descriptive purpose, the main task for the near future is to develop an array of conceptual frameworks that can be put to work in organizing formal descriptions and in converting suitable portions of them into effective descriptions. As long as one works under the aegis of these methodological limitations, the following maxim needs to be kept in mind: The only thing that a formal model can capture is the form of something. Whether form is of the essence in the case of the human psyche is in fact an ancient and a still important question, but not one that I hope to answer just yet. The patent answer that presents itself is to keep the question open, and to continue exploring all of the available options.

One benefit of this openness is that it permits the exploration of the thinking mind's connection with information.

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | My mither sent me to the town,
|-
| width="5%" |   || To warp a plaiden wab;
|-
| colspan="2" | But the weary, weary warpin o't
|-
| width="5%" |   || Has gart me sigh and sab.
|-
| colspan="2" align="right" | — Robert Burns, ''To the Weaver's Gin You Go'', [CPW, 307]
|}

The reason for my interest in signs, a reason that accounts for their part in inquiry and helps to explain their role in AIR, is that I take signs to be identical with all that is able to appear in awareness, or all that can be a content of consciousness. This amounts to saying that the ostensible analogy between signs and contents of consciousness, and thus between texts and streams of consciousness, is a potential identity. Speaking with respect to their potentiality, I would like to suggest that signs are identical with the possible contents of consciousness, and that the contents of consciousness all have the characters of potential signs. The broadest conceivable definition of what constitutes a "sign" leads to the broadest conceivable definition of what constitutes a "text", and so one is led to the idea that the whole stream of consciousness belonging to a person or a community, not just the miniscule fraction of it that happens to get written down in the conventional arrays of characters, can literally be regarded as a text.

What appears in awareness is a case of what one calls "phenomena", and a study that considers what can be a content of consciousness is called a "phenomenology". This means that

This is not the place to argue for the full strength of the proposed identity:

<pre>
Phenomena = Appearances in Awareness
= Contents of Consciousness
= Signs.
</pre>

Indeed, if one conceives consciousness to be a continuum, more exactly, if one considers every connected field (and stream) of consciousness to be a continuous manifold, as it very likely is, then this argument would depart from the realm of discrete signs and finite texts that is proper to computational models.

No matter how carefully the terms are qualified, allowing the equations to apply in purely formal and wholly potential senses, the argument for the soundness of this joint identification is by no means easy, presents the danger of leading this discussion far afield, if not astray, and is, in any case, not really needed to achieve the aims of the present work. Fortunately, while the full strength of the identity is not required for the present application, it can continue to serve as a useful analogy.

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | A bonie, westlin weaver lad
|-
| width="5%" |   || Sat working at his loom;
|-
| colspan="2" | He took my heart, as wi a net,
|-
| width="5%" |   || In every knot and thrum.
|-
| colspan="2" align="right" | — Robert Burns, ''To the Weaver's Gin You Go'', [CPW, 307]
|}

This is not the place to argue for my particular way of seeing things, whose rationale ultimately depends on the integral relationship between the pragmatic style of phenomenology and the pragmatic theory of signs. There is still too much potential for misunderstanding between the writer that is due merely to possible differences in the uses of words, and not to any matters of substance. Until these ideas can be fully developed, the relation between signs and contents of consciousness, or the relation between texts and streams of consciousness, can still be treated as a useful analogy.

The array of what appears in awareness and the condition of what can be a content of consciousness is the range and quality of "phenomena". To study what is able to appear in awareness and to contemplate what could be a content of consciousness is to consider "phenomena" in general. A study that treats of phenomena, whether in their widest generality or restricted in a particular way, is appropriate to call a "phenomenology". There are many different styles of phenomenology, in spite of the factious pretenses of universality that are likely to be part and parcel of any style that is particular enough to find favor with a party of individual agents.

From the pragmatic point of view, there is a close relation between phenomena and signs.

The style of phenomenology that is needed for this work is the subject of a later discussion. Here, I make only the remarks that are needed for orientation.

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | I sat beside my warpin wheel,
|-
| width="5%" |   || And ay I ca'd it roun.
|-
| colspan="2" | But every shot and every knock,
|-
| width="5%" |   || My heart it gae a stoun.
|-
| colspan="2" align="right" | — Robert Burns, ''To the Weaver's Gin You Go'', [CPW, 307]
|}

The pragmatic idea about phenomena is that all phenomena are signs of significant objects, except for the ones that are not. In effect, all phenomena are meant to appear before the court of significance and are deemed by their very nature to be judged as signs of potential objects. Depending on how one chooses to say it, the results of this evaluation can be rendered in one of the following ways:

<ol style="list-style-type:decimal">

<li><p>Some phenomena are in fact signs of significant objects. That is, they turn out to exist in a certain relation, one that is formally identical to a sign relation, wherein they denote objects that are important to the agent in question, an agent that thereby becomes the interpreter of these signs.</p></li>

<li><p>Some phenomena fail to be signs of significant objects, however much they initially appear to be. In this event, the failure can be accounted for in either one of two ways:</p></li>

<ol style="list-style-type:lower-latin">

<li><p>Some phenomena can fail to be signs of any objects at all. This amounts to saying that what appears is not really a sign at all, not really a sign of any object at all.</p></li>

<li><p>All phenomena are signs in some sense, even if only granted a default, nominal, or token designation as signs, but some signs still fail to qualify as signs of significant objects, because the objects they signify are not important to the agents in question.</p></li>

</ol></ol>

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | The moon was sinking in the west,
|-
| width="5%" |   || Wi visage pale and wan,
|-
| colspan="2" | As my bonie, westlin weaver lad
|-
| width="5%" |   || Convoy'd me thro the glen.
|-
| colspan="2" align="right" | — Robert Burns, ''To the Weaver's Gin You Go'', [CPW, 307]
|}

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | But what was said, or what was done,
|-
| width="5%" |   || Shame fa' me gin I tell;
|-
| colspan="2" | But Oh! I fear the kintra soon
|-
| width="5%" |   || Will ken as weel's myself!
|-
| colspan="2" align="right" | — Robert Burns, ''To the Weaver's Gin You Go'', [CPW, 307]
|}

=====5.1.2.8. The Formative Tension=====

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | A lassie all alone, was making her moan
|-
| width="5%" |   || Lamenting our lads beyond the sea: —
|-
| colspan="2" | “In the bluidy wars they fa', and our honor's gane an a',
|-
| width="5%" |   || And broken hearted we maun die.”
|-
| colspan="2" align="right" | — Robert Burns, ''As I Stood by Yon Roofless Tower'', [CPW, 570]
|}

An incidental purpose of this reconnaissance is to point out the more problematic features of interpretation, and especially the properties of sign comprehension that are observed to be salient in natural settings. It is these more problematic aspects of interpretation that ultimately place the greatest demands on artificial intelligence research, at any rate, if it is going to support an understanding of the ways that human beings use real languages in real practice, or if it is ever going to supply a medium for modeling the ways that human beings develop an ability to produce and to understand real texts.

I am sensitive to the possibility that there are many features in the situation of the present inquiry, perhaps due solely to my description, that make it appear like an artificial, an implausible, or a specialized set of circumstances. I begin by discussing a number of these features, after which I argue that they are really quite typical of any situation where one has to interpret a problematic text in a problematic language. A text or a language is ''problematic'', of course, only in relation to a prospective interpreter. To clarify this sense of the word ''problematic'' it helps to introduce the following definitions.

# A "fully interpretive language" (FIL) is one whose attributes are fully filled out in all three directions of natural language use, to wit, along syntactic, semantic, and pragmatic dimensions.
# A "fully interpretive grammar" (FIG), more commonly referred to as the "full grammar" of a language, is a body of knowledge that generates or specifies a FIL, incorporating the full details of its syntax, semantics, and pragmatics.
# A "problematic text" is one that seems to make some sort of sense, but whose meaning, if any, is not entirely clear on first reading.
# A "problematic language" is one whose "full grammar" is not yet available for articulation by the user in question, no matter how well the user is able to employ the language itself. It ought to be obvious that all "natural languages" are problematic for their customary users.

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | As I stood by yon roofless tower,
|-
| width="5%" |   || Where the wa'flow'r scents the dewy air,
|-
| colspan="2" | Where the houlet mourns in her ivy bower,
|-
| width="5%" |   || And tells the midnight moon her care:
|-
| colspan="2" align="right" | — Robert Burns, ''As I Stood by Yon Roofless Tower'', [CPW, 570]
|}

One of the functions of the epitext that I am weaving through the main text of this discussion is to keep constantly before the mind the more problematic features of natural languages, to illustrate the diversity of their utilities, and to notice a few of the elaborative, figurative, and imaginative devices that are not so easily formalized.

It is true that poems and programs share a conscious definition, coming to their collective senses under the head of ''effective description''. At least, it ought to be clear to the impartial observer that there is a close kinship between ''the words that do'' to engender action, that start it toward the hope of eventual success, and ''the words that fit'' a finite ambition, that point it toward the realms of eternal truth, and that this relation amounts to an accord that unites their intentions or a resonance that serves to bind their performance into a concerted whole, so I leave the reader to rede which is which and to judge on a case by case basis what the individual occasion demands.

It is also true that abstract category theory, in the ways that it affords a view of formal analogies, is largely a study of mathematical metaphors. But aside from all that, there is much else that mathematicians, poets, and programmers are bound to see alike and ought to share in common.

Under many natural circumstances, the only way to unravel the meaning of a problematic text is to place it in the field of influence of a FIL, typically as embodied in a variety of different interpreters, and to see how it is led to develop along the prevailing lines of interpretive force. In corresponding circumstances, approached in a complementary fashion, the only way to uncover the structure of a problematic language is to scatter a sample of signs and texts throughout its field of influence, and then to observe how these literal test particles are led to develop along interpretive lines, and if there is a coherent sensibility in force.

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | The winds were laid, the air was still,
|-
| width="5%" |   || The stars they shot along the sky,
|-
| colspan="2" | The tod was howling on the hill,
|-
| width="5%" |   || And the distant-echoing glens reply.
|-
| colspan="2" align="right" | — Robert Burns, ''As I Stood by Yon Roofless Tower'', [CPW, 570]
|}

One of the abiding tasks of artificial intelligence research is to figure out how natural languages do what they do for the human mind. This task amounts to articulating the FILs that humans actually use, and thus to arrive at the ''fully interpretive grammars'' (FIGs) that generate these FILs, shaping their syntax, semantics, and pragmatics. The main tool that one has in this task is ''formalization'', the process that devises formal models for the ongoing processes of interpretation.

Until one develops a battery of formal methods for exploring fields of interpretive influence and for tracing lines of interpretive force, one tends to be impelled by signs without understanding how or why one is moved by them, to wallow around in interpretive phenomena with little control over what develops, and to wander aimlessly through domains of apparent significance and evident meaning with no insight into their underlying structures and generative forms. Consequently, an attempt to avoid all formalization, though it appears at first sufficient to the gaining of experience, is not sufficient to the gaining of understanding, and therefore ultimately leads to the impoverishment of experience itself.

Unfortunately, there are equally pernicious tendencies that arise in the attempt to formalize experience and thus to arrive at formalized models. There is the tendency, in pursuing formalizations of a difficult subject, to settle on a premature formalization, that is, a narrowly circumscribed set of models, or an overly simplistic typology for addressing the topic, and then, in a vain attempt to avoid further difficulties by dictating to the subject how it ought to behave, to think that a partially successful formalization gives one the right to bar the subject from leaving the charmed circle swept out by its survey, or else to think that one can afford to ignore all aspects of the subject that do not fit within it. This temptation seems to arise on a recurring basis in the history of every formal science, being so well known from the dawn of awareness that its pattern is emblazoned in myth under the name of ''Procrustes''.

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | The burn, adown its hazelly path,
|-
| width="5%" |   || Was rushing by the ruin'd wa',
|-
| colspan="2" | Hasting to join the sweeping Nith,
|-
| width="5%" |   || Whase roaring seemed to rise and fa'.
|-
| colspan="2" align="right" | — Robert Burns, ''As I Stood by Yon Roofless Tower'', [CPW, 570]
|}

Generally speaking, formalizations constitute an indispensable utility in adapting the intellect to its environment and in helping it to build up relatively stable systems of organized knowledge. But it takes a ''force'' to make things ''fit'', since the formal picture is partial to its own bias, and the intellectual image distorts the world as much as it adapts to it.

The ''violence'' that is involved in a formalization, that is, the degree of its departure from nature, is observed to become especially acute each time that a specialized community of inquirers succeeds in developing a fresh array of formal methods and formal models, and thereby acquires a standardized resource that appears to equip them with an expanded range of formal powers. For those inquirers, these are competencies whose winning is not easy and whose salve they do not wish to give up, as they think they might by thinking too much about the lineaments of its ligaments and the limitations of its liniments. Consequently, one finds that the following sort of situation typifies the state of formal inquiry: Against every conscious caution that the formalization is only partial and in spite of every conscientious concession that the object domain is vastly more complex that any representation can contain, the tendency persists, once a formalization is fixed, to treat it just as if it were already and always would be perfectly adequate to the object.

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | The cauld blae North was streaming forth
|-
| width="5%" |   || Her lights, wi hissing, eerie din:
|-
| colspan="2" | Athort the lift they start and shift,
|-
| width="5%" |   || Like Fortune's favors, tint as win.
|-
| colspan="2" align="right" | — Robert Burns, ''As I Stood by Yon Roofless Tower'', [CPW, 570]
|}

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | Now, looking over firth and fauld,
|-
| width="5%" |   || Her horn the pale faced Cynthia rear'd,
|-
| colspan="2" | When lo! in form of minstrel auld
|-
| width="5%" |   || A stern and stalwart ghaist appear'd.
|-
| colspan="2" align="right" | — Robert Burns, ''As I Stood by Yon Roofless Tower'', [CPW, 570]
|}

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | And frae his harp sic strains did flow,
|-
| width="5%" |   || Might rous'd the slumbering Dead to hear,
|-
| colspan="2" | But O, it was a tale of woe
|-
| width="5%" |   || As ever met a Briton's ear!
|-
| colspan="2" align="right" | — Robert Burns, ''As I Stood by Yon Roofless Tower'', [CPW, 570]
|}

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | He sang wi joy his former day,
|-
| width="5%" |   || He, weeping, wail'd his latter times:
|-
| colspan="2" | But what he said — it was nae play!
|-
| width="5%" |   || I winna ventur't in my rhymes.
|-
| colspan="2" align="right" | — Robert Burns, ''As I Stood by Yon Roofless Tower'', [CPW, 570]
|}

=====5.1.2.9. The Vehicle of Communication : Reflection on the Scene, Reflection on the Self=====

=====5.1.2.10. (7)=====

=====5.1.2.11. (6)=====

=====5.1.2.12. Recursions : Possible, Actual, Necessary=====

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | Is there a whim-inspired fool,
|-
| colspan="2" | Owre fast for thought, owre hot for rule,
|-
| colspan="2" | Owre blate to seek, owre proud to snool?
|-
| width="5%" |   || Let him draw near;
|-
| colspan="2" | And owre this grassy heap sing dool,
|-
| width="5%" |   || And drap a tear.
|-
| colspan="2" align="right" | — Robert Burns, ''A Bard's Epitaph'', [CPW, 220]
|}

The purpose of a sign, for instance, a name, expression, program, or text, is to denote and possibly to describe an object, for instance, a thing, situation, mode of being, or activity in the world. In cases of practical interest, the object is usually very complex and the sign is usually very simple. Indeed, the intention of the whole descriptive enterprise is take objects as complex and as subtle as possible and to arrive at signs as simple and as concrete as the agent can conceive of fashioning to describe that object. Not surprisingly, the value of this exercise to the agent that carries it out is measured by the degree of difference in the apparent complexities of the object and the sign, or the proportion of success in this project is the measure of its value to the agent involved in it. In the cases of ultimate interest, the sorts of objects that the agent is charged to describe begin with something like the natural and social world itself, moves on to the natural and social language that avails itself to describe this world, and ends up with the natural and social mind that evolves in association with this language and with this world. In effect, a ''trialogue'', a three way dialogue or a threefold dialectic.

When the reality to be described is infinitely more complex than the typically finite resources that an agent has to describe it, then any number of elliptic, multiple, and repeated uses of these resources are bound to occur, leading to the strategies of approximation, abstraction, and recursion, respectively. All of these techniques have in common the fact that a ''systematic ambiguity'' in the use of signs is introduced and tolerated, necessitating a new order of context sensitivity, discernment, intelligence, or just plain good sense in the conduct of interpretations. A ''systematic ambiguity'' or a ''controlled equivocation'' occurs when the same sign is used for many different things or when the same sign is used at many different stages of a process.

Although the elliptic strategy of approximation is tantamount to simply ''leaving off'' from the effort to describe a difficult object, in effect, ''throwing up one's hands'' in exasperation, exhaustion, supplication, or surrender, by this means hoping to escape from the self imposed part of the requirement to describe it more closely, and finally ''giving up'' the attempted description with the significance of the data already recorded, no matter how much the ''broken off'' approach ''falls short'' of its goal, the closely related strategies of abstraction and recursion are rather more persistent in their tries at describing the object.

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | Is there a Bard of rustic song,
|-
| colspan="2" | Who, noteless, steals the crowds among,
|-
| colspan="2" | That weekly this area throng?
|-
| width="5%" |   || O, pass not by!
|-
| colspan="2" | But, with a frater-feeling strong,
|-
| width="5%" |   || Here, heave a sigh.
|-
| colspan="2" align="right" | — Robert Burns, ''A Bard's Epitaph'', [CPW, 220]
|}

In effect, so long as an agent sticks to the object and persists in the purpose of describing it / this kind of object, ... In other words, an agent is forced to resort to the stratagems of abstraction and recursion, where the same sign is used for many different objects and when the same sign is used to mark the progress of an activity at many different stages of its process, respectively. The underlying principle involved here is a kind of ''pragmatic pigeonhole principle''.

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | Is there a man, whose judgment clear
|-
| colspan="2" | Can others teach the course to steer,
|-
| colspan="2" | Yet runs, himself, life's mad career,
|-
| width="5%" |   || Wild as the wave? —
|-
| colspan="2" | Here pause — and, thro the starting tear,
|-
| width="5%" |   || Survey this grave.
|-
| colspan="2" align="right" | — Robert Burns, ''A Bard's Epitaph'', [CPW, 220]
|}

A ''resilient enough system of interpretation'' (RESOI), if driven to the point of distraction by the task of describing an inexhaustibly complex reality, makes several strategies available to its interpretive agent, either for preventing its being driven over the edge or for recovering from the inevitable lapses of attention that nevertheless happen to occur. The most salient of these strategies can be organized for discussion in the following manner:

'''Approximation.''' In resorting to approximation, one accepts the variety of natural bounds that apply to one's capacities for significant denotation, acknowledges the practical constraints that affect one's abilities for attending to detail and retaining exact records, and acts accordingly. This means recognizing the limitations of one's capacity for attention, recording the amounts that one can at the levels of accuracy that are feasible, and restricting one's intentions appropriately to capturing an aspect of one's object or representing a fraction of its reality.

'''Abstraction.''' In resorting to abstraction, one is trying to escape the limitations of a ''strict democracy'' in one's representations, otherwise known as the "one object, one sign" rule. Abstraction occurs when the same sign is used to refer to many different things, often conceived to form a class or a set of objects. In effect, abstraction introduces a common name or a general concept that denotes each individual object in a multitude of particular objects. Typically and most effectively, this comes about in recognition of a common attribute, a general feature, or a universal property that all of these objects share, giving the process of abstraction the beneficial side effect that the abstract sign can be newly re interpreted as referring to the abstract property in question.

Depending on the kinds of entities that are covered by an abstraction and the orders of logical complexity that are involved in this coverage, abstractions can be classified according to their domains of application and qualified according to their manners of construction and derivation. The next topics for discussion are two varieties of abstraction, called ''recursion'' and ''polymorphism'', that are especially important for the purpose of building computational models of interpretation and that deserve special mention in the present inquiry.

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | The poor inhabitant below
|-
| colspan="2" | Was quick to learn and wise to know,
|-
| colspan="2" | And keenly felt the friendly glow,
|-
| width="5%" |   || And softer flame;
|-
| colspan="2" | But thoughtless follies laid him low,
|-
| width="5%" |   || And stain'd his name!
|-
| colspan="2" align="right" | — Robert Burns, ''A Bard's Epitaph'', [CPW, 220]
|}

'''Recursion''' occurs when the same sign is used at different stages of a designated process, referring to different amounts of work to be done, marking the different amounts of work already done that establish the different settings of work going on, and often appearing at different levels of the agenda, charge, or mission that plots out this process. When a process is described by a text, that is, by a recorded agenda, outline, program, recipe, or script, then the recursion is typically reflected in the recursive sign's appearance at different structural levels of the text that serves to recapitulate or specify the process. For instance, a recursive sign can show up initially in the heading and then turn up at least one more time in the body of a text that codes a specification of a procedure, a text that formulates a definition of a function, or a text that constitutes a program, routine, or subprogram.

'''Polymorphism''' is a type of higher order abstraction that occurs when the same sign is used to denote elements of many different conceptual classes or objects of many distinct logical types. Comprehending the possible options calls for many alternative "conventions of intention", many heterogeneous "directions of connotation", and many splintered if still overlapping "moments of interpretation" to sort out the profusion of senses that is engendered. In the intermediate time frame, this type of diversity can appear to require a panoply of intellectual conceptions to organize the resulting multitude of meanings and to demand a variety of connotative planes to arrange their separate senses across, but it ultimately leads to a richer idea of the original aim or the intended object, as the potential for interpretation can be attributed to it.

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | Reader, attend! whether thy soul
|-
| colspan="2" | Soars Fancy's flights beyond the pole,
|-
| colspan="2" | Or darkling grubs in this earthly hole,
|-
| width="5%" |   || In low pursuit;
|-
| colspan="2" | Know, prudent, cautious, self control
|-
| width="5%" |   || Is wisdom's root.
|-
| colspan="2" align="right" | — Robert Burns, ''A Bard's Epitaph'', [CPW, 220]
|}

One of the aims of this work as a whole is to explain the use of sign relations in the interpretation of complex texts, for instance, this sentence, this paragraph, this entire work, just to name a few of the most obvious examples among the many that are conceivable. As the text I produce to explain the pragmatic theory of sign relations is itself just a sign in a sign relation to which the theory is intended to apply, the encounter with self reference, in both the senses of a self referent text and a self referent writer, cannot be avoided. Among the questions that this encounter brings in its train are the issues of self indicating signs, texts, and interpreters, bringing the following topics to a head:

# '''Indexical signs''' are signs that indicate their own interpretive context.
# '''Reflexive signs''' are signs that indicate themselves or their issuer.
# '''Recursive signs''' are signs that embody, enclose, or invoke themselves, that count themselves among their own parts or that include references to these parts within their own compositions, for instance, texts that incorporate references to their own headings, subtitles, or titles.

=====5.1.2.13. Ostensibly Recursive Texts=====

=====5.1.2.14. (3)=====

=====5.1.2.15. The Freedom of Interpretation=====

=====5.1.2.16. The Eternal Return=====

=====5.1.2.17. (1)=====

=====5.1.2.18. Information in Formation=====

=====5.1.2.19. Reflectively Indexical Texts=====

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | Now Robin lies in his last lair,
|-
| colspan="2" | He'll gabble rhyme, nor sing nae mair;
|-
| colspan="2" | Cauld poverty wi hungry stare,
|-
| width="5%" |   || Nae mair shall fear him;
|-
| colspan="2" | Nor anxious fear, nor cankert care,
|-
| width="5%" |   || E'er mair come near him.
|-
| colspan="2" align="right" | — Robert Burns, ''Elegy on the Death of Robert Ruisseaux'', [CPW, 268]
|}

A ''reflectively indexical text'' (RIT) is a text that refers to any aspect of its actual or intended ''context of interpretation'' (COI). With respect to a text, a COI is everything that is conceived to embody its prospects for communication, incorporating the dynamics to actuate the text along with the media to situate the text. For instance, a RIT can refer to any of the formal roles that play a part in actualizing the meaning of a text, in particular, it can refer to the roles of the agents that collaborate to bring its sense to life, in effect, the agents that work to ''animate'' it. A special case of a RIT, distinguished as a ''reflexively indexical text'', is one that adverts to its issuer. Depending on whether its references to a COI are ''explicit'' or ''implicit'', a RIT is called an ERIT or an IRIT, respectively. Because the same RIT can make many different references, explicit and implicit, to many different aspects of a contemplated COI, the associated attributions do not lead to mutually exclusive categories but merely to overlapping qualifications, any number of which a RIT can possess in parallel, inclusively and independently.

Aside from the usual arrays of messages that a text is meant to convey, a RIT has something to say about the ''communication situation'' where it finds itself engaged, where it happens to fall whether the possibility of such an occasion falls within its original intention or not, and where it summons agents to fall in line with its images of things and its patterns of action even if they fail to suit the occasions of their invocations. But more than that, a RIT can indicate the ''pragmatic setting'' where it has a call to be understood, where it is designed to evolve one or more clear meanings, and where it presses agents to render these meanings ever more effective in practice.

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | To tell the truth, they seldom fash'd him,
|-
| colspan="2" | Except the moment that they crush'd him;
|-
| colspan="2" | For sune as chance or fate had hush'd 'em,
|-
| width="5%" |   || Tho e'er sae short,
|-
| colspan="2" | Then wi a rhyme or sang he lash'd 'em,
|-
| width="5%" |   || And thought it sport.
|-
| colspan="2" align="right" | — Robert Burns, ''Elegy on the Death of Robert Ruisseaux'', [CPW, 268]
|}

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | Tho he was bred to kintra-wark,
|-
| colspan="2" | And counted was baith wight and stark
|-
| colspan="2" | Yet that was never Robin's mark
|-
| width="5%" |   || To mak a man;
|-
| colspan="2" | But tell him, he was learn'd and clark,
|-
| width="5%" |   || Ye roos'd him then!
|-
| colspan="2" align="right" | — Robert Burns, ''Elegy on the Death of Robert Ruisseaux'', [CPW, 268]
|}

=====5.1.2.20. (4)=====

=====5.1.2.21. (5)=====

=====5.1.2.22. (6)=====

=====5.1.2.23. (7)=====

=====5.1.2.24. (8)=====

=====5.1.2.25. The Discursive Universe=====

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | Wee sleekit, cow'rin, tim'rous beastie,
|-
| colspan="2" | O, what a panic's in thy breastie!
|-
| colspan="2" | Thou need na start awa sae hasty,
|-
| width="5%" |   || Wi bickering brattle!
|-
| colspan="2" | I wad be laith to rin an chase thee,
|-
| width="5%" |   || Wi murdering pattle!
|-
| colspan="2" align="right" | — Robert Burns, ''To a Mouse, On Turning Her Up in Her Nest with the Plough'', [CPW, 131]
|}

This project began with the aim of articulating an aspect of intelligent activity, namely, the inquiry into inquiry that appears to be implied in the very ability to do inquiry, to learn from the impressions of passing experience, and to reason about their indications for future experience. Inspired by the enunciation of these high aims the work proceeds in a top down fashion, though of course the rubble of previous experience is always there to suggest the orders of material to expect at the bottom. This mode of investigation, when it works, amounts to a recursive form of conceptual analysis, starting from the barest conception of its aim, seeking the conditions necessary for the possibility of its actualization, trying to determine the functional components that allow it to operate in principle, undertaking to shore up the practical supports that permit it to prosper in reality, and working to alleviate the practical obstacles that impact on its implementation in adverse ways.

A particular agent does what appears to be necessary at each moment in a succession of moments in order to achieve a particular aim, and hopes that what appears to be necessary to an agent who follows a given path cannot be totally immaterial to what is actually necessary in general. The relationship between apparent necessity and actual necessity is the topic of another discussion later in this work, so I leave it till then. At this point, it only needs to be observed that an apparent necessity constitutes a real force on the agent who observes it, in other words, that it constrains the acts of the agent to whom it appears necessary. Given the freedom of intellect that comes from the reflective criticism of particular developments, a particular agent's particular inquiries are hopefully conceived in such a way as to work toward necessary truths.

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | I'm truly sorry man's dominion
|-
| colspan="2" | Has broken Nature's social union,
|-
| colspan="2" | An justifies that ill opinion,
|-
| width="5%" |   || Which makes thee startle
|-
| colspan="2" | At me, thy poor, earth born companion,
|-
| width="5%" |   || An fellow mortal!
|-
| colspan="2" align="right" | — Robert Burns, ''To a Mouse'', [CPW, 131]
|}

Given the nature of particular agency, along with the circumstance that an inquiry must be carried out by a particular agent, an inquiry is apt to proceed in ways that are far from being absolutely necessary, and it is bound to wander on paths that fail to be optimal in the use of time. But this fact — the fact that many departures from necessity are likely to affect the progress of any particular inquiry, and the fact that such contingencies, deficiencies, and facticities are almost sure to apply to one's present inquiry — although its likelihood in general is frequently suspected by a reflective agent and its certainty in theory is probably apparent to a critical agent, its import is usually not clearly ''known'', not in the detailed sense that its application to the moment in question is available to the agent who needs to act on it, and not in the pressing sense that its bearings on the consequences for experience are apparent to the agent who ought to be concerned about their subsequent effects. But the widespread suspicion that what appears to be necessary is not always actually and absolutely necessary, however much it is likely to verge on the truth, remains completely vague in that form, and it does not conjure up enough of an objection to deter action on what appears to be necessary, not unless a concrete alternative also appears.

In this way one is able to see the form of short term independence, the apparent indifference and the seeming lack of correlation that persists in the meantime, between actual necessities and apparent necessities. The apparent necessity continues to subsist as a factitious matter, no matter how grave it appears to the agent who falls within its orbit and no matter how much it constrains the circumstantial actions of the agent to whom it in fact appears necessary. A lack of actual necessity does not prevent an apparent necessity from continuing to appear just as if it were called for. Conversely, a lack of apparent necessity in no way impedes an actual necessity from continuing to rule the total situation. With all due respect to apparent necessities, the fact of their actual facticity is perfectly capable of holding true, however much these very conditions are able to constrain the actions of the particular agents to whom they appear necessary. Moreover, the factitious nature and the virtual force that are severally attributed to an apparent necessity are just as apparently independent of each other, at least, in medias res. The facticity of an apparent necessity continues to hold in fact, however forcefully it actually succeeds in compelling the activities of an agent. The force of an apparent necessity continues to stay in effect, in spite of its actual facticity, right up until the time when it no longer appears to be necessary.

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | I doubt na, whyles, but thou may thieve;
|-
| colspan="2" | What then? poor beastie, thou maun live!
|-
| colspan="2" | A daimen icker in a thrave
|-
| width="5%" |   || 'S a sma request;
|-
| colspan="2" | I'll get a blessin wi the lave,
|-
| width="5%" |   || An never miss't!
|-
| colspan="2" align="right" | — Robert Burns, ''To a Mouse'', [CPW, 132]
|}

What I hope that my discussion of The LOS leads to is an inkling of the type of dialogue that is capable of taking place between a formal domain and the informal context in which it lives, the world in which it is born, continues to grow, and even forces to evolve along with its development, the setting at times in which it lies dormant, remaining restively inert for years or simply sleeping through the appointed phases of the night, the fold in which it is able to be reborn, to come to a new life, and to arise afresh. The form of this dialogue is one that suggests the name for itself of a ''discursion'', a word that is coined to carry a wealth of various possibilities: a ''dialectic recursion'' or a ''recursive dialogue'', a ''recursive analysis'' (RA) or a ''recursive excursion'' (RE), perhaps the very form of ''recursive inquiry'' (RI) that admits of its decomposition into one or another ''recursive undertaking'' (RU) and thereby maintains the very form of its own constitution. This array of acronyms serves to stake out a ready field of discursive research and exploration, one that is open in certain directions to unformalized possibilities of experience, in a sense or in essence, to its own future.

But the question remains whether sign-bearing agents can act, at least, as if they are able to be aware of their bearing as one component of a coherent, competent, and complete code of conduct, even a form of life. And the question continues how interpreters can acquire their faculties for the conscientious development and the deliberate elaboration of the factors that affect their own interpretive activities, in sum, how they can reflect on the factual contingencies that affect their own sign use, on the facticity of the circumstances that constrain these uses, and on the factors that determine the facility of the conditions that lead up to these uses, and then act on the results of all these reflections to make improvements in all these factors. In this way of broaching the subject of reflection, I am forced to drop it almost immediately, with the aim of starting afresh at another point and approaching the topic again, the next time from another direction.

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | Thy wee bit housie, too, in ruin!
|-
| colspan="2" | Its silly wa's the win's are strewin!
|-
| colspan="2" | An naething, now, to big a new ane,
|-
| width="5%" |   || O foggage green!
|-
| colspan="2" | An bleak December's win's ensuin,
|-
| width="5%" |   || Baith snell an keen!
|-
| colspan="2" align="right" | — Robert Burns, ''To a Mouse'', [CPW, 132]
|}

At times one enters a state of mind that seems so rich in possibilities, teems with so many avenues of interesting departures, and unlocks such veins of unsuspected wealth in the world of ideas that one wants to be sure to revisit it again, in order to explore the rest of the thoughts that seem likely to unfold from its locus, its nexus, and its treasury. But the only way that one can be sure to return to anything like the same state of mind is to put reminders of it all about, at every other point that one later passes through and in the vicinity of every locus the neighborhood of which one is likely to visit. Now a state that one experiences at a former time is not always possible to experience again, but it may be possible to make nearby passes or to approach arbitrarily close to the essence of the exact experience at a host of future times. Then consider a manifold of possible states of mind as forming a space that possesses its conceivable extension. And so one gets these ideas: (1) of a manifold that is suffused with the idea of a manifold that is suffused just so, (2) of a manifold that is suffused with more or less accurate ideas of itself, and (3) of a manifold that is suffused with its own idea just far enough that it can serve to maintain the orbits of the agents that pass through it in suffusion with the very idea of doing so.

If the writer can din the reader into an awareness that the repetition of a word does not imply the repetition of a thought, that the repetition of a sign does not imply the repetition of any idea, that the repetition of a state does not mean its repetition forever, then this repetition serves its purpose, however close it verges on absurdity.

The discussion arrives at the question of signs and texts that signify, beyond their ostensible denotations and their obvious connotations, the characters of their authors, the features of their intended readers, and much more besides about the nature of their joint adventures, whatever their levels of participation in them, the processes of communication.

If the question of the interpreter that is signified by a sign reduces to the question of the interpretation that is signified by that sign, and if this reduces to the question of the interpretant that is signified by the sign, then one arrives at the circumstance of sign that relates to its interpretant along several paths.

"effective descriptions and finite texts" (EDAFTs)

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | Thou saw the fields laid bare an waste,
|-
| colspan="2" | An weary winter comin fast,
|-
| colspan="2" | An cozie here, beneath the blast,
|-
| width="5%" |   || Thou thought to dwell,
|-
| colspan="2" | Till crash! the cruel coulter past
|-
| width="5%" |   || Out thro thy cell.
|-
| colspan="2" align="right" | — Robert Burns, ''To a Mouse'', [CPW, 132]
|}

At this juncture the discussion comes face to face with a type of text, many of whose signs are subject to different levels of interpretation. Besides denoting characters and creatures of legend and myth that form at first sight the subjects of the text, describing their features, and depicting their various adventures, they also appear to be amenable to recursive or self referent interpretations or to suggest an extra sense for themselves. Indeed, they seem designed to serve an added intention of their author, namely, to say something about the aims of the author, the attributes of the audience, whether hoped or feared, and the nature of the whole attempt to communicate.

Just to indicate the types of self reference that are being contemplated, it helps to introduce a number of informal definitions. Let any suitable set of entities {writer, sign, reader} be called a ''linkage'' of the sign in question, and let any suitable set of entities {writer, text, reader} be called a ''linkage'' of the text in question. In either of these uses, let the subset of entities {writer, reader} be called the ''link'' of that linkage, and let the elements of a link be called its ''ends'' or ''termini''. At present, the situations of interest are those in which all of the signs in a text, at least, those that denote anything at all, are considered to share the very same link, which they all bear in common with their text. From now on, this condition is taken for granted unless it is otherwise expressly noted. Given a sign within a text, the union of their linkages is a set of entities {writer, sign, text, reader} that is useful to call a ''nocking'' of the sign. Together with the specification of a sign relation that suits a particular condition of interpretation, these constructs go toward defining a ''communication situation'', an ''interpretive setting'', or a ''pragmatic frame'' for the sign or the text in question.

Naturally, these constructions require a lot more information about the details of a given interpretation in a given situation in order to pin them down exactly, but this is enough to rough out their general ideas. Their main use in the current setting is simply to provide a ready way of talking about the properties of certain kinds of complex texts, as they are become subject to certain kinds of ''loopy'', ''recursive'', or ''self-referent'' interpretations.

If a sign within a text is interpreted as making any kind of denotative reference to its own nocking, namely, to the appropriate set of entities {writer, sign, text, reader}, its elements, or its properties, then it is useful to consider this sign and this text as being self referent in the broad sense that they refer to accessory or instrumental aspects of the pragmatic frame itself, and thus can be said to have an ''internal aim''. This can happen whether or not a sign within a text denotes any object beyond its nocking, and thus can be said to have an ''external aim''.

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | That wee bit heap o leaves an stibble,
|-
| colspan="2" | Has cost thee monie a weary nibble!
|-
| colspan="2" | Now thou's turn'd out, for a' thy trouble,
|-
| width="5%" |   || But house or hald,
|-
| colspan="2" | To thole the winter's sleety dribble,
|-
| width="5%" |   || An cranreuch cauld!
|-
| colspan="2" align="right" | — Robert Burns, ''To a Mouse'', [CPW, 132]
|}

In the pragmatic theory of signs it is often said, “The question of the interpreter reduces to the question of the interpretant.” If this is true then it means that questions about the special interpreters that are designated to serve as the writer and the reader of a text are reducible to questions about the particular sign relations that independently and jointly define these two interpreters and their process of communication. The assumptions and the implications that are involved in this maxim are best explained by retracing the analysis that leads to this reduction, setting it out in the following stages:

In the pragmatic theory of signs it is often said, “The question of the interpreter reduces to the question of the interpretant.” If this is true then it means that questions about the special interpreters that are designated to serve as the writer and the reader of a text are reducible to questions about the particular sign relations that independently and jointly define these two interpreters and their process of communication. The assumptions and the implications that are involved in this maxim are best explained by retracing the analysis that leads to this reduction, setting it out in the following stages:

<ol style="list-style-type:decimal">

<li><p>By way of setting up the question of the interpreter, it needs to be noted that it can be asked in any one of several modalities. These are commonly referred to under a variety of different names, for instance:</p></li>

<ol style="list-style-type:lower-alpha">

<li><p>What may be: the "prospective" or the "imaginative";<br>
also: the contingent, inquisitive, interrogative, optional, provisional, speculative, or "possible on some condition".</p></li>

<li><p>What is: the "descriptive" or the "indicative";<br>
also: the actual, apparent, definite, empirical, existential, experiential, factual, phenomenal, or "evident at some time".</p></li>

<li><p>What must be: the "prescriptive" or the "imperative";<br>
also: the injunctive, intentional, normative, obligatory, optative, prerequisite, or "necessary to some purpose".</p></li>

</ol></ol>

It is important to recognize that these lists refer to modes of judgment, not the results of the judgments themselves. Accordingly, they conflate under single headings the particular issues that remain to be sorted out through the performance of the appropriate judgments, for instance, the difference between an apparent fact and a genuine fact. In general, it is a difficult question what sorts of relationships exist among these modalities and what sorts of orderings are logically or naturally the best for organizing them in the mind. Here, they are given in one of the possible types of logical ordering, based on the idea that a thing must be possible before it can become actual, and that it must become actual (at some point in time) in order to qualify as being necessary. That is, being necessary implies being actual at some time or another, and being actual implies being possible in the first place. This amounts to thinking that something must be added to a condition of possibility in order to achieve a state of actuality, and that something must be added to a state of actuality in order to acquire a status of necessity.

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | But Mousie, thou art no thy lane,
|-
| colspan="2" | In proving foresight may be vain:
|-
| colspan="2" | The best laid schemes o mice an men
|-
| width="5%" |   || Gang aft agley,
|-
| colspan="2" | An lea'e us nought but grief an pain,
|-
| width="5%" |   || For promis'd joy!
|-
| colspan="2" align="right" | — Robert Burns, ''To a Mouse'', [CPW, 132]
|}

All of this notwithstanding, it needs to be recognized that other types of logical arrangement can be motivated on other grounds. For example, there are good reasons to think that all of one's notions of possibility are in fact abstracted from one's actual experiences, making actuality prior in some empirically natural sense to the predicates of possibility. Since a plausible heuristic organization is all that is needed for now, this is one of those questions that can be left open until a later time.

<ol style="list-style-type:decimal" start="2">

<li><p>Taking this setting as sufficiently well understood and keeping these modalities of inquiry in mind, the analysis proper can begin. Any question about the character of the interpreter that is acting in a situation can be identified with a question about the nature of the process of interpretation that is taking place under the corresponding conditions.</p></li>

<li><p>Any question about the nature of the process of interpretation that is taking place can be identified with a question about the properties of the interpretant that follows on a given sign. This is a question about the interpretant that is associated with a sign, in one of several modalities and as contingent on the total context.</p></li>

</ol>

In summary: The question of the interpreter that is signified to act reduces to the question of the interpretation that is signified to occur, and thus to the question of the interpretant that is signified to follow the given sign under the given conditions. Expanding over the various modalities: The question of the interpreter reduces to the question of the interpretation that is determined, designed, or depicted to occur, and this in turn reduces to the question of the interpretant that is indicated, intended, or imagined to be associated with the given sign.

To follow this reduction in stages, the character of the interpreter that can be signified in some modality to be acting in a situation is identified with the nature of the process of interpretation that can be signified in that same modality to be taking place in that same situation, and this is the matter of the kind of interpretant that can be signified in that same modality to be following on the sign that is given in that same situation.

{| align="center" border="0" cellpadding="0" cellspacing="0" width="90%"
| colspan="2" | Still thou art blest, compar'd wi me!
|-
| colspan="2" | The present only toucheth thee:
|-
| colspan="2" | But och! I backward cast my e'e,
|-
| width="5%" |   || On prospects drear!
|-
| colspan="2" | An forward, tho I canna see,
|-
| width="5%" |   || I guess an fear!
|-
| colspan="2" align="right" | — Robert Burns, ''To a Mouse'', [CPW, 132]
|}

=====5.1.2.26. (7)=====

=====5.1.2.27. (6)=====

=====5.1.2.28. (5)=====

=====5.1.2.29. (4)=====

=====5.1.2.30. (3)=====

=====5.1.2.31. (2)=====

=====5.1.2.32. (1)=====

===5.2. Reflective Inquiry===

====5.2.1. Integrity and Unity of Inquiry====

One of the very first questions that one encounters in the inquiry into inquiry is one that challenges both the integrity and the unity of inquiry, a question that asks: “Is inquiry one or many?” By this one means two things:

# Concerning the integrity of inquiry: How are the components and the properties of inquiry, as identified by analysis, integrated into a whole that is singly and solely responsible for its results, and as it were, that answers for its answers in one voice? These qualities of unanimity and univocity are necessary in order to be able to speak of an inquiry as a coherent entity, whose nature it is to have and to hold the boundaries one finds in or gives to it, rather than being an artificial congeries of naturally unrelated elements and features. In other words, this is required in order to treat inquiry as a systematic function, that is, as the action, behavior, conduct, or operation of a system.
# Concerning the unity of inquiry: Is the form of inquiry that is needed for reasoning about facts the same form of inquiry that is needed for reasoning about actions and goals, duties and goods, feelings and values, guesses and hopes, and so on, or does each sort of inquiry — aesthetic, ethical, practical, speculative, or whatever — demand and deserve a dedicated and distinctive form? Although it is clear that some degree of modulation is needed to carry out different modes of inquiry, is the adaptation so radical that one justly considers it to generate different forms, or is the changeover merely a matter of mildly tweaking the same old tunes and draping new materials on the same old forms?

If one reflects, shares the opinion, or takes the point of view on experimental grounds that inquiry begins with uncertainty, then each question about the integrity and the unity of inquiry can be given a sharper focus if it is re-posed as a question about the integrity and the unity of uncertainty, or of its positive counterpart, information.

Accordingly, one is led to wonder next: Is uncertainty one or many? Is information one or many? As before, each question raises two more: one that inquires into the internal composition of its subject, or the lack thereof, and one that inquires into the external diversity of its subject, or the lack thereof. This reflection, on the integrity and the unity, or else the multiplicity, of uncertainty and information, is the image of the earlier reflection, on the facts of sign use. Once more, what appears in this reflection is so inconclusive and so insubstantial that there is nothing else to do at this point but to back away again from the mirror.

To rephrase the question more concretely: Is uncertainty about what is true or what is the case the general form that subsumes every species of uncertainty, or is it possible that uncertainty about what to do, what to feel, what to hope, and so on constitute essentially different forms of inquiry among them? The answers to these questions have a practical bearing in determining how usefully the presently established or any conceivable theory of information can serve as a formal tool in different types of inquiry.

Another way to express these questions is in terms of a distinction between ''form'' and ''matter''. The form is what all inquiries have in common, and the question is whether it is anything beyond the bare triviality that they all have to take place in some universe of inquiry or another. The matter is what concerns each particular inquiry, and the question is whether the matter warps the form to a shape all its own, one that is peculiar to this matter to such a degree that it is never interchangeable with the forms that are proper to other modes of inquiry.

====5.2.2. Apparitions and Allegations====

Next I consider the preparations for a phenomenology. This is not yet any style of phenomenology itself but an effort to grasp the very idea that something appears, and to grasp it in relation to the something that appears. I begin by looking at a sample of the language that one ordinarily uses to talk about appearances, with an eye to how this medium shapes one's thinking about what appears. A close inspection reveals that there are subtleties issuing from this topic that are partly disclosed and partly obscured by the language that is commonly used in this connection.

* An ''apparition'', as I adopt the term and adapt its use to this context, is a property, a quality, or a respect of appearance. That is, it is an aspect or an attribute of a phenomenon of interest that appears to arise in a situation and to affect the character of the phenomenal situation. Apparitions shape themselves in general to any shade of apperception, assumption, imitation, intimation, perception, sensation, suspicion, or surmise that is apt or amenable to be apprehended by an animate agent.

* An ''allegation'', in the same manner of speaking, is any description or depiction, any expression or emulation, in short, any verbal exhalation or visual emanation that appears to apprehend a characteristic trait or an illuminating trace of an apparition.

The terms ''apparition'' and ''allegation'' serve their purpose in allowing an observer to focus on the sheer appearance of the apparition itself, in assisting a listener or a reader to attend to the sheer assertion of the allegation itself. Their application enables an interpreter to accept at first glance or to acknowledge at first acquaintance the reality of each impression as a sign, without being forced to the point of assuming that there is anything in reality that the apparition is in fact an appearance of, that there is anything in reality that the allegation is in deed an adversion to, or, as people commonly say, that there is anything of substance "behind" it all.

Ordinarily, when one speaks of the ''appearance'' of an object, one tends to assume that there is in reality an object that has this appearance, but if one speaks about the ''apparition'' of an object, one leaves more room for a suspicion whether there is in reality any such object as there appears to be. In technical terms, however much it is simply a matter of their common acceptations, the term ''appearance'' is said to convey slightly more ''existential import'' than the term ''apparition''. This dimension of existential import is one that enjoys a considerable development in the sequel.

If one asks what apparitions and allegations have in common, it seems to be that they share the character of signs. If one asks what character divides them, it is said to be that apparitions are more likely to be generated by an object in and of itself while allegations are more likely to be generated by an interpreter in reaction to an alleged or apparent object. Nevertheless, even if one agrees to countenance both apparitions and allegations as a pair of especially specious species of signs, whose generations are differentially attributed to objects and to interpreters, respectively, and whose variety runs through a spectrum of intermediate variations, there remains a number of subtleties still to be recognized.

For instance, when one speaks of an ''appearance'' of a sign, then one is usually talking about a ''token'' of that type of sign, as it appears in a particular locus and as it occurs on a particular occasion, all of which further details can be specified if required. If this common usage is to be squared with calling apparitions a species of signs, then talk about an ''appearance'' of an apparition must have available to it a like order of interpretation. And thus what looks like a higher order apparition, in other words, an apparition of an apparition, is in fact an even more particular occurrence, specialized appearance, or special case of sign. At this point I have to let go of the subject for now, since the general topic of ''higher order signs'', their variety and interpretation, is one that occupies a much broader discussion later on in this work.

Any action that an interpreter takes to detach the presumed actuality of the sign from the presumed actuality of its object, at least in so far as the sign appears to present itself as denoting, depicting, or describing a particular object, remains a viable undertaking and a valuable exercise to attempt, no matter what hidden agenda, ulterior motive, or intentional object is conceivably still invested in the apparition or the allegation. If there is an object, property, or situation in reality that is in fact denoted or represented by one of these forms of adversion and allusion, then one says that there is a basis for acting on them, a justification for believing in them, a motivation for taking them seriously, a reason for treating them as true, or a foundation that is capable of lending support to their prima facie evidence.

Once the dimension of existential import is recognized as a parameter of interpretation, for example, as it runs through the spectrum of meanings that the construals of ''apparitions'' and ''appearances'' are differentially scattered across, then there are several observations that ought to be made about the conceivable distributions of senses:

# In principle, the same range of ambiguities and equivocalities affects both of the words ''apparition'' and ''appearance'' to the same degree, however much their conventional usage tilts their individual and respective senses one way or the other.
# Deprived of its existential import, the applicational phrase ''appearance of an object'' (AOAO) means something more akin to the adjectival or analogous phrase ''object-like appearance'' (OLA). Can it be that the mere appearance of the preposition ''of'' in the application "P of Q" is somehow responsible for the tilt of its construal toward a more substantial interpretation, one with a fully existential import?
# Interpreting any apparition, appearance, phenomenon, or sign as an ''appearance of an object'' is tantamount to the formation of an abductive hypothesis, that is, it entertains the postulation of an object in an effort to explain the particulars of an appearance.
# The positing of objects to explain apparitions, appearances, phenomena, or signs, to be practical on a regular basis, requires the preparatory establishment of an ''interpretive framework'' (IF) and the concurrent facilitation of an ''objective framework'' (OF). Teamed up together, these two frameworks assist in organizing the data of signs and the impressions of ideas in connection with the hypotheses of objects, and thus they make it feasible to examine each ''object-like appearance'' and to convert each one that is suitable into an ''appearance of an object''.

At this point it ought to be clear that the pragmatic theory of signs permits the ''whole of phenomenal reality'' (WOPR) to be taken as a sign, perhaps of itself as an object, and perhaps to itself as an interpretant. The articulation of the exact sign relation that exists is the business of inquiry into a particular universe, and this is a world whose existence, development, and completion are partially contingent on the character, direction, and end of that very inquiry.

The next step to take in preparing a style of phenomenology, that is, in acquiring a paradigm for addressing apparitions or in producing an apparatus for dealing with appearances, is to partition the space of conceivable phenomena in accord with several forms of classification, drawing whatever parallel and incidental lines appear suitable to the purpose of oganizing phenomena into a sensible array, in particular, separating out the kinds of appearances that one is prepared to pay attention to, and thus deciding the kinds of experiences that one is ready to partake in, while paring away the sorts of apparitions that one is prepared to ignore.

It may be thought that a phenomenology has no need of preparation or partition, that the idea is to remain openly indiscriminate and patently neutral to all that appears, that all of its classifications are purely descriptive, and that all of them put together are intended to cover the entire range of what can possibly show up in experience. But attention is a precious resource, bounded in scope and exhausted in detail, while the time and the trouble that are available to spend on the free and the unclouded observation of phenomena are much more limited still, at least, in so far as it concerns finite agents and mortal creatures, and thus even the most liberal phenomenology is forced to act on implicit guidelines or to put forward explicit recommendations of an evaluative, a normative, or a prescriptive character, saying in effect that if one acts in certain ways, in particular, that if one expends an undue quantity of attention on the "wrong" kinds of appearances, then one is bound to pay the price, in other words, to experience unpleasant experiences as a consequence or else to suffer other sorts of adverse results.

This observation draws attention to the general form of constraint that comes into play at this point. Let me then ask the following question: What is the most general form of preparation, partition, or reparation, of whatever sort of disposition or structure, that I can imagine as applying to the whole situation, that I can see as characterizing its experiential totality, and that I can grasp as contributing to its ultimate result? For my own part, in the present situation, the answer appears to be largely as follows.

As far as I know, all styles of phenomenology and all notions of science, whether general or special, either begin by adopting an implicit recipe for what makes an apparition worthy of note or else begin their advance by developing an explicit prescription for a "worthwhile" appearance, a rule that presumes to dictate what phenomena are worthy of attention. This recipe or prescription amounts to a critique of phenomena, a rule that has an evaluative or a normative force. As a piece of advice, it can be taken as a ''tentative rule of mental presentation'' (TROMP) for all that appears or shows itself, since it sets the bar for admitting phenomena to anything more than a passing regard, marks the threshold of abiding concern and the level of recurring interest, formulates a precedence ordering to be imposed on the spectra of apparitions and appearances, and is tantamount to a recommendation about what kinds of phenomena are worth paying attention to and what kinds of shows are not worth the ticket — in a manner of speaking saying that the latter do not repay the price of admission to consciousness and do not earn a continuing regard.

The issue of a TROMP ("tentative rule of mental presentation") can appear to be a wholly trivial commonplace or a totally unnecessary extravagance, but realizing that a choice of this order has to be made, that it has to be made at a point of development where no form of justification of any prior logical order can be adduced, and thus that the choice is always partly arbitrary and always partly based on aesthetic considerations, ethical constraints, and practical consequences — all of this says something important about the sort of meaning that the choice can have, and it opens up a degree of freedom that was obscured by thinking that a phenomenology has to exhaust all apparitions, or that a science has to be anchored wholly in bedrock.

If it appears to my reader that my notion of what makes a worthwhile appearance is tied up with what I can actually allege to appear, and is therefore constrained by the medium of my language and the limits of my lexicon, then I am making the intended impression. One of the reasons that I find for accepting these bounds is that I am decidedly less concerned with those aspects of experience that appear in one inconsistent and transient fashion after another, and I am steadily more interested in those aspects of experience that appear on abiding, insistent, periodic, recurring, and stable bases. Since I am trying to demonstrate how inquiry takes place in the context of a sign relation, the ultimate reasons for this restriction have to do with the nature of inquiry and the limited capacities of signs to convey information.

Inquiry into reality has to do with experiential phenomena that recur, with states that appear and that promise or threaten to appear again, and with the actions that agents can take to affect these recurrences. This is true for two reasons: First, a state that does not appear or does not recur cannot be regarded as constituting any sort of problem. Second, only states that appear and recur are subject to the tactics of learning and teaching, or become amenable to the methods of reasoning.

There is a catch, of course, to such a blithe statement, and it is this: How does an agent know whether a state is going to appear, is bound to recur, or not? To be sure, there are hypothetically conceivable states that constitute obvious problems for an agent, independently of whether an instance of them already appears in experience or not. This is the question that inaugurates the theoretical issue of signs in full force, raises the practical stakes that are associated with their actual notice, and constellates the aspect of a promise or a threat that appears above. Accordingly, the vital utility of signs is tied up with questions about persistent appearances, predictable phenomena, contingently recurrent states of systems, and ultimately patterned forms of real existence that are able to integrate activity with appearance.

In asking questions about integral patterns of activity and appearance, where the category of action and the category of affect are mixed up in a moderately complicated congeries with each other and stirred together in a complex brew, it is helpful on a first approximation to "fudge" the issue of the agent a bit, in other words, to "dodge", "fuzz", or "hedge" any questions about the precise nature of the agent that appears to be involved in the activities and to whom the appearances actually appear. This intention is served by using the word "agency" in a systematically ambiguous way, namely, to mean either an individual agent, a community of agents, or any of the actions thereof. In this vein, the following sorts of questions can be asked:

# What appearances can be recognized by what agencies to occur on a recurring basis? In other words, what appearances can be noted by what agencies to fall under sets of rules that describe their ultimate patterns of activity and appearance?
# What appearances can be shared among agents and communities that are distributed through dimensions of culture, language, space, and time?
# What appearances can be brought under the active control of what agencies by observing additional and alternative appearances that are associated with them, that is, by acquiring and exploiting an acquaintance with the larger patterns of activity and appearance that apply?

There is a final question that I have to ask in this preparation for a phenomenology, though it, too, remains an ultimately recurring inquiry: What form of reparation is due for the undue distribution of attention to appearance? In other words, what form of reform is called on to repair an unjust disposition, to remedy an inadequate preparation, or to adjust a partition that is not up to par? Any attempt to answer this question has occasion to recur to its preliminary: What form of information does it take to convince agents that a reform of their dispositions is due?

As annoying as all of these apparitions and allegations are at first, it is clear that they arise from an ability to reflect on a scene of awareness, and thus, aside from the peculiar attitudes that they may betray from time to time, they advert to an aptitude that amounts to an inchoate agency of reflection, an incipient faculty of potential utility that the agent affected with its afflictions is well-advised to appreciate, develop, nurture, and train, in spite of how insipid its animadversions are alleged to appear at times. This marks the third time now that the subject of reflection has come to the fore. Paradoxically enough, no increment of charm appears to accrue to the occasion.

A good part of the work ahead is taken up with considering ways to formalize the process of reflection. This is necessary, not just in the interest of those apparitions that are able to animate reflection, or for the sake of those allegations that are able to survive reflection, but in order to devise a regular methodology for articulating, bringing into balance with each other, and reasoning on the grounds of the various kinds of reflections that naturally occur, the apparitions that arise in the incidental context of experience plus the allegations that get expressed in the informal context of discussion. Later discussions will advance a particular approach to reflection, bringing together the work already begun in previous discussions of ''interpretive frameworks'' (IFs) and ''objective frameworks'' (OFs), and constructing a compound order or a hybrid species of framework for arranging, organizing, and supporting reflection. These tandem structures will be referred to as ''reflective interpretive frameworks'' (RIFs).

Before the orders of complexity that are involved in the construction of a RIF can be entertained, however, it is best to obtain a rudimentary understanding of just how the issues associated with reflection can in fact arise in ordinary and unformalized experience. Proceeding by this path will allow us to gain, along with a useful array of moderately concrete intuitions, a relatively stable basis for comprehending the nature of reflection. For all of these reasons, the rest of this initial discussion will content itself with a sample of the more obvious and even superficial properties of reflection as they develop out of casual and even cursory contexts of discussion, and as they make themselves available for expression in the terms and in the structures of a natural language medium.

====5.2.3. A Reflective Heuristic====

In a first attempt to state explicitly the principles by which reflection operates, it helps to notice a few of the tasks that reflection performs. In the process of doing this it is useful to keep this figure of speech, where the anthropomorphic ''reflection'' is interpreted in the figure of its personification, in other words, as a hypostatic reference that personifies the reflective faculty of an agent.

One of the things that reflection does is to look for common patterns as they appear in diverse materials. Another thing that reflection does is to look for variations in familiar and recognized patterns. These ideas lead to the statement of two aesthetic guidelines or heuristic suggestions as to how the process of reflection can be duly carried out:

:: Try to reduce the number of primitive notions.

:: Try to vary what has been held to be constant.

These are a couple of ''aesthetic imperatives'' or ''founding principles'' that I first noticed as underlying motives in the work of C.S. Peirce, informing the style of thinking that is found throughout his endeavors (Awbrey & Awbrey, 1989). It ought to be recognized that this pair of imperatives operate in antagonism or work in conflict with each other, each recommending a course that strives against the aims of the other. The circumstances of this opposition appear to suggest a mythological derivation for the faculty of reflection that is being personified in this figure, as if it were possible to inquire into the background of reflection so deeply as to reach that original pair of sibling rivals: Epimetheus, Defender of the Same; Prometheus, Sponsor of the Different.

Aesthetic slogans and practical maxims do not have to be consistent in all of the exact and universal ways that are required of logical principles, since their applications to each particular matter can be adjusted in a differential and a discriminating manner, taking into account the points of their pertinence, the qualities of their relevance, and the times of their salience. Nevertheless, the use of these heuristic principles can have a bearing on the practice of logic, especially when it comes to the forms of logical expression and argumentation that are available for use in a particular language, specialized calculus, or other formal system. Although one's initial formulations of logical reasoning, in the shapes that are seized on by fallible and finite creatures, can be as arbitrary and as idiosyntactic as particular persons and parochial paradigms are likely to make them, a dedicated and persistent application of these two heuristic rudiments, whether in team, in tandem, or in tournament with each other, is capable of leading in time to forms that subtilize and universalize, at the same time, the forms initially taken by thought.

====5.2.4. Either/Or : A Sense of Absence====

<br>
{| align="center" border="0" cellpadding="0" cellspacing="0" style="text-align:left; width:90%"
| colspan="2" | Che faro senza Euridice ?
|-
| colspan="2" | Dove andro senza il mio ben ?
|-
| colspan="2" | Che faro ? Dove andro ?
|-
| colspan="2" | Che faro senza il mio ben ?
|-
| colspan="2" |  
|-
| colspan="2" | What can I do, with Eurydice gone?
|-
| colspan="2" | And whither go, without my dearest love?
|-
| colspan="2" | What can I do? And whither go?
|-
| colspan="2" | What can I do without my dearest love?
|-
| width="50%" |  
| Gluck, ''Orfeo ed Euridice'', [Glu, 74]
|}
<br>

While I'm on the subject of imperatives and maxims, of how they often come in pairs that appear to strive both pro and con each other, and of how they are able to make a kind of sense, whether in conjunction or in alternation, without having to be logically consistent with one another, …

Absent a sense of what is good a creature is lost. And since this sense is only a feeling, a groping, a hint, an inkling, all in all a wandering shade, it can only be a sign of what is good, and a fallible sign at that. Does a ''sense'' of what's good ever contend with a sense of what's ''good''? If the mind is inclined to emphasize the turn of the phrase that it is bound to hold nearer to itself, namely, the mind's own “sense”, then it seems to indicate an aesthetic sensibility. If the mind is disposed to stress the part of speech that it places more dearly in association with its object, namely, the mind's own “good”, then it seems to take on an ethical intention. But this is just shadow play. The live question is: What is there in this state of disharmony that speaks to its absence?

====5.2.5. Apparent, Occasional, and Practical Necessity====

{| align="center" cellpadding="0" cellspacing="0" width="90%"
|
<p>In the present state of things a man abandoned to himself in the midst of other men from birth would be the most disfigured of all. Prejudices, authority, necessity, example, all the social institutions in which we find ourselves submerged would stifle nature in him and put nothing in its place. Nature there would be like a shrub that chance had caused to be born in the middle of a path and that the passers by soon cause to perish by bumping into it from all sides and bending it in every direction.</p>
|-
| align="right" | Rousseau, ''Emile, or On Education'', [Rou1, 37]
|}

I appear to be entering an apparent arena where it is apparently necessary to approach every appearance of an absolute concept, and so every apperception of an abstract idea, with an air of apprehension, to attend as assiduously as I am able to the factitious aspect of every concept approached, every fact asserted, every idea approximated, and every predicate applied, as each advances from a potentially articulate to a palpably artificial species of phenomenon, and thus to appreciate for myself the quality of an apparition that seems to affect everything that I apprehend, and also to apprise others of the attitude that I find I am advised to adopt, by attempting to append to every article of note an appraisal of the aura of evanescence and instability that appears all about it, to assign to every point of application a device to reflect the animate artifice that is pressured to suit its address to the subject, to attach to every point of articulation a reflex of the animating instinct that is appealed to for flexing the armament of its attack on the theme, to affix an array of verbal hedges all around it to serve as an appellate apparatus and to assure an attitude of appropriate reserve toward it.

As I enter this allegorical and apparitional context, I find that every attempt at articulate expression is affected with an array of afflictions. In this abode of seeming allegations, this abyss of teeming apparitions, every point I try to make is quickly surrounded by a brace of blooming confusions and rapidly swarmed over by a mass of buzzing diffusions.

Here, the qualifiers "alleged by", "alleged", "allegedly", in allegiance with the qualifiers "appears to", "apparent", "apparently", and arranged alongside all of their associated, derivative, and equivalent modifiers, can literally be distributed to any part of speech, any phrase of any sentence, or any phase of discourse that I can think to assemble or venture to articulate.

Here, the admonitions "alleged" and "apparent" can arguably be applied to any term, any premiss, and any argument that appears to enter an arena of discussion or that afterwards appears to arouse attention.

Here, the amplifications that appear almost able to assert themselves here, that array themselves at all available points of articulation and arraign as arrant adventures all attempts at advancing any appreciable amount beyond appearances and allegations, that avail themselves of all the available veils of allusion to the vanity of appearances, but only accomplish another order of obfuscation, that are able to accumulate in any account that attempts to appreciate all that allegedly appears in it, all the allied assonances of asinine alliterations and that is augmented in accord with all that is actually fit to print, ascends to an altitude of such arrogance that it assumes the ability to arrest all the associates of its nominal constants and of the point of overpowers its pronounced variables, as the latter are so typically represented by the deceptively percussive decussation of the variable name "X", appear to excise out of existence the very objects that it aims to mark in the forms of its syntax.

As I continue to pursue the problems that remain of interest to me here, I am led to increase the manifold of ways that are available to converge on each object, to insert a growing multitude of signs into the medium, and to introduce new points of articulation into the developing text. Each of these developments appears to arise in a natural fashion from the intentions that I bring to this work and as I bring them to bear on each object in view, as I search for a way to catch at least a fractured image of its more glaring aspects and as I strive to settle on a way to pin down at least a fragmentary inkling of its more striking features. If I aim to bring home this catch in the net of those few terms that I can fix in the forms of its sieve and fasten in the figures of its syntax, then I need to adjust the scope of the modifiers that are affixed from the frames of its paradigms and afforded by the folds of its inflections. But each new sign that I adduce, in the instant that it starts to afford a point of attachment, one that seems sufficient to suspend the orders of variation that appear to me, in that same moment it also occasions a point of departure, one that seems necessary to pursue through orders of variation that are barely hinted at in my present imagination, and each element of which promises to serve as a conceptual peg that whole new orders of conceivable changes can be pinned on.

As a result of these developments, an initially admirable attempt at clarification issues in a luminescent haze of modulations that affects every object in sight with a spectral host of modifiers, that glosses over the original indictment of every object of investigation, that limns every outline of a potential content with a dubious aura of charismatic nuances, that surrounds every figure with an array of apprehensions without quite arresting any detail of observation, and that obscures the interior features of every shape with a sheer but sketchy silhouette.

The reverberant perturbations that stem from each new predicate added to the account appear to interfere with the very mode of action or the very state of being that it attempts to delineate about its subject, where the numinous veils of adumbration that evolve about every object ascend to levels of amplitude that appears to collapse the very objects, acts, and facts that they began with the aim to connote, and finally, where the corresponding moment of coruscation that precipitate about every object finally seems to crush it beneath the weight of its own encrustation, to corrupt the sense of the original signs, to corrode the very object of their intention, and to scatter the remains of the object in a chorus of vacillations that detonate every intention to denote.

In this way it is possible to discover, if nothing else, a few of the ways that reflection can go astray. It appears to be clear from the drift that is evident in this particular style of investigation that there do indeed exist "modes of dispersive reflection" (MODR's) that "murder to dissect" the objects of their investigation. Under the influence of these styles, an initially admirable attempt at clarification appears to issue in a fog of glosses, a luminous haze of interpolations, a numinous cloud of nuances, a questionable array of qualifications, all of which threaten to blot out the original object of inquiry.

Try to imagine a religious icon, an object of veneration in many ways — to many it is a mystery in its own right, to many it is a wonder that they invest in it, and to some it is meaningful solely for the sake of what it represents to the understanding — now broken to pieces by the catastrophes of nature, the invading infidel, or the indifferent vandal, now scattered by hordes of iconoclasts, now gathered again and hoarded by troops of souvenir seekers, here and there exploited for use as raw materials in a host of new constructions, leaving the chips in the dust and the immovable chunks to fall where they may, or casting what's left in the waterway, now and again finding the pieces partly encrusted with barnacles and partly worn smooth by the actions of waves through time, until they barely bear a likeness to anything anyone ever bore in mind. What chance is there now of anyone re assembling the resemblance once more, of fitting the pieces together in the way they are meant to be?

This is the situation of humanity after the "Destruction of Babel", if one takes this fable as a metaphorical way of accounting for a current condition that is real enough, the evident lack of communication that prevails among the host of purported communities. This is not just a matter of linguistic diversity, but a question of fundamental beliefs. If the legend is interpreted with a due measure of discernment, then the downcast state of understanding that it purports to explain is not so much a matter of superficial differences in syntax as it is concerned with the deeper semantics of ultimate meanings.

Humanity survives in what appears to be an abject state, with regard to its fondest hopes and with respect to its projected ideals if not with reference to its literal origins, that is, in relation to the founding meanings that it appears to invest in all of its most basic intentions. Each portion of humanity takes the share of value that is disbursed to it as their collective host is dispersed into a rout of value systems. Taking each fragment of meaningful value as it receives it from this encounter and from this deliverance, it appears as if each fraction of humanity is deliberately restricted to a dissipative way of acting for ever after. Given the numerous "common koines" that are its lot, its loot, its boot, and its strapping, the abiding community of interests appears to be hobbled by the limited extent that these means afford it to purchase a meaningful expression for itself in the market of ideas. For all of these reasons and more, humanity appears to operate in what amounts to a degraded condition, at least, as it stacks up against the potential that humanity conceivably has for actualizing common ideals, implementing shared meanings, and realizing globally distributed values.

At this point one chances on a new source of power in symbols, signs that allow of being cobbled together in just the condition that they arrive and in just the mode that they derive from incidental sources, in all their partially eroded shapes and variously polished textures, without being forced to fit exactly in any form of pre arranged setting. In this way one is forced, as it were, by the ravages of time, and in a rather paradoxical fashion, to accept responsibility for an extra degree of flexibility and a novel measure of freedom. More than other types of signs, namely, in contrast to icons and indices that retain more or less independent arrays of formal and material connections, respectively, with their objects, symbols require the living actuality of an interpreter to read between the lines and to fill in the mortar between the more static building blocks of discourse.

There has to be a way to alleviate the tensions that are vaulted away in this suspension of signs without recanting the significance of the sense that their union is intended to intensify, a way to bear the overbearing turgidity of the result that is expressed in their coagulate composition without precipitating the full collapse of the subtended circumspection, a way to clarify the oppressive turbidity of the medium that is stirring to carry this tedium without embroiling the odium in the melody past all hope of its ultimate redemption, a way to distill the solutions that are synthesized in this distribution of moduli to termini without despoiling the tribute of the lesson that all are concerted to spell out in unison, a way to redress the grievances that remain disconsolate in their levies without leaving a pan of their balancing act to wait upon imponderables, a way to revisit the guarded commentaries that are billetted at, around, and through every locus that rises to a point of note in this discourse, a way to trim the edges of the hedges that trim each note of vacillation just as it begins to border on broaching any point that falls into view, a way to unify the manifold of apparent sensitivities to appearance that are likely to be displayed in the complexion and the countenance of this evolving expression and that ought to find their signs manifested in any sensible account of its conduct, its demeanor, its meaning, or its mien, but without numbering up to infinity the signs of apparent sensitivities that could express an appearance in the evolution of this expression and without numbing down into oblivion the sensitivity to apparent sensation that the expression of this evolution is adapted or designed to develop, and thus I am charged to go in search of the likeliest ways that appear.

{| align="center" cellpadding="0" cellspacing="0" width="90%"
|
<p>From time to time some brief and insubstantial reflection arose concerning the instability of the things of this world, whose image I saw in the surface of the water, but soon these fragile impressions gave way before the unchanging and ceaseless movement which lulled me and without any active effort on my part occupied me so completely that even when time and the habitual signal called me home I could hardly bring myself to go.</p>
|-
| align="right" | Rousseau, ''Reveries of the Solitary Walker'', [Rou2, 87]
|}

It is just when the whole phenomenal world threatens to dissolve into a texture of appearances that this show of nature reveals a secret kinship with the very texts of human discourse that interweave its pattern — that interlace its threads and interpret its suggestions, here supplying a neutral backing for the bolder displays of natural phenomena, there embroidering a new elaboration or a wholly superfluous decoration on the brocades that nature affords, that try to catch up the raveled clues of phenomena in the variegated weaves of their own local colors, that vie to sew up the knots of the world within their nets of rival description, and that venture to convert a patchwork of piecewise sensible events into a manifold of eventual sense, no matter how transient and tentative it proves itself to be — and thus the whole show of nature saves itself through the artifice of recognizing this kinship.

After a state of affairs advances this far, it is easy to see what an unwieldy nuisance it is to have to keep on inserting the reminders of appearance on the scene of awareness, what a cumbersome annoyance burdensome constant that might be inserted on apparitional reminders of appearance / to save the mementos of mortal frailty that no earthly agent has sufficient forbearance to attend on all the apparitions of appearance that are possible to spy on the scene of phenomena, and … Rather than trying to maintain a constant appeal to the apparent status of everthing the mind posits and instead of bothering to preserve a continual admonition to remember the frailties of mortal mentality, … When it becomes unwieldy to maintain a constant appeal to appearances and annoying to preserve a continual admonition of past the point of becoming a constant annoyance and a continual nuisance, one can settle on a where the resonances and the resplendences that are set up among the sights and the sounds of the signs that appear all seem to break up the very solidity of the objects themselves — but only seems to do so — it must be an aberration of the medium that makes them seem to shimmer.

Let this cadenza then be descanted: (1) that the necessities of a live interpretation can never be discounted, (2) that the interruptions of a faithful interpreter are never truly out of order, and (3) that to each (a) hint of allegation, (b) mark of appearance, or (c) note of attainder a dutiful interpreter is resoundingly enjoined to adjoin the rejoinders: "Says who?", "Sees who?", or "Cui bono?", respectively.

The rule is often invoked: To make a virtue out of necessity. In the present arena, already ruled over by modes of apparition and allegation, I am obliged to reconstitute apparent necessities as apparent virtues: (1) by examining the very form of rhetorical expediency that appears to be forced on me at this point, (2) by considering the practical lessons that can be drawn from the lines of its force, and (3) by contemplating the possibility that a genuinely valid form of logical argument can in fact be extracted from the raw materials of this and similar examples.

====5.2.6. Approaches, Aspects, Exposures, Fronts====

A large part of this project is devoted to the construction of various frameworks, as objects that are intended to satisfy a set of abstract requirements or partial specifications.

Over and above the specialized properties that go toward distinguishing the structural and the functional approaches from each other, and apart from the levels of detail that go to make up their particular instances, it is useful to consider the common form of activity that appears to be involved in both approaches, and thus to abstract the form of a “front”. In general, a “front” is an abstract form of organization that appears to embody itself or manages to realize itself in a concrete mass of material activities, and thus to constellate a pattern of action in space and time. This explains how the image of a “front” is relevant to various ways of approaching an object, and thus it indicates the sense of the metaphor, as far as it goes. But when it comes to the kind of a “front” that finds itself configured in a particular way of approaching the construction of an intended object, then there are more specific features that remain to be described.

As a form of activity that possesses a definite direction and perhaps even a deliberate purpose, a front marks the initial organization of an otherwise confused mass of material actions and momentary transits into a moderately coherent movement toward a common end or a shared goal. When a front is regarded as a partial envisioning of its intended object, a prospective view of the possibilities that are being afforded for its construction, and a proximal approach to many critical questions about the object, for instance, whether and how an object that satisfies the intended description can be constructed, then this front is clearly seen to constitute a form of inquiry in its own right.

Regarded as a form of inquiry, a front arrays itself into a gradual succession of agencies, faculties, or processes. Broadly taken, these divide into two parts, which can be personified in the following terms:

# The “van” exposes the generative ideas that come to the fore in shaping a front. In this role, the van is exposed to a host of material instances that it is forced to face with some ambivalence, since they afford not only a field of real opportunities for the advance of the front but also a range of obstructive challenges to its continued viability. The duty of agents in the van is to try to enunciate as clearly as possible the principles that determine the constitution of the front, the virtues of which ideas are in all likelihood responsible for inspiring their allegiance to this front. Actions that belong to the van, in regard to the ways that their logical arrangements, spatial placements, and temporal successions can be put in relationship to each other, are typically found to work best if they can (a) keep to the leading edge of the front, (b) stay as far as possible ahead of the game, (c) finish their part of the work before the main body of activities in the front gets going to cloud the issue, and (d) vest their results on the crest of the wave, where they are the easiest to find in a pinch.
# The “ruck” collects the supporting activities of the front that (a) stand behind its gradual advance, (b) contribute to its incremental development, and (c) maintain the continuity of its automatic functions.

As forms of partial and proximal approach, the structural and functional fronts of inquiry, considered in connection with the trains of supporting activity that stand behind their gradual advance, and taken in light of the waves of successive investigation that are bound to follow in their causal wakes, are constitutionally required to interrogate each other's prerogatives and even prospectively to cover the selfsame jurisdiction. Of course, it is almost inevitable that the persistent advances of these independently principled inquiries will eventually run across each other, since they criss-cross the same regions of concern over and over again. Over time, as these contrasting frontal systems come to meet and start to pass through each other, no matter whether they find themselves in one common accord or whether they are found at cross purposes to each other, no matter whether it is a mutual facilitation or a counterposed reluctance and resistance that their regimes of effrontery induce in one another, they are most likely determined to continue with their separate progressions until, sooner or later, they intersect each other at every point of interest under survey, and accordingly transect their common space in a way that yields a coordinate system for the RIF as a whole.

In the previous paragraph I tried to give a graphic description of how a coordinate system for an area of discussion can arise from conceptual considerations and develop through what are “principally” logical forms of analysis. If the whole development seems a bit obtuse in the beginning, and remains oblique until the very end, when it finally becomes obvious what is already transpiring, then that is just the way it often occurs. It frequently happens that one person, working at one time, presents a formally defined domain, as it appears from one point of view, and then another person, or the same person working at another time, presents a formally different domain, or one that appears from another perspective, and only a bit late does it occur to anyone that the underlying domains referred to are partially the same, or perhaps wholly coincident spaces of objects. Only then, as if an afterthought to this step of synthesis, does it become abundantly clear that inferences comprising a whole new level of implications can derive from the superimposition of the various analytic frameworks, that is, from the fact that these different kinds of properties apply to each and every object in the composite view.

====5.2.7. Synthetic A Priori Truths====

{| align="center" cellpadding="0" cellspacing="0" width="90%"
|
<p>Nature, we are told, is only habit. What does that mean? Are there not habits contracted only by force which never do stifle nature? Such, for example, is the habit of the plants whose vertical direction is interfered with. The plant, set free, keeps the inclination it was forced to take. But the sap has not as a result changed its original direction; and if the plant continues to grow, its new growth resumes the vertical direction.</p>
|-
| align="right" | Rousseau, ''Emile, or On Education'', [Rou1, 39]
|}

There is a particular line of thinking, incidental to this construction, that is useful to draw out and to develop for the bearing it has and the perspective it gives on a long standing puzzle, namely, the question of ''synthetic a priori'' truths. These are the kinds of truths that usually require considerable efforts of discovery and invention just to realize the truth of, and yet that are commonly felt after the fact to have always been completely destined, evident, foregone, and necessary. It is a matter of controversy in some circles whether truths of this order really exist, or, if truths of any such character do exist, then whether they really fall under the exact terms of the given description. But the “thrill of discovery” that marks their actual experience is real enough in practice, in logic, in mathematics, and in other formal studies, where even the most purely deductive conclusions do not seem so wholly foregone in their ways that one can afford to forgo the joys and the trials of their proofs. Thus, it follows that the persistence of this experience, as felt, needs to be appreciated in itself, no matter what scheme of theory one selects to explain it or else to explain it away.

Let me abstract, for the moment, from the structural and functional axes of the current construction, describing any development along analogous lines as a process of “coordination”, and referring to the form of what results as an “axial coordinating system” (ACS), with axes to be named. What I want to highlight here is the typical progression of experiences that an agent passes through in the process of developing any instance in the form of an ACS, starting from (1) the performance of the separate analyses, working through (2) the synthesis of their combined results, and finally moving on to (3) the derivation of the novel implications on a freshly refurbished stage of inference. Making these abstractions not only yields a clearer view of the relevant structures involved in the process but it also develops a generalized picture of the coordination project that is much more flexible in the present use and increasingly adaptable to future applications.

With an eye to the generic features of my paradigmatic coordination process, and with the abstract idea of an ACS in hand, let me return to the immediate application. As I indicated, one of the benefits that I hope to extract from a study of this form of emergent coordination, taking in its process and its result, is to clarify the problem of SAP propositions in scientific reasoning, and thus to derive a measure of insight into the forms of sapience that depend on them. Unless the ostensibly fruitful nature of SAP propositions evaporates into thin air with their exposure to the heat of reflection and unless their status as advertised entirely boils away with the resolution of their problematic features, then their analysis can help to rationalize the role of SAP propositions in scientific knowledge, as they arise through inquiry, as they enter into one's compendia of belief or knowledge, and as they contribute to the skills whereby one builds an overall grasp of truth.

I think that the sequence of experiential realizations that I depicted in my reconstruction of a developing faculty for coordination, no matter whether it is regarded as a process or as a result, not only can account for many of the paradoxical features of SAP truths but also can explain the impressions that typically occur in the process of achieving them. First, it explains why the full recognition of the supposedly ''a priori'' status does not occur until after the synthetic step is finished, that is, until after the separate analytic perspectives are integrated and after the object domain is reconstituted under their freshly combined views. Further, it explains why this wholly reconstructive and retrospective vision, but one that constitutes a newly coherent mode of perception and a slightly elevated perspective, then appears to look on what was never anything but a pre-established domain. Finally, it explains why the appearance or the apparition of anything non analytic contributing to the mix, the very impression that there were ever any truths beyond the manifestly deductive variety, momentarily fades out of sight in the evanescent manner of a transient illusion, at least, until the need of some exigency calls once again for the power of a synthetic capacity.

In any case, the effects that one typically experiences in going through these steps of coordination and in bringing about the instrumentality of the corresponding ACS are remarkably similar in many of their most puzzling features to those that are involved in the experiential process and the moment of realization that one comes to expect in the discovery of what is commonly called an item of ''synthetic a priori'' knowledge.

It is at this point that one is forced to distinguish the order of being from the order of knowing, and once again, within the order of knowing to distinguish the order of discovery from the order of justification. If a recursive analysis leads one only to make explicit an assumption that one has implicitly taken for granted up until that time, then, no matter which way one chooses to proceed from that point, calling that assumption into question or continuing to believe it, the process of explication itself still reflects a measure of progress.

====5.2.8. Priorisms of Normative Sciences====

Let me start with some questions that continue to puzzle me, in spite of having spent a considerable spell of time pursuing their answers, and not for a lack of listening to the opinions expressed on various sides. I first present these questions as independently of the current context as I possibly can, and then I return to justify their relevance to the present inquiry.

The questions that concern me concern the relationships of identity, necessity, or sufficiency that can be found to hold among three classes of properties or qualities that can be attributed to or possessed by an agent, and conceivably passed from one agent to another. The relevant classes of properties or possessions can be schematized as follows:

: '''Teachings''' — learnings, lessons, disciplines, doctrines, dogmas, or things that can be taught and learned, transmitted and received.

: '''Understandings''' — articles of knowledge, items of comprehension, bits of potential wisdom that form the possession of knowledge.

: '''Virtues''' — aspects of accomplished performance, attainments of demonstrated achievement, qualities of accomplishment, completion, excellence, mastery, maturity, or relative perfection, ''grits'' or integrities that form the exercise of art, justice, and wisdom.

The category of ''teachings'', as a whole, can be analyzed and divided into two subcategories:

# There are ''disciplines'', which involve elements of action, behavior, conduct, and instrumental practice in their realization, and thus take on a fully evaluative, normative, prescriptive, or procedural character.
# There are ''doctrines'', which are properly restricted to realms of attitude, belief, conjecture, knowledge, and speculative theory, and thus take on a purely descriptive, factual, logical, or declarative character.

The category of ''virtues'' can be subjected to a parallel analysis, but here it is not so much the domain as a whole that gets divided into two subcategories as that each virtue gets viewed in two alternative lights:

# With regard to its qualities of action, execution, and performance.
# As it affects its properties of competence, knowledge, and selection.

The reason for this difference in the sense of the analysis that applies to each is that it is one of the better parts of virtue to bring about a synthesis between action and knowledge in the very actuality of the virtue itself.

At this point one arrives at the general question:

: What is the logical relation of virtues to teachings?

In particular:

<ol style="list-style-type:lower-latin">

<li>Does one category necesarily imply the other?</li>

<li>Are the categories mutually exclusive?</li>

<li>Do they form independent categories?</li>

</ol>

Are virtues the species and teachings the genus, or perhaps vice versa? Or do virtues and teachings form domains that are essentially distinct? Whether one is a species of the other or whether the two are essentially different, what are the features that apparently distinguish the one from the other?

Let me begin by assuming a situation that is plausibly general enough, that some virtues can be taught, symbolized as <math>V \land T\!</math>, and that some cannot, symbolized as <math>V \land \lnot T\!</math>. I am not trying to say yet whether both kinds of cases actually occur, but merely wish to consider what follows from the likely alternatives. Then the question as to what distinguishes virtues from teachings has two senses:

# Among virtues that are special cases of teachings, <math>V \land T\!</math>, the features that distinguish virtues from teachings are known as ''specific differences''. These qualities serve to mark out virtues for special consideration from amidst the common herd of teachings and tend to distinguish the more exemplary species of virtues from the more inclusive genus of teachings.
# Among virtues that transcend the realm of teachings, <math>V \land \lnot T\!</math>, the features that distinguish virtues from teachings are aptly called ''exclusionary exemptions''. These properties place the reach of virtues beyond the grasp of what is attainable through any order of teachings and serve to remove the orbit of virtues a discrete pace from the general run of teachings.

In either case it can always be said, though without contributing anything of substance to the understanding of the problem, that it is their very property of ''virtuosity'' or their very quality of ''excellence'' that distinguishes the virtues from the teachings, whether this character appears to do nothing but add specificity to what can be actualized through learning alone, or solely through teaching, or whether it requires a nature that transcends the level of what can be achieved through any learning or teaching at all. But this sort of answer only begs the question. The real question is whether this mark is apparent or real, and how it ought to be analyzed and construed.

Assuming a tentative understanding of the categories that I indicated in the above terms, the questions that I am worried about are these:

<ol style="list-style-type:decimal">

<li>Did Socrates assert or believe that virtue can be taught, or not?<br>In symbols, did he assert or believe that <math>V \Rightarrow T\!</math>, or not?</li>

<li>Did he think that:</li>

<ol style="list-style-type:lower-latin">

<li>knowledge is virtue, in the sense that <math>U \Rightarrow V\!</math>?</li>

<li>virtue is knowledge, in the sense that <math>U \Leftarrow V\!</math>?</li>

<li>knowledge is virtue, in the sense that <math>U \Leftrightarrow V\!</math>?</li></ol>

<li>Did he teach or try to teach that knowledge can be taught?<br>In symbols, did he teach or try to teach that <math>U \Rightarrow T\!</math>?</li></ol>

My current understanding of the record that is given to us in Plato's Socratic Dialogues can be summarized as follows:

At one point Socrates seems to assume the rule that knowledge can be taught, <math>U \Rightarrow T\!</math>, but simply in order to pursue the case that virtue is knowledge, <math>V \Rightarrow U\!</math>, toward the provisional conclusion that virtue can be taught, <math>V \Rightarrow T\!</math>. This seems straightforward enough, if it were not for the good chance that all of this reasoning is taking place under the logical aegis of an indirect argument, a reduction to absurdity, designed to show just the opposite of what it has assumed for the sake of initiating the argument. The issue is further clouded by the circumstance that the full context of the argument most likely extends over several Dialogues, not all of which survive, and the intended order of which remains in question.

At other points Socrates appears to claim that knowledge and virtue are neither learned nor taught, in the strictest senses of these words, but can only be ''divined'', ''recollected'', or ''remembered'', that is, recalled, recognized, or reconstituted from the original acquaintance that a soul, being immortal, already has with the real idea or the essential form of each thing in itself. Still, this leaves open the possibility that one person can help another to guess a truth or to recall what both of them already share in knowing, as if locked away in one or another partially obscured or temporarily forgotten part of their inmost being. And it is just this freer interpretation of ''learning'' and ''teaching'', whereby one agent catalyzes not catechizes another, that a liberal imagination would yet come to call ''education''. Therefore, the real issue at stake, both with regard to the aim and as it comes down to the end of this inquiry, is not so much whether knowledge and virtue can be learned and taught as what kind of education is apt to achieve their actualization in the individual and is fit to maintain their realization in the community.

How are these riddles from the origins of intellectual history, whether one finds them far or near and whether one views it as bright or dim, relevant to the present inquiry? There are a number of reasons why I am paying such close attention to these ancient and apparently distant concerns. The classical question as to what virtues are teachable is resurrected in the modern question, material to the present inquiry, as to what functions are computable, indeed, most strikingly in regard to the formal structures that each question engenders. Along with a related question about the nature of the true philosopher, as one hopes to distinguish it from the most sophisticated imitations, all of which is echoed on the present scene in the guise of Turing's test for a humane intelligence, this body of riddles inspires the corpus of most work in artificial intelligence, if not the cognitive and the computer sciences at large.

{| align="center" cellpadding="0" cellspacing="0" width="90%"
|
<p>Reason alone teaches us to know good and bad. Conscience, which makes us love the former and hate the latter, although independent of reason, cannot therefore be developed without it. Before the age of reason we do good and bad without knowing it, and there is no morality in our actions, although there sometimes is in the sentiment of other's actions which have a relation to us.</p>
|-
| align="right" | Rousseau, ''Emile, or On Education'', [Rou_1, 67].
|}

Aesthetics, ethics, and logic are categorized as ''normative sciences'' because they pursue knowledge about the ways that things ought to be, their objects being beauty, justice, and truth, respectively. It is generally appreciated that there are intricate patterns of deep and subtle interrelationships that exist among these subjects, and among their objects, but different people seem to intuit different patterns, perhaps at different times. At least, it seems that they must be seeing different patterns of interrelation from the different ways that they find to enact their insights and intuitions in customs, methods, and practices. In particular, one's conception of science, indeed, one's whole approach to life, is determined by the ''priorism'' or the ''precedence ordering'' that one senses among these normative subjects and employs to order their normative objects. This Section considers a sample of the choices that people typically make in building up a personal or a cultural ''priorism of normative sciences'' (PONS).

For example, on the modern scene, among people trained to sport all of the modern fashions of scientific reasoning, it is almost a reflex of their modern identities to echo in their doctrines, if not always to follow in their disciplines, those ancients who taught that "knowledge is virtue". This means that to know the truth about anything is to know how to act rightly in regard to it, but more yet, to be compelled to act that way. It is usually understood that this maxim posits a relation between the otherwise independent realms of knowledge and action, where knowledge resides in domains of signs and ideas, and where action presides over domains of objects, states of being, and their changes through time. However, it is not so frequently remembered that this connection cuts both ways, causing the evidence of virtue as exercised in practice to reflect on the presumption of knowledge as possessed in theory, where each defect of virtue necessarily reflects a defect of knowledge.

In other words, converting the rule through its contrapositive yields the equivalent proposition "evil is ignorance", making every fault of conduct traceable to a fault of knowledge. Everyone knows the typical objection to this claim, saying that one often knows better than to do a certain thing while going ahead and doing it anyway, but the axiom is meant to be taken as a new definition of knowledge, ruling overall that if one really, really knows better, then one simply does not do it, by virtue of the definition. This sort of reasoning issues in the setting of priorities, putting knowledge before virtue, theory before practice, beauty and justice after truth, or reason itself before rhyme and right.

It is not that reason sees any reason to disparage the just deserts that it places after or intends to diminish the gratifications that it defers. Indeed, it aims to give these latter values a place of honor by placing them more in the direction of its aims, and it thinks that it can take them up in this order without risking a consequential loss of geniality. According to this rationale, it is the first order of business to know what is true, while purely an afterthought to do what is good.

It is not too surprising that reason assigns a priority to itself in its own lists of aims, goods, values, and virtues, but this only renders its bias, its favor, its preference, and its prejudice all the more evident. And since the patent favoritism that reason displays is itself a reason of the most aesthetic kind, it thus knocks itself out of its first place ranking, the ranking that reason assumes for itself in the first place, by dint of the prerogative that it exercises and in view of the category of excuse that it uses, from then on deferring to beauty, to happiness, or to pleasure, and all that is admirable in and of itself, or desired for its own sake. This self-demotion of reason is one of the unintended consequences of its own argumentation, that leads it down the garden path to a self-deprecation. It is an immediate corollary of reason trying to distinguish itself from the other goods, granting to itself an initially arbitrary distinction, and then reflecting on the unjustified presumption of this self-devotion. This condition, that reason suffers and that reason endures, is one that continues through all of the rest of its argumentations, that is, unless it can find a better reason than the one it gives itself to begin, or until such time as it can show that all good reasons are one and the same.

So the maxim "knowlege is virtue", in its modern interpretation, at least, leads to the following results. It makes just action, right behavior, and virtuous conduct not merely one among many practical tests but the only available criterion of knowledge, reason, and truth. Sufficient criterion? If a conceptual rule is the only available test of some property, then it must be an essential criterion of that property. This conceives the essence of knowledge to lie in a conception of action. This maxim can be taken, by way of its contrapositive, as a pragmatic principle, positing a rule to the effect that any defect of virtue reflects a defect of knowledge. This makes truth the ''sine qua non'' of justice, right action, or virtuous conduct, that is, it makes reason the ''without which not'' of morality. Since virtuous conduct is distinguished as that action which leads to what we call ''beauty'', ''beatitude'', or ''happiness'', by any other name just that which is admirable in and of itself, desired for its own sake, or sought as an end in itself, whether it is only in the conduct itself or in a distinct product that the beauty is held to abide, this makes logic the sublimest art. (Why be logical? Because it pleases me to be logical.)

{| align="center" cellpadding="0" cellspacing="0" width="90%"
| <p>It depends on what the meaning of the word "is" is.</p>
|-
| align="right" | President William Jefferson Clinton, August ?, 1998.
|}

Of course, there is much that is open to interpretation about the maxim "knowledge is virtue". In particular, does the copula "is" represent a necessary implication (<math>\Rightarrow\!</math>), a sufficient reduction ("is only", <math>\Leftarrow\!</math>), or a necessary and sufficient identification (<math>\Leftrightarrow\!</math>)?

====5.2.9. Principle of Rational Action====

{| align="center" cellpadding="0" cellspacing="0" width="90%"
|
<p>Knowledge systems are just another level within this same hierarchy, another way to describe a system. … The knowledge level abstracts completely from the internal processing and the internal representation. Thus, all that is left is the content of the representations and the goals toward which that content will be used. As a level, it has a medium, namely, knowledge. It has a law of behavior, namely, if the system wants to attain goal ''G'' and knows that to do act ''A'' will lead to attaining ''G'', then it will do ''A''. This law is a simple form of rationality — that an agent will operate in its own best interests according to what it knows.</p>
|-
| align="right" | Allen Newell, ''Unified Theories of Cognition'', [New, 48–49].
|}

How does this ancient issue, concerning the relation of reason, to action, to the good that is overall desired or intended, transform itself through the medium of intellectual history onto the modern scene? In particular, what bearing does it have on the subjects of artificial intelligence and systems theory, and on the object of the present inquiry? As it turns out, in classical cybernetics and in systems theory, and especially in the parts of AI and cognitive science that have to do with heuristic reasoning, the transformations of the problem have tarried so long in the vicinity of a singular triviality that the original form of the question is nearly unmistakable in every modern version. The transposition of the theme <math>(\text{Reason}, \text{Action}, \text{Good})\!</math> into the mode of <math>(\text{Intelligence}, \text{Operation}, \text{Goal})\!</math> can make for an interesting variation, but it does not alter the given state of accord or discord among its elements and does nothing to turn the lock into its key.

How do these questions bear on the present inquiry? Suppose that one is trying to understand something like an agency of life, a capacity for inquiry, a faculty of intelligence, or a power of learning and reasoning. For starters, ''something like'' is a little vague, so let me suggest calling the target class of agencies, capacities, faculties, or powers that most hold my interest here by the name of ''virtues'', thereby invoking as an offstage direction the classical concepts of ''anima'' and ''arete'' that seem to prompt them all. What all of these virtues have in common is their appearance, whether it strikes one on first impression or only develops in one's appreciation through a continuing acquaintance over time, of transcending or rising infinitely far beyond all of one's attempts to construct them from or reduce them to the sorts of instrumentalities that are much more basic, familiar, mundane, ordinary, simpler, in short, the kinds of abilities that one already understands well enough and is granted to have well under one's command or control. For convenience, I dub this class of abilities, that a particular agent has a thorough understanding of and a complete competency in, as the ''resources'' of that agent.

The language of ''virtues'' and ''resources'' gives me a way to express the main problem of this inquiry, indeed, the overriding challenge that is engaged in every round of effective analysis and functional modeling. I emphasized the ''apparent transcendence'' of virtues because the hope is often precisely that this appearance will turn out to be false, not that the virtue is false in any of the properties that it seems to have, but that the awesome aspect of its unapproachability can be diminished, and that a way opens up to acquire this virtue by means of the kinds of gradual steps that are available to a fallible and a finite agent.

If I had my own choice in the matter I would proceed by using the words ''knowledge'' and ''understanding'' as synonyms, deploying them in ways that make them refer to one and the same resource, roughly corresponding the Greek ''episteme'', and thus guaranteeing that the faculty they denote is teachable. But others use these terms in ways that make one or the other of them suggest a transcendental aptitude more akin to ''wisdom'', and thus amounting to a virtue extending in the intellectual direction whose very teachability is open to question. Keeping this variety of senses and understandings in mind, it is advisable to be flexible in one's usage.

Virtue involves, not just knowing what is the case and knowing what can be done in each case, but knowing how to do each thing that can be done, knowing which is the best to do in a given case, and finally, having the willingness to do it.

What are the features that are really at stake in the examination of these admittedly paradigmatic and even parabolic examples? There are two ways that virtues appear to transcend the limitations of effectively finite and empirically rational resources and thus appear to distinguish themselves from teachings and understandings, that is, from the orders of disciplined conduct and doctrinal knowledge that bind themselves too severely to the merely mechanical ritual and the purely rote recitation.

# In their qualitative aspect, virtues appear to combine characters of act and will that appear to be lacking in the simple imputations of knowledge alone. In particular, virtues appear to display qualities of persistent action, efficient volition, the will to actually do the right thing, and the willingness to keep on doing the right thing on each occasion that arises. Thus, virtues appear to possess a live performance value that is not guaranteed by simply knowing the right thing to do and to say, indeed, they appear to have a unique and irreproducible mix of qualities that goes beyond the facts circumscribed by any name and thus that goes missing from the ordinary interpretation of its meaning.
# In their quantitative aspect, virtues appear to be infinitely far beyond the grasp of discrete, finite, and even rational resources.

====5.2.10. The Pragmatic Cosmos====

This Section outlines the general idea of a ''priorism of normative sciences'' (PONS) and it presents the particular PONS that I will refer to as the ''pragmatic cosmos''. This is the precedence ordering for the normative sciences that best accords with the pragmatic approach to inquiry, incidentally framing and introducing the order of normative sciences that I plan to deploy throughout the rest of this work. From this point on, whenever I mention a PONS without further qualification, it will always be one or another version of a pragmatic PONS that I mean to invoke, all the while taking into consideration the circumstance that its underlying theme still leaves a lot of room for variation in the carrying out of its live interpretation.

Roughly speaking, in regard to the forms of human aspiration that are exercised in normative practices and studied in the normative sciences, the study of states or things that satisfy agents is called ''aesthetics'', the study of actions that lead agents toward these goals or these goods is called ''ethics'', and the study of signs that indicate these actions is called ''logic''. Understood this way, logic involves the enumeration and the analysis of signs with regard to their ''truth'', a property that only makes sense in the light of the actions that are indicated and the objects that are desired. In other words, logic evaluates signs with regard to the trustworthiness of the actions that they indicate, and this means with respect to the utility that these indications exhibit in a mediate relationship to their objects. As an appreciative study, logic prizes the properties of signs that allow them to collect the scattered actions of agents into coherent forms of conduct and that permit them to indicate the general courses of conduct that are most likely to lead agents toward their objects.

From this "pragmatic" point of view, logic is a special case of ethics, one that is concerned with the conduct of signs, and ethics is a special case of aesthetics, one that is interested in the good of actual conduct. Another way to approach this perspective is to start with the ''good'' of anything and to work back through the maze of actions and indications that lead to it. An action that leads to the good is a good action, and this puts the questions of ethics among the questions of aesthetics, as the ones that contemplate the goods of actions. A sign that indicates a good action, that shows a good way to act, is a good sign, and this puts the domain of logic squarely within the domain of aesthetics. Moreover, thinking is a sign process that moves from signs to interpretant signs, and this makes thinking a special kind of action. In sum, the questions that logic takes up in its critique of good signs and good thinking are properly seen as special cases of aesthetic and ethical considerations.

The circumstance that the domain of logic is set within the domain of ethics, which is further set within the domain of aesthetics, does not keep each realm from rising to such a height in another dimension that each keeps a watch over all of the domains that it is set within. In sum, the image is that of three cylinders standing on their concentric bases, telescopically extending to a succession of heights, with the narrowest the highest and the broadest the lowest, rising to the contemplation of the point that virtually completes their perspective, just as if wholly sheltered by the envelope of the cone that they jointly support, no matter what its ultimate case may be, whether imaginary or real, rational or transcendental.

Logic has a monitory function with respect to ethics and aesthetics, while ethics has a monitory function solely with respect to aesthetics. By way of definition, a ''monitory function'' is a duty, a role, or a task that one discipline has to watch over the practice of another discipline, checking the feasibility of its intentions and its proposed operations, evaluating the conformity of its performed operations to its intentions, and, when called for, reforming the faith, the feasance, or the fidelity of its acts in accord with its aims. A definite attitude and particular perspective are prerequisites for an agent to exercise a monitory role with any hope or measure of success. The necessary station arises from the observation that not all things are possible, at least, not at once, and especially that not all ends are achievable by a fallible creature within a finite creation. Accordingly, the agent of a monitory faculty needs to help the agency that is involved in the effort or the endeavor it monitors to observe the due limits of its proper arena, the higher considerations, and the inherent constraints that force a fallible and finite agent to choose among the available truths, acts, and aims.

To recapitulate the pragmatic ''priorism of normative sciences'' (PONS):

Logic, ethics, and aesthetics, in that order, cannot succeed in any of their aims, whether they turn to contemplating the natures of the true, the just, and the beautiful, respectively, for their own sakes, whether they turn to speculating on the certificates, the semblances, or the more species tokens of these goods, as they might be utilized toward a divergent conception of their values, or whether they convert from the one forum to the other market, and back again, in an endless series of exchanges, that is, unless their prospective agents possess the initial capital that can only be supplied by competencies at the corresponding intellectual virtues, and until they are willing to risk the stakes of adequately generous overhead investments, on orders that are demanded to fund the performance of the associated practical disciplines, namely, those that are appropriate to the good of signs, the good of acts, and the good of aims in themselves. In sum, the domains and the disciplines of logic, ethics, and aesthetics, in that order, are placed so aptly in regard to one another that each one waits on the order of its watch and each one maintains its own proper monitory function with respect to all of the ones that follow on after it.

Why do things have to be this way? Why is it necessary to impose a PONS, much less a pragmatic PONS, on the array of goods and quests? If everyone who reflects on the issue for a sufficient spell of time seems to agree that the Beautiful, the Just, and the True are one and the same in the End, then why is any PONS necessary? Its necessity is apparently relative to a certain contingency affecting the typical agent, namely, the contingency of being a fallible and finite creature. Perhaps from a ''God's Eye View'' (GEV), Beauty, Justice, and Truth all amount to a single Good, the only Good there is. But the imperfect creature is not given this view as its realized actuality and cannot contain its vision within the ''point of view'' (POV) that is proper to it. Even if it sees the possibility of this unity, it cannot actualize what it sees at once, at best being driven to work toward its realization measure by measure, and that is only if the agent is capable of reason and reflection at all.

The imperfect agent lives in a world of seeming beauty, seeming justice, and seeming truth. Fortunately, the symmetry of this seeming insipidity can break up in relation to itself, and with the loss of the objective world's equipoise and indifference goes all the equanimity and most of the insouciance of the agent in question. It happens like this: Among the number of apparent goods and amid the manifold of good appearances, one soon discovers that not all seeming goods are alike. Seeming beauty is the most seemly and the least deceptive, since it does not vitiate its own intention in merely seeming to achieve it, and does not destroy what it reaches for in merely seeming to grasp it.

Monitory functions, as a rule, tend to shade off in extreme directions, on the one hand becoming a bit too prescriptive before the act, whether the hopeful effects are hortatory or prohibitory, and on the other hand becoming much too reactionary after the fact, whether the tardy effects are exculpatory or recriminatory. In the midst of these extremes, that is, within the scheme of monitory functions at large, it is possible to distinguish subtler variations in the nuances of their action that work toward the accomplishment the same general purpose, but that achieve it with a form of such gentle urging all throughout the continuing process of gaining a good, that affect a promise of such laudatory rewards, and that afford an array of incidental senses of such ongoing satisfaction, even before, while, and after the aimed for good is effected, that this class of moderate measures is aptly known as ''advisory functions'' (AFs).

In the process of noticing what is necessary and what is impossible, and in distinguishing itself from the general run of monitory functions, an AF is able to adapt itself to get a better grip on what is possible, to the point that it is eventually able to make constructive suggestions to the agent that it monitors, and thus to give advice that is both apt and applicable, positive and practical, or usable and useful. If this is beginning to sound familiar, then it is not entirely an accident. As I see it, it is from these very grounds that the facility for ''abductive simile'' or the faculty of ''abductive synthesis'' (AS) first arises, to wit, just on the horizon of monitory observation and just on the advent of advisory contemplation that an agent of inquiry, learning, and reasoning first acquires the ''quasi'' ability to regard one thing just as if it were construed to be another and to consider each thing just inasmuch as it haps to be like another.

In the abode of the monitor I thus discover the first clues I can grasp as to how the ''abductive bearing'' (AB) of hypothetical reasoning can be bound together from the primitive elements of the most uncertain states that the mind can ever know. To my way of thinking, this derivation of ABs from the general conduct of monitory duties and the specific ethos of advisory roles, all as pursuant to the PONS, seems to strike a chord with the heart of wonder beating at the core of every agent of inquiry, and accordingly to fashion an answer to the central query, in the words of William Shakespeare: "Where is fancy bred?" Beyond the responsibility to continue driving the cycle of inquiry and to keep on circulating the fresh communication of provisional answers, this form of speculation on the origin of the AB points out at least one way whence these faculties of guessing widely but guessing well can lead me from the conditions of amazement, bewilderment, and consternation that the start of an inquiry all but constantly finds me in.

The anchoring or the inauguration of an ''abductive bearing'' (AB) within the operations of an ''advisory function'' (AF), and the ensconcement or the installation of this positively constructive advisory, in its turn,within the office of an irreducibly negative monitory function, one that watches over the active, aesthetic, and affective aspects of experience with an eye to the circumstance that not all goods can be actualized at once — this array of inferences from the apical structure of the PONS ought to suffice to remind each agent of inquiry of how it all hinges on the affective values that one feels and the effective acts that one does.

In principle, therefore, logic assumes a purely ancillary role in regard to the ethics of active conduct and the aesthetics of affective values. On balance, however, logic can achieve heights of abstraction, points of perspective, and summits of reflection that are otherwise unavailable to a mind embroiled in the tangle of its continuing actions and immersed in the flow of its current passions. By rising above this plain immersion in the dementias swept out by action and passion, logic can acquire the status of a handle, something an agent can use in its situation to avoid being swept along with the tide of affairs, something that keeps it from being swept up with all that the times press on it to sweep out of mind. By means of this instrument, logic affords the mind an ability to survey the passing scene in ways that it cannot hope to imagine while engaged in the engrossing business of keeping its gnosis to the grindstone, and so it becomes apt to adopt the attitude that it needs in order to become capable of reflecting on its very own actions, affects, and axioms.

====5.2.11. Reflective Interpretive Frameworks====

<br>

{| align="center" border="0" cellpadding="0" cellspacing="0" style="text-align:left; width:90%"
| colspan="2" | Tell me, good Brutus, can you see your face?
|-
| colspan="2" |  
|-
| colspan="2" | No, Cassius, for the eye sees not itself
|-
| colspan="2" | But by reflection, by some other things.
|-
| colspan="2" |  
|-
| colspan="2" | 'Tis just;
|-
| colspan="2" | And it is very much lamented, Brutus,
|-
| colspan="2" | That you have no such mirrors as will turn
|-
| colspan="2" | Your hidden worthiness into your eye,
|-
| colspan="2" | That you might see your shadow. …
|-
| colspan="2" |  
|-
| colspan="2" | Into what dangers would you lead me, Cassius,
|-
| colspan="2" | That you would have me seek into myself
|-
| colspan="2" | For that which is not in me?
|-
| colspan="2" |  
|-
| colspan="2" | Therefor, good Brutus, be prepared to hear.
|-
| colspan="2" | And since you know you cannot see yourself
|-
| colspan="2" | So well as by reflection, I, your glass,
|-
| colspan="2" | Will modestly discover to yourself
|-
| colspan="2" | That of yourself which you yet know not of.
|-
| width="50%" |  
| ''Julius Caesar'', 1.2.53–72
|}

<br>

The rest of this Section (?), continuing the discussion of formalization in terms of concrete examples and extending over the next 50 (?) Subsections (?), details the construction of a ''reflective interpretive framework'' (RIF). This is a special type of sign theoretic setting, illustrated in the present case as based on the sign relations A and B, but intended more generally to constitute a fully developed environment of objective and interpretive resources, in the likes of which an ''inquiry into inquiry'' can reasonably be expected to find its home.

An inquiry into inquiry necessarily involves itself in various forms of self application and self reference. Even when the ''inquiree'' and the ''inquirer'', the operand inquiry and the operant inquiry, are conceived to be separately instituted and disjointly embodied in material activity, they still must share a common form and enjoy a collection of definitive characteristics, or else the use of a common term for both sides of the application is equivocal and hardly justified. But this depiction of an inquiry into inquiry, if it is imagined to be valid, raises a couple of difficult issues, of how a form of activity like inquiry can be said to apply and to refer to itself, and of how a general form of activity can be materialized in concretely different processes, that is, represented in the parametrically diverse instantiations of its own generic principles. Before these problems can be clarified to any degree it is necessary to develop a suitable framework of discussion, along with a requisite array of conceptual tools. This is where the construction of a RIF comes in.

{| align="center" cellpadding="0" cellspacing="0" width="90%"
| <p>And now the investigation itself is under investigation.</p>
|-
| align="right" | — President Clinton, August 17, 1998
|}

The task of building a RIF is here approached on two fronts, structural and functional. The structural approach looks to the formal constitution of the framework itself, with an eye to the static logical relationships that potentially exist among its objective and its interpretive elements, that is, the abstract relations that can be permitted through the medium of its use to be brought to light, to be recognized on future occasions, and to be signified to a community of observant and interpretive agents. The functional approach looks to the dynamic and effective conduct of a typical reflective interpreter, with an eye to the medium of operational resources that support its activity, and it seeks to discover amid this defrayal the workings of the act of reflection that makes it all possible.

{| align="center" cellpadding="0" cellspacing="0" width="90%"
|
<p>I was, at that time, in Germany, whither the wars, which have not yet finished there, had called me, and as I was returning from the coronation of the Emperor to join the army, the onset of winter held me up in quarters in which, finding no company to distract me, and having, fortunately, no cares or passions to disturb me, I spent the whole day shut up in a room heated by an enclosed stove, where I had complete leisure to meditate on my own thoughts.</p>
|-
| align="right" | — Rene Descartes, ''Discourse on Method'', [Des1, 35]
|}

=====5.2.11.1. Principals vs. Principals=====

{| align="center" cellpadding="0" cellspacing="0" width="90%"
|
<p>So it is that these old cities which, originally only villages, have become, through the passage of time, great towns, are usually so badly proportioned in comparison with those orderly towns which an engineer designs at will on some plain that, although the buildings, taken separately, often display as much art as those of the planned towns or even more, nevertheless, seeing how they are placed, with a big one here, a small one there, and how they cause the streets to bend and to be at different levels, one has the impression that they are more the product of chance than that of a human will operating according to reason.</p>
|-
| align="right" | — Rene Descartes, ''Discourse on Method'', [Des1, 35]
|}

Once this much is said and done, one comes to a realization of the fact that a principle, as a point of logic, is not always a principal in the orders of causes, prizes, rights, or times. Even if the word ''principly'', coined to mean ''in principle'', is well formed in principle to serve that purpose, even if it is quickly struck from a readily available syntactic material, strikes true to the types of an adjective or an adverb, easily fills in for a much more cumbersome prepositional phrase, and fills out a formerly empty slot in the language, and even if tenders itself in print as the gentler equal to impress its point, still, it clangs in speech to the point that it is likely to be irrevocably confused with the sound of the already established word ''principally'', and so, by dint of a certain ''phonological exclusion principle'', the expression of its intention in this way is subject to being excised from the language, bowing out in preference to the accidental antecedents that it arrives to find already prevailing on the scene. In sum, an essentially abstract idea can be inhibited from a particular manner of elaboration on what are purely contingent, developmental, evolutionary, and historical grounds.

Here is a problem that vies with the question of the chicken or the egg, asking which of these firsts comes first: the principal or the principle. Without being able to say which first comes to mind, it may be possible to tell, in point of time, which first entered the lists of language or came to express itself in speech, at least, on the assumption that the PEP has import for this case, and that the first item to enter the lexicon blocks the full inflection of the later entry.

From this case of a broken analogy, this example of a missing point of symmetry, or this paradigm of a defective paradigm, in short, from the mere fact that the noun ''principle'' fails in its distribution of uses to fill out the available patterns and to become as fully inflected as the noun ''principal'', it is possible to draw a surprising number of lessons:

# It happens that accidents of personal, cultural, or evolutionary history can abrade the facility with which one reflects on essences. Accidental properties of one's linguistic and mental constitution can supply the array of means that one has available to approach the most extreme questions, those concerned with original and ultimate meanings. Accidents of history operate to shape and polish, to impair and repair the faculties of reflection, the instruments of language and mind that one uses to reflect on questions of abstract, eternal, formal, ideal, or invariant form, to contemplate general schemes of categories for objects, and to consider matters of fundamental principle. For good or ill, an accumulation of accidents impacts on the character of one's reflection, innately marking or marring the equanimity with which one thinks about the arrays of otherwise indifferent and equally likely alternatives.
# A phonological exclusion principle need not apply in syntactic cases or pragmatic situations where interpreters are reliably discerning enough to adequately resolve the textual, verbal, and vocal ambiguities. For instance, it would not matter that the physical signals represented by the words ''principally'' and ''principly'' fail to be discriminated by the ear of the interpreter if the mind of the interpreter, informed by the practical and the syntactic contexts of their sundry utterances, and guided by the innate sense of what makes sense in each situation, could be relied on to chose the proper interpretation.
# This example of a broken analogy or a defective paradigm, and the problem of converting it to instructive uses and positive advantages, brings up the related but more general puzzle that is commonly known as the ''problem of learning from negative examples''. By this is meant, not just being informed by defective or imperfect examples, or learning from examples that are associated with negatively valued consequences, but inducing the laws that apply to a situation from the events that never occur within it.

Returning to the topic of reflection, as approached on the ''structural'' and ''functional'' fronts …

Progress on the ''functional'' or the ''operational'' front can be made by taking up once again the informal calculus of applicational operators that I used at the beginning of the current chapter to annotate the analysis of inquiry, by taking further steps toward formalizing this calculus, and by representing the operation of reflection within it. Progress on the structural front can be made by ...

A RIF is intended to formally allow for a specific area of reflection on experience. I want to use reflection as a bona fide form of observation, in the dual sense that one reflects on happenings in the outside world and again on a range of experiences in one's inner world. Moreover, I want to treat reflection as a genuinely empirical form of observation, perfectly capable of making mistakes in the data and the descriptions it provides, but provisionally able to supply the materials that are needed for building up true theories about the reflected domains of experience.

In historical perspective, there is an array of contentious issues that generally arises in this connection to obstruct the carrying out of any such intention. At times the liberty of reflection is simply proscribed as being out of bounds for the aims of empirical and objective science, at least, if it continues to form a source of data and ideas, then the custom is never to credit the source. At times the region of reflection obtrudes so far from within the preserve of a purely private interest to impose on the realm of a properly public concern that little remains to be seen of the world outside, and no room is left over for the forum of concrete reason to proceed in its own right according to its own lights. Since the next subsection, dedicated to the phenomenology of reflection, takes up this host of issues in great detail, a brief discussion of their bearing on the task of building a RIF is all that is needed at this point.

In order to constitute a RIF as an empirical framework, in other words, as a formal apparatus that can serve to facilitate experiential inquiry, it is necessary to rehabilitate the operation of reflection as a genuine form of experiential observation, one that is capable of generating contingent, defeasible, falsifiable, or hypothetical descriptions of what it reflects on, where the field of view for reflection encompasses everything it is given or gains a power to reflect on, including activities in the external world, affective impressions and motivational impulses that arise in the realm of feeling and drives, and the more or less controlled conduct of reflection itself. To do this, it is necessary, in turn, to achieve a resolution of and to reach an understanding on two points:

# First, one needs to recognize that ''empirical'' necessarily implies ''experiential'' and ''experimental'', but that none of these terms is limited of necessity to implying ''external'', at least, not in the sense of an externality that is exclusive of all felt experience.
# Finally, one needs to separate the practice of reflection from the herd of ''incorrigibles'' it is liable to be confounded with on the modern scene, to sort it out from the "bad company" of its former associates and all their pretensions to (a) immediacy of inference, (b) impeccability of insight, (c) infallibility of introspection, (d) unimpeachability of intuition, and (e) incognizability of reality.

If the reader reflects that I seem to be trying to make reflection out to be a curious sort of hybrid creation, akin on the one side to primitive forms of observation, but akin on the other side to sophisticated forms of contemplation, and that this makes it the main constitutional problem of its temperament to control its own hubris in such a way that it can keep itself from weening over to excessive degrees in either direction, then the reader reflects correctly. If reflection keeps to this middle course, then, whatever its natural disposition or original inclination might be, it can still enjoy the effective qualities and formal virtues that belong to both realms of experience, mediating between the real and the rational, and joining the sensory to the intellectual.

A historical perspective also shows that the task of arrogating modest powers to reflection without reaching over into insupportable realms of imagination is apparently a difficult balancing act for the human mind to maintain. For reasons that will soon become obvious, I find it useful to describe this as the ''cartesian polarity'' problem.

Whereas other operations of mental life have to be forced to the point of examining their own conduct, even tricked into it, it is almost the reflex of reflection to reflect once again on its own action, at least, to reflect on whatever array of intermediate insights or whatever sample of partial results it finds itself able to pin down in memorable signs and texts. And yet, aside from the statement of this vacuity, it seems pointless for reflection to personify itself, often to the point of impersonating itself, and difficult for it to find anything interesting to say, without bringing in something else, some other matter to reflect on. Thus, I do not want to fall into the narcissistic trap of thinking that internal reflection, or introspection, is the only source of knowledge that is certain and true, but neither do I want to vanish in the echoistic dissipation of clinging to external reflection, or reflectances, as the only source of inspiration.

=====5.2.11.2. The Initial Description of Inquiry=====

In order to form a more cogent sense of the direction I must take from this point on, I need to make a concise review of the questions that have been raised and the assumptions that have been laid down so far. I can begin this review with the question of the '''''Criterion''''', and after developing a critical theme that connects it with its antecedent topics, work backward to the questions of the '''''Problem''''' and the '''''Method'''''.

Under the question of a Criterion (§ 1.1.3) a critical assumption was taken to serve as a guiding hypothesis for this inquiry and to provide a tentative model for its conduct:

According to my current understanding of inquiry, and the tentative model of inquiry that will guide this project, the criterion of an inquiry's competence is how well it succeeds in reducing the uncertainty of its agent about its object. (§ 1.1.3).

It is time to examine this hypothesis a bit more carefully, to review the bearing it has on inquiry and the role it has in an inquiry into inquiry. In this regard, the guiding model elected here needs to be interrogated under the twin lights of how it can apply to inquiry in general while it continues to ply itself fully subject to a specific inquiry into inquiry. In its most general implications, the question is: How can any principle profess to master all inquiry whatsoever, and how can the use of such a rule pretend to serve any inquiry at all, at least, so long it waits on the outcome of its own examination and has to operate under a cloud of suspicion that is pursuant to a particular inquiry into inquiry?

To me it seems ''intuitively certain'' (IC) that the purpose of inquiry is to reduce the uncertainty of its agent about its object, and that it does this by increasing the clarity of the signs that the agent possesses with respect to the object. For future reference, let me detach the predicate of this observation and refer to it as the ''initial description'' (ID) of inquiry.

'''ID.''' The purpose of an inquiry is to reduce the uncertainty of its agent about its object, and it does this by increasing the clarity of the signs that the agent possesses with respect to the object.

This ID depicts inquiry in general in terms of its object in general and it allows for more specific inquiries to have more particular objects. And yet, as if by a manner of reflective reflex, no sooner do I express this insight in the form of an ''intuitively certain hypothesis'' (ICH) than I begin to suspect it, to have doubts about the certainty of its truth, and to worry about the clarity of its expression.

On reflection, the ID of inquiry, for all the quality of an ICH that once affected it, at least enough to make me identify with it, starts to find itself in another light, much less IC and much more H, and it begins to appear to me as a nearly indifferent object of contemplation, something else to think about, nothing more. The text of its expression, that I took the time to weave its carefully picked signs into, presents itself to my view as an alien object, composed of almost senseless characters, as if designed to ensnare my mind in a medium of false images rather than to liberate my thinking by means of a clear and distinct truth.

This array of doubts, suspicions, and worries is not so much due to the ID of inquiry itself, that I continue to maintain my sympathies with and to preserve my own recognizance of, as it is on account of and for the sake of the other notions that it raises, whose certainty and clarity it cannot rise above, or so it seems. In other words, whatever ''certainty'' and ''clarity'' may be, it seems sure that the certainty and the clarity of the ID of inquiry cannot be greater than the certainty and the clarity, respectively, of the notions of agency, certainty, clarity, objectivity, and significance that the ID of inquiry invokes, involves, and implicates. But how do I know this, and, indeed, do I really know this?

{| align="center" cellpadding="0" cellspacing="0" width="90%"
|
<p>But, like a man who walks alone, and in the dark, I resolved to go so slowly, and to use such caution in all things that, even if I went forward only very little, I would at least avoid falling.</p>
|-
| align="right" | — Rene Descartes, ''Discourse on Method'', [Des1, 39]
|}

If the ID of inquiry marks a step that I plan to take in an inquiry into inquiry, then I need to raise the following questions about it. How can I tell if it is a step in the right direction? Expressed in other ways: How can I test the utility of this step with the information that I find available to me at my present state of knowledge, or within reach of it? How can I come to know, short of rashly staking my whole enterprise on its trustworthiness, whether it establishes a foothold on a viable path, not just marking a point on a feasible path of investigation, but lying a discrete and reasonable distance in the direction of its goal, and thus being capable of leading toward a state that is ''ultimately certain'' (UC) as to what it represents about inquiry?

If all the themes are aptly initialized, then the end of inquiry is met when a condition of equanimity, balance, or harmony is achieved between the facts IC and the facts UC, in other words, when everyone sees the point that each one is trying to make and when everyone understands the line that each one is meant to get across.

If the simple entertainment of a simple hypothesis is enough to change the course of an inquiry in an irremediable, irreversible, or irrevocable way, and in a manner that makes a pragmatically significant difference to the outcome, then all hope is lost of discovering a robust method or developing a self corrective procedure for inquiry, and thus of giving to inquiry an adaptive and evolving form. This means that inquiry is not a feasible endeavor for agents of a fallible sort unless they are capable of taking up a flexible attitude and maintaining the following sorts of stance:

# To contemplate a diversity of hypotheses with a reasonable measure of immunity against their actual effects, that is, while remaining moderately well insulated from the consequences these hypotheses would have in execution.
# To entertain a wildly incorrect but still corrigible guess with a respectable level of impunity, that is, while preserving the overall ideals of a reparable harmony in their own systems of belief and while operating without loss of ultimate geniality in the community of inquiry at large. Still, it is best not to play on forms of dissonance that do not come tempered with at least an idea of how to atone for themselves in time.

Imagine that an agent begins in a state of almost complete ignorance or near total uncertainty about inquiry, such as might be associated with the mere possession of the name ''inquiry'' or reflected by an encounter with another type of nominal pointer to the topic of inquiry in general. Against this background of ''original sinescience'' and relative to this condition of initial innocence, the ID of inquiry does indeed appear to give an impression of saying something more definite about inquiry, and thus it does seem to increase the agent's knowledge in some measure, whether in certainty, clarity, or distinctness I cannot say for sure at this point. But how can an agent tell if this appearance is real?

How can I address the array of questions "How?" that I raise as I reflect on the ID of inquiry? The only response I can think of that answers the arraignment of all these challenges is simply to continue with the inquiry itself. The ID of inquiry has something to say about the ''how'' of inquiry, expressing its suggestions in terms of the phrase that concerns itself with "increasing the clarity of the signs that the agent possesses with respect to the object". But the implicit charge of this ID, to "clarify the signs that the agent has of the object", is not entirely unambiguous in and of itself, and until it can be rendered free of ambivalence in its own right it is difficult to entertain an effective action on its behalf. On further reflection, it becomes apparent that the charge to clarify signs contains the potential of being interpreted in at least two ways, and thus, for the purposes of effective action, requires a further clarification. Consequently, toward the end of action on the charge yet another inquiry and yet another clarification, this time concerning the character of the charge itself, becomes due.

In accord with this analysis of inquiry as a process of clarification, and of clarification in turn as a process that operates on sign relations, the next few paragraphs consider various interpretations of the clarification task, initiating the process of comparing and contrasting their elements, and ultimately seeking to classify their variety. This discussion notices one general feature that all types of clarification process appear to have in common and it discerns another general feature that splits the genus of clarification processes into a couple of broad moieties or species.

Inquiry, considered as a process of clarification, is the chief way that a sign relation can grow and develop in service to the life of its agent. If not assured as the principal way, at least, while the jurisdictions of automatic adaptation, oblique evolution, and random ramification are yet uncharted and unassessed, it is probably still the most principled way that sign relations have of adapting and evolving to meet the objectives of interpretive agents in their given environments of needs and objects. By way of a general comparison, then, all reasonable interpretations of the clarification task involve the augmentation of sign relations by the addition of ''elementary sign relations'', that is, ordered triples of the form <math>(o, s, i).\!</math>

Treating the process of clarification as one that affects the growth and development of a sign relation, even if constrained to the medium of its syntactic domain, there is, of course, an overwhelming diversity of ways that one can imagine an arbitrary sign relation as growing through time. No matter whether it restrains its labors to the monotonic annexation of ever more triples <math>(o, s, i)\!</math> to the masses of data already accumulated or whether it liberates the full deliberations of a discursive process, thus invoking the ebb and flow of corrective, editorial, reflective, remedial, and reversible processes, not every mode of growth or development that can occur in a sign relation has a bearing on reducing the uncertainty of an agent about an object or has the effect of promoting the clarity of the given signs.

With regard to inquiry as clarification, and clarification in turn as the evolution of a sign relation, it does not matter whether one views it as a process of exploration and discovery, taking place in a preconceived cartesian space <math>O \times S \times I\!</math> and seeking to find clearer signs for each known object, or whether one views it as a process of creation and invention, staking out the syntactic parts of elementary sign relations <math>(o, s, i),\!</math> following the directions of transient clarity to the signs of maximal achievable clarity, making and testing novel combinations with an eye toward present objects, and picking out the clearest indications for inclusion in one's current sign relation.

To review: Inquiry depends on clarification, and clarification depends on the augmentation or the evolution of sign relations in various ways. In order to stay within the realms of possibility that are accessible to computational processes and covered by computational models, it is best to look for varieties of clarification process that are tantamount to recursive forms of development in sign relations, those that one can contemplate being carried out by a recursively defined growth process. Even working under these constraints, there is still an amazingly large variety of different ways that the ''eking out'' of initial sign relations and the ''imping out'' of fledgling sign relations can proceed.

To expand: Inquiry depends on clarification, and clarification depends on the augmentation or the evolution of sign relations in directions that serve the interests and help to achieve the intentions of their agents. If this is true then it must be possible to say something about the ways that sign relations figure into the interests and intentions of agents. In this connection, the desire to relate sign relations to the objectives of interpretive agents touches on the topics of what are normally called the ''normative sciences'', namely: aesthetics, ethics, and logic. Since the style of pragmatic thought that I am using puts a distinctive twist on the way that these three disciplines are regarded in relation to each other, it is necessary for me to attach a slight gloss on this point. Along the way, related concerns about the topic of "justification" make a natural appearance, and this allows me to set down some initial thoughts about the ''forms of justification'' (FOJs) that I contemplate using in this work.

=====5.2.11.3. An Early Description of Interpretation=====

Insofar as the analysis of etymological and other associations existing among familiar and other words can lead up in time to a proper analysis of the underlying concepts involved, the discussion just given can be taken in the spirit with which it is offered, as a body of suggestions about the derivation of an abductive faculty from an advisory function. But insofar as the sole themes that exhibit any novelty about them are easy to lose in the cacophony of verbal clutter that frequently ensues, it may be advisable, one last time, in as summary a refrain as possible, to recapitulate the chief points of originality to which this work gives fanfare. The originality I advert to appears to arise from Aristotle:

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<p>Words spoken are symbols or signs (''symbola'') of affections or impressions (''pathemata'') of the soul (''psyche''); written words are the signs of words spoken. As writing, so also is speech not the same for all races of men. But the mental affections themselves, of which these words are primarily signs (''semeia''), are the same for the whole of mankind, as are also the objects (''pragmata'') of which those affections are representations or likenesses, images, copies (''homoiomata'').</p>
|-
| align="right" | (Aristotle, ''On Interpretation'', i.16<sup>a</sup>4–9, p. 115)
|}

Of all the early intimations of the sign relation that have come down to the present time, this depiction is the clearest and the fullest, in the level and the lucidity of the details that it articulates, of any I know. There will be ample occasion to return to its ideas on a recurring basis throughout the rest of this work. For the moment, there are especially a couple of themes, traceable to this locus and running through its text, that I want to draw out and to emphasize, not only for the message that they convey for the sake of the immediate inquiry, but for their manner of bearing it into the current context of applications.

The assumptions and implications of Aristotle's account that are relevant here are first listed briefly, after which all of the points in turn are explored in full detail.

# Assumptions of constancy
# Compound of fractures
# Cognitive coin/currency/ideas are derivative in relation to affective basis/realm/ideas
# Concrete material presented by emotions: affects and motives

On Aristotle's account, there is a special path of relationship that can be traced in either direction between the pragmatic objects (''pragmata'') that attract, concern, or interest the mind and the symbolic signs that the mind exploits to express its feelings or its thoughts, and this route passes through the ''pathemata'', literally, the affects that these objects impress on the mind (''psyche''). Picturesquely, the image overall is that of a coin, which is cast, minted, or stamped in a die, mold, or template and then split into fragments that serve as tokens of mutual recognition if and when they are ever again caused “to be cast or thrust together” (''symballein''). Aside from the nuance that distinguishes artificial and cultural symbols (''symbola'') from biological and natural signs (''semeia''), the signs that mediate peoples' cognitive processes can be seen to arise as fragments of likenesses of their objects. And so it is appropriate to dub this picture of sign relations as the ''iconoclastic theory of signs''.

A reasonably useful correspondence is formed between the sign relations of Aristotle and Peirce, respectively, by fashioning the following rough linkages:

# Associate the categories of ''objects'' in both accounts.
# Collapse for the moment the distinction between ''signs spoken'' and ''signs written'' and put them together in association with a maximally generic class of ''signs''.
# Incorporate ''pathemata'' as a particular species of mental conditions, determinations, or entities, that is, as a special class of modes of bearing and modes of being affecting the mind, and put them in association with a special class of ''interpretant signs''.

It is only the last part of this likely correspondence that leads to much difficulty and requires a closer examination to untangle its complexities. In accord with the lines of its suggestion ''affections'' and ''impressions'' are brought together and given their places within a maximally generic class of affective states, cognitive conditions, conceptual formations, ideal forms, motive dispositions, potentially existent ''states of mind'', and virtually existent ''things in mind''. Under these and other terms, a catch-all category is formed that ranges through the entire panorama of mental dispositions and mental entities, encompassing the full spectrum of determinations in the medium of the mind, all of which, for maximum convenience and generality can be referred to most simply as ''ideas''. Granted this endowment of ideas, the suggestion is made to associate all of these ideas, at least, for the purposes of a formal treatment, with the pragmatic domain of ''interpretant signs''.

This recasting of Aristotle's model of interpretation into the molds of Peirce's theory of sign relations, given that the former retains a body of concrete material that the latter casts off from its abstract forms, barely roughs out an interpretation of the theory in terms of the model. And yet this interpretation is still a bit forced in the character of its suggested associations, and it still faces a number of objections to the likelihood of its truth, its utility, or its other potential virtues ever becoming actualized. These objections indicate a daunting array of more obscure obstacles that line up behind and loom into the distance beyond the one or two more obstinate obstructions that make themselves obvious at present, all of which inveigh, inveigle, and weigh against carrying through the completion of a viable work along the lines projected.

Many of the problems with this trial interpretation, in it stands in its currently tentative state of development, can be traced to the fact that I have not fully disentangled the formal and the material aspects of the category of pathemata, or else, I have not distinguished the properties they display by virtue of their roles in a relation from the properties they exhibit by virtue of their inherent natures. Indeed, as the issue now appears to me on reflection, I have not examined thoroughly enough the relation between the two distinctions I just supposed, that between form and matter, and that between relation and essence, nor even asked how and whether any such distinctions can be made on a reliable basis.

One of the things that one means by ''matter'' is the mass of instances that one uses to illustrate a common ''form''. In this sense of the terms, the distinction between form and matter has been implicitly relevant to this work, if not especially salient within it, ever since it entered on the course of discussing the process and the product of formalization in terms of a selection of concrete examples. One of the reasons that one chooses to present a formal subject in the medium of its own materials, that is, by means of examples, is that this very form is not yet grasped abstractly enough, that is, by means of clear and distinctive definitions. This choice can be due to the presence of difficulties and obstructions, on the part of the presenter or the presentee, that stand in the way of a more comprehensive form of understanding being achieved all at once.

In the light of these considerations, one can see that this discussion has stayed immersed in the matter of its topic for quite some time now. Indeed, it is fair to describe the current trajectory of this inquiry as “a matter in search of its form”. A distinction between form and matter, if it matters to anyone, is not already finished, but yet to be formed. And yet, no sooner does it form itself in a particular way than it seems, at least with regard to itself, that it was always meant to be that way. And so one encounters a form of distinction that evolves in coincidence with a form of life, taking the essence of its own formation to be a kind of ''synthetic a priori''. Several factors now conspire to make the relevance of this distinction become more acute at this point.

The whole class of affections, cognitions, dispositions, impressions, and motivations of the mind that I include in the category of ''ideas'' is the smallest collection I can form that remains connected in its associations, that is, as a class of signs whose interpretations lead in and to itself. But giving this host of ideas a fixed and a simple name, no matter how well it means to unify the manifold of impressions under a simple term, and no matter how well it manages to organize the array of associations under a specious concept of unity, does nothing to dispel the blooming complexity, the boiling diversity, the booming heterogeneity, and the bubbling incongruity of the ideas themselves, which seem to spite every attempt to incorporate their livelier qualities and to regulate their ongoing varieties in the almost purely nominal forms of integrity that mere words can provide.

A purely formal study of sign relations could proceed unhindered from this point, at least, unencumbered by the material aspects of its subject, if only it had a clear definition of what a sign relation is meant to be. And yet, from a purely formal perspective, almost any axioms are worth pursuing, as long as they pick out an interesting class of formal objects and even so long as they manifest a certain elegance in their own right.

It is frequently claimed that the virtues of ''elegance'' and ''interest'' are qualities that abstract axioms and formal objects can possess quite independently of their anchorage in, bearing on, connection to, or any other crassly pragmatic relationship with a ''ground'', such as might be constituted by an additional application to a concrete subject or by an extraneous utility in a practical matter. I am tempted to agree, though I suspect that there are further difficulties and greater paradoxes still hiding within the word ''independently''. In any case, if the axioms that one selects to characterize sign relations are intended to have empirical applications, objective justifications, and practical motivations, to be justified by reasons that go beyond the contemplation of abstract forms for their own sakes, to be motivated by purposes that reach beyond the purer forms of aesthetic enjoyment, the self justifying and self seeking forms of entertainment, exercise, experiment, and exploration, or the riskier forms of speculation against the chances of their future utility, and until the essential features of a general definition can be grasped, it will be necessary to preserve the data of their material occurrences for whatever insight it can afford into the qualities that determine the worth of signs, and it will be prudent to continue mining the material evidence for exemplary instances of their actual use.

Before this discussion can proceed much further, there are a couple of seeming distinctions that I find myself in need of trying to make real, or otherwise of knowing the reason why they cannot be made in reality. There are reasons why I emphasize the ''seeming'' and ''trying'' aspects of this situation:

# It is not entirely clear to me whether these apparent distinctions, as commonly described and as usually intended, are capable of being maintained in reality, as opposed to what little significance they have while posed on the grounds of mere imagination.
# Even if the corresponding forms of distinction are capable of being established in a clear, a proper, and a successful manner, it remains an open question to me at this point how this ought to be done in practice, and the experiences I have had in previous attempts lead me to believe that each of these tasks is far more difficult than it appears at first.

In the process of carrying out this inquiry I find it necessary to make a type of rhetorical transition that can be cast in the stereotyped form: “In the process of doing ''x'' I find it necessary to do ''y''.”

=====5.2.11.4. Descriptions of the Mind=====

In the process of interpreting Aristotle's text on interpretation as an early account of sign relations, I invoked a number of distinctions that appeared to be called for to guide the interpretation of the text and to form what I sense to be the most reasonable interpretation of its terms. All of these distinctions are drawn from common practice and are usually assumed to be easy to make in any case. And yet, on reflection, I find that I have little or no faith in their advertised properties or in my right to take them for granted.

For instance, consider the distinction between form and matter. When I reflect on the question of how and whether I can make this distinction, I find myself hard pressed to tell whether the distinction itself is a formal distinction, a material distinction, or something else entirely. Trying to pin it down between the first two cases, it seems to be either, or both, and yet neither, depending on the light that I choose to throw on the question, to consider the alternatives by, and to interrogate the usual answers under. If it is either, then the significance of the other is diminished to nothing, as all that the opposite side of the divide can amount to is sliced away by a gradual slippage down the apposite slope, until the significance of the entire distinction appears but to disappear. If it is both, then it violates the exclusiveness that is usually assumed to hold between the two sides of the distinction. If it is neither, then it invalidates the exhaustiveness that is usually assumed to apply to the distinction between form and matter. Whatever the case, I am called on to assume something unusual. Indeed, it seems I am forced to recognize a ''tertium quid'' or a ''third something'', in other words, an option that can supply a novel alternative to the choice between form and matter, a category that the Greeks only hint at obscurely or obliquely allude to under the name of an ''entelechy'', and something that I can well nigh call the ''interpretive'' case.

By way of guidance in this innovation, or this novel effort to capture interpretation in and of itself, I adduce two texts that help to show the way that the relationship between form and matter has often been seen.

The first text illustrates the use of this distinction in the context of a psychological investigation.

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<p>We describe one class of existing things as substance (''ousia''); and this we subdivide into three: (1) matter (''hyle''), which in itself is not an individual thing; (2) shape (''morphe'') or form (''eidos''), in virtue of which individuality is directly attributed; and (3) the compound of the two.</p>

<p>Matter is potentiality (''dynamis''), while form is realization or actuality (''entelecheia''), and the word actuality is used in two senses, illustrated by the possession of knowledge (''episteme'') and the exercise of it (''theorein'').</p>

<p>Bodies seem to be pre-eminently substances, and most particularly those which are of natural origin; for these are the sources from which the rest are derived.</p>

<p>But of natural bodies some have life and some have not; by life we mean the capacity for self sustenance, growth, and decay. Every natural body, then, which possesses life must be substance, and substance of the compound type.</p>

<p>But since it is a body of a definite kind, viz., having life, the body cannot be soul, for the body is not something predicated of a subject, but rather is itself to be regarded as a subject, i.e., as matter.</p>

<p>So the soul (''psyche'') must be substance (''ousia''') in the sense of being the form (''eidos'') of a natural body (''soma''), which potentially (''dynamei'') has life. And substance in this sense is actuality (''entelecheia'').</p>
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| align="right" | Aristotle, ''De Anima'', II.i.412<sup>a</sup>6–412<sup>a</sup>22
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<p>The soul, then, is the actuality of the kind of body we have just described. But actuality has two senses, analogous to the possession of knowledge and the exercise of it.</p>

<p>Clearly actuality in our present sense is analogous to the possession of knowledge; for both sleep and waking depend upon the presence of soul, and waking is analogous to the exercise of knowledge, sleep to its possession but not its exercise.</p>

<p>Now in one and the same person the possession of knowledge comes first. The soul may therefore be defined as the first actuality of a natural body potentially possessing life; …</p>

<p>So one need no more ask whether body and soul are one than whether the wax and the impression it receives are one, or in general whether the matter of each thing is the same as that of which it is the matter; for admitting that the terms unity and being are used in many senses, the paramount sense is that of actuality.</p>

<p>We have, then, given a general definition of what the soul is: it is substance in the sense of a formula; i.e., the essence of such and such a body [a natural body potentially possessing life].</p>

<p>Suppose that an implement, e.g., an axe, were a natural body; the substance of the axe would be that which makes it an axe, and this would be its soul; suppose this removed, and it would no longer be an axe, except equivocally. …</p>

<p>If the eye were a living creature, its soul would be its vision; for this is the substance in the sense of formula of the eye. But the eye is the matter of vision, and if vision fails there is no eye, except in an equivocal sense, as for instance a stone or painted eye. …</p>

<p>The waking state is actuality in the same sense as the cutting of the axe or the seeing of the eye, while the soul is actuality in the same sense as the faculty of the eye for seeing, or of the implement for doing its work. …</p>

<p>It is also uncertain whether the soul as an actuality bears the same relation to the body as the sailor to the ship.</p>
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| align="right" | Aristotle, ''De Anima'', II.i.412<sup>a</sup>22–413<sup>a</sup>9
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The second text illustrates the use of an analogous distinction between form and matter within the context of a logical investigation.

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<p>And how can one know the certainty of demonstrations except by examining the argument in detail, the form and the matter, in order to see if the form is good, and then if each premiss is either admitted or proved by another argument of like force, until one is able to make do with admitted premisses alone?</p>
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| align="right" | Leibniz, ''Theodicy'', [Leib, 89]
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Take once again the distinction between form and matter, and allow me to say that this distinction is ''interpretive'' in character or nature. This gives me the option of saying that it is formal in some cases but material in other cases. It can all depend on choice and circumstance. If I interpret it as formal then certain things follow. If I interpret it as material then other things follow. But I can rest with calling it interpretive, leaving it to the moment to actualize what is most fitting. If I interpret it as interpretive, which amounts to a way of holding any further decision in suspense, then I am choosing to remain all the while constantly aware of the circumstances and the conditions that affect the actualization of this distinction as either form or matter, or else to experience the consequences of failing to do so.

But then, on marking any distinction, a moment's reflection brings me to ask: “Who or what makes this distinction that I mark?” And whether I say that it is I, or you, or whoever else agrees in marking it with us, whose activity constitutes the making of this distinction, or whether I think it is someone other or something else that makes this distinction that all of us merely mark and remark, and whether it is decided in the end that the maker is always coincident or sometimes distinct in regard to the marker, then I find myself still having to ask: “How and why is this distinction being marked, in particular, what side or sides, with respect to each other, are its maker and its marker on?”

For these reasons, it is necessary to use an indirect strategy in order to approach the questions of these distinctions that I want to consider. The ostensible distinctions are first described in very rough terms and introduced in the ways that they are naturally and usually thought of. Thus, without taking for granted the clarity, fidelity, sensibility, or validity of their formulations, the distinctions are initially presented in the terms by which they are commonly indicated, intended, suggested, or regarded as being established. This manner of approach is demanded in order to keep from assuming, if at all possible, the prior worth of the very formulations that are being examined and tested, and it tries to make it a separate question whether these intentions to distinguish can continue to be maintained in the very same terms and formulations. Once this preliminary investigation is carried through to a conclusion, positive or negative, I can then return to analyze more carefully and more generally the whole process of making such distinctions.

# The rough idea of one distinction is to sort the properties of things into two categories: the “properties that things have” versus the “properties that things are given”. More specifically, and in reference to a typical agent, the former class is intended to include the properties that things have, in and of themselves, independently of any agent, whereas the latter class is intended to include the properties that things are given by an agent. Now, it is clear that the common usage of words like ''have'' and ''give'' leaves a wide range of ambiguity still remaining that needs to be resolved by the right interpretation. (essential vs imputed properties)
# The rough idea of another distinction is to sort signs and ideas, no matter whether they are considered severally or together, into two categories: “signs and ideas as they actually occur” versus “signs and ideas in an abstract vacuum”. More specifically, and with reference to a particular class, community, or population of interpretive agents, the former category is intended to include signs and ideas as they actually occur among these agents, for example, as actualized, embodied, implemented, operationalized, or realized, whether consciously or not, among human beings, whereas the latter category leaves unanswered the question of embodiment and is therefore open to any suggestion as to how these signs and these ideas are intended to be conceived. (empirical and material vs theoretical and formal signs and ideas)

[Alternative text?] The form of distinction that I need at this point tagging “signs and ideas as they actually occur”, for example, as actualized, embodied, and realized, whether consciously or not, in human beings, and leaving “signs and ideas in a vacuum” untagged by any special mark.

Suppose I need to draw a distinction, that marks out a special dominion from its more general domain, but I want to be careful to emphasize the inclusion of the species within the genus, as much as their separation. So let me paint the distinction this way, that it overlays a distinctive tincture on the species but not on the genus, and thus it highlights the special dominion as it resides within the grounds of its generic domain. Given this special form of distinction, it deserves to be given a name. It is fitting to describe it as a ''partial functional distinction'' (PFD), by way of recognizing the partial function that assigns a special value to the species, but not necessarily any value to the rest of the genus. A better nickname, more compact than the verbose description and more mnemonic than the acronym, is served by coining the term ''distincture''. In this context, let the species that is distinguished by a particular distincture be referred to as the ''content'' of that distincture, and let the remains of the genus be referred to as the ''discontent'' or even as the ''distent'' of that distincture. Notice that a distinction, as it is ordinarily understood, has at least two distinctures associated with it, where each especially values what the other does not necessarily value, and where the content of either one is the discontent of the other one. It should also be noted that I have not said anything yet about these partial functions being computable, only that they are conceivable.

I use the terms ''figure'', ''ground'', and [?] to indicate the ''species'', the ''remains'', and the ''genus'', respectively, of a PFD or ''distincture''. Notice that the figure and the ground are not treated symmetrically, but that each element of the figure is given an extra feature, a mark of attention or a tincture of distinction, that it does not have before the distinction is made, even if it is nothing more than a recognition or a representation, explicitly given by an agent to an element, of a feature that the element already possesses. Notice the asymmetry that occurs between the treatment of the figure and the treatment of the ground. Given this special form of distinction, and appreciating this asymmetry, …

There are several ways of studying sign relations that avoid the realm of affects and motives, at least, they seem to get around it for a while, thereby obviating the problems of delving into this refractory material. A purely combinatorial approach …

It seems to me that every ''impression'' has something in the way of an ''impulse'' about it, in other words, that every affective condition is analogous, equivalent, or identical in some sense to a motive disposition. For this reason, I comprehend the category of pathemata to comprise a generic class of ''double duty ideas'' that one is free to interpret in either a passive or an active sense, in short, as ''affects'' or ''motives'', respectively. To sum up the understanding of these terms that I reach at this point: The category of pathemata, encompassing both affective impressions and motive impulses, can be treated as a species of ideas. Ideas constitute a genus of mental conditions, dispositions, or entities that are theoretically tantamount to ''mental signs''. Whereas ideas can be understood as ''signs in the mind'', it is perhaps best to regard them as ''signs of the mind'', or even as ''the mind in signs'', that is, as all the ways that the mind conducts itself and continues to live in signs.

Affects and motives, by way of giving them a conventional placement in the larger class of ideas that inhabit the ''mind'' of a particular agent, can be seen as belonging somewhat closer to the ''body'' of their agent, since they are especially concerned with maintaining the health and the life of that body, and they preserve an interest in the viability, the vitality, and the overall well being of the particular agent concerned. Accordingly, whatever else the signs called ''pathemata'' are about, they are partly about the body of functions and structures that are required to maintain their agent in a viable form. No matter what other objects their signals of conditions and their suggestions of actions may have, they are partially intended to serve a particular form of materially constituted agent and to help it preserve its own accustomed nature.

To form a better sense of how affects and motives fit within the category of ideas, or mental signs, and of how they can be located within a suitable domain of interpretant signs, I give these pathemata the somewhat arbitrary collective name of ''motives or themes'' (MOTs), intended to suggest little more than the common coin of emotions and motivations, and then I quickly divide this overly bulky, burdensome, and burgeoning class of meanings into three subdomains:

# The ''ostensibly ultimate interpretants'' (OUIs) comprise whatever aspects of affirmative and definite sense are available to these MOTs. They constitute the ultimate meanings that appear to be achievable and affirmable at a given moment in the development of an interpreter or in the evolution of an interpretation. At any given time, they seem to be the ultimate interpretants that all properly directed mental processes are tending toward, and yet none of this stands in the way of diverse interpretants being taken as the ultimate achievables at other times.
# The ''almost never objective notions'' (ANONs)
# The ''never objective notions'' (NONs)

Talk of mental impressions, whether taken literally, as being the forms that objects are imagined to impress on the mind, or taken figuratively, as bearing the information that objects are recognized to transfer into the medium afforded by the mind, is frequently criticized as a metaphor that leads to many false, misleading, specious, or spurious impressions. For instance, the very idea of a mental impression is often censured on the grounds that it promotes an offshoot of illegitimate ''instructivist'' notions, themselves the outgrowth of commonly discredited ''essentialist'' doctrines, teaching something to the effect that objects have the power to “instruct” the mind about their essential natures or true properties, thereby directly, literally, and materially imbuing, informing, infusing, and “instructuring” the mind, not just ''about'' their actual characters, but fully ''in'' their ideal forms and wholly ''with'' their real natures.

Obviously, construing the word ''impression'' in that strict a fashion, hobbling it to senses that remain as naively literal as they favor the purely material, is bound to raise a welter of absurdities in the mind. The notion that an object itself can provide instruction in its nature is not invidious in itself. It is only the refractory implication that often accompanies it, the uncritical, unexamined, and unreflective assumption that this degree of directness affords a quality of infallibility to what nature teaches, insofar as its lessons can be imparted to the finite mind. Regarded in this light, the fallacies imputed to essentialist doctrines and instructivist notions are not essential to the elements of validity that a charitable interpretation could find in them, but only variations on the same old theme, the pervasive illusion that any mode of infallible cognition is available to a finite mind.

What can be said about this fundamental fallacy, to wit, the presumption of infallibility that so persistently appears to affect the minds of the very agents who least deserve to claim its prerogative with any justice, that so frequently appears to remain as incorrigible as it is devoted to preserve its unregenerate state? If one inquires into the origin of this delusion and into the source of its effects on the mind, then the issue can be divided into two branches, distinguishing the mechanisms of its operation from the motives of its enterprise, the ''how'' from the ''why''. When it comes to the mechanisms that are capable of accomplishing the effects of this illusion, it seems to be through the creations of the imagination and the ingenuities of speculative thought, in short, through the inventions of wishful thinking that it manages to maintain itself. When it comes to the motives that can be held responsible for mounting the measures of effort that the work on this web of deception requires, they seems to harbor their most fugitive aspiration in the overwhelming need to find some perfection somewhere. This is tantamount to a desire or a disposition, affecting the conduct of agent who falls subject to it, that expects a promise of perfect certainty at some point or other, and thus insists on placing a measure of absolute trust in a point of dogma or a rule of method.

Wherever there is an prevailing need to believe that one already knows, to think with respect to a question that the answer is already found, then there will be no genuine inquiry occurring in that direction.

If it pertinent to characterize the kind of agent that behaves this way, it is almost as if the agent imagines that it cannot actually be, neither begin to act, nor continue to act, without such a guarantee of certainty. But what kind of surety would that be, but another specious certificate? Perhaps this character of conduct is due to an excessive sensitivity to the “irritation of doubt” or an exaggerated intolerance for existing in a state of uncertainty. But no amount of finite intuition can purchase an instruction so authoritative that it can ever be interpreted as infallible. In summary, if essentialism and its oftentimes corollary instructivism are interpreted perversely, that is, taking at face value the excessively literal images and the extremely material senses that their underlying thematic metaphors are able to bring to mind, then they are definitely capable of leading to ridiculous conclusions, but if they are interpreted in figurative and formal manners, then there may be a sufficient amount of interpretive elbow room for them to convey a measure of sense.

In this work I intend to give a liberal interpretation to the issue of what kinds of forms are able to impress their shapes on the mind. I consider it likely that they can take the forms, among other things, of probability distributions, in other words, patterns of amplitude, density, frequency, intensity, or likely value that can be predicated of events in the world or impressions in the mind with equal felicity. If these forms are still too concretely cast, then still more formal forms are available. Abstracting from the contents of a strictly probabilistic interpretation, one is left with functions from domains of elementary events articulated in the world or existential experiences affecting the mind, respectively, to ranges of a common value, the height of which has the sole utility of indicating to different degrees the diverse elements under its dominion. In the resulting spaces of functions, forms of dispensation or patterns of distribution accumulate over the domains of external events and the domains of internal experiences, respectively, like crowns of foliage above the branches of the corresponding trees. It is trees like these, nothing more literal, that provides the material for mental impressions. In summary, the medium of functional forms is able to furnish a common ground for the exchange of information between events and experiences and to supply a mode of comparison that connects any domain of objects with any domain of ideas.

Granted the liberality of this interpretation, and given the looseness of this resulting constructions, while recognizing the general fallibility of all the mind's affections and impressions, and also remembering that the essences of some substantial objects are formal, relational, or structural rather than absolute, literal, or material, the force of common objections to essential instructions is blunted or diffused on these points, that is, the potential charges of essentialist and instructivist fallacies become defused in their application to these forms of distributed interpretation.

Given that reasonable interpretations are available for the language of mental impressions, that serve to make talk about mental impressions at least potentially sensible, then what explains the very real problems that nevertheless continue to arise from this usage? As far as I can detect, the real problem with the supposition of an instructive mechanism of transfer is a steady bias, arising especially in certain corners of the modern scene, toward material rather than formal interpretations of the forms, the patterns, or the shapes that are taken as being impressed on the mind.

=====5.2.11.5. Of Signs and the Mind=====

In the process of trying to clarify my initial description of inquiry, I worked my way back through several modern animadversions, credulous and critical at turns, to an original classical source for many of the ideas that are involved in it. In the process of attempting to understand this text, encountered as a foundation stone in the discussion of sign relations, I found myself invoking, almost reflexively, a number of distinctions, for instance, that between ''figure'' and ''letter'', as they are used to mark a manner of interpretation, and that between ''form'' and ''matter'', as they concern the content of an indication, and each distinction in its turn seemed to be necessary just in order to outline a sufficient indication of what I sense to be a proper reading of this text.

But the mere formation and the occasional invocation of these words, no matter how familiar the sound of them, is of little benefit to my reader if I cannot explain the sense of them, whether long established meanings or any new gleanings that I intend, and the instrumentality intended for these distinctions can be of little use to anyone unless I can say, with regard to each conceivable distinction, how it is made or how I make it. In accordance with this reflection on the making of distinctions, I am thus led to ask: “Who makes these distinctions, and how are they made? Are they made before us, by us, or after us?”

I think I began innocently enough, with no predominating desire either to dispatch or else to vindicate any particular line of thought, but simply to trace the effects of certain ideas, and this means tracking them down to their sources, in whatever places they are to be found, whether ancient or modern, as well as trying to deduce or to foresee their consequences in theory or in action, whether for good or for ill. And now this form of investigation, more like a process of divestiture, brings me to an array of questions that I have only the slightest clues how to answer.

All of the leads that I do have arise from noticing the things that are left still lying about, the remains of the case that led to the inquiry. This includes, beyond the intentional products of focal investigations, the artifacts of analysis that still disturb the development of a clear and complete picture, the circumstantial evidence that fails to fit my nearest approximation to the facts, and the incidental residues that are supposed to have their sole function only in servicing the machinery of method, and yet that seem to litter the grounds around my present site.

For example, it was among this array of clutter on the workbench that I first noticed the distinction between form and matter. It appears as an element or tool that I have been using all along without much reflection on its qualities. And yet there is nothing simple in the way it appears, on the one hand, as a constituent of construction, on the other hand, as an instrument of interpretation. And then, the moment I start to reflect on it, to become aware of the contingent facticity of the distinction, as if for the very first time, I find I must confess that I have lost all my formerly implicit faith in the trustworthiness, the unquestioned utility, and the ultimate validity of even this apparently innocuous distinction. There is nothing then to do about it but to begin again, to examine the worth of this now hypothetical distinction, to see whether the old trust in it can be reconstructed, whether a new justification for it needs to be devised, whether anything like it has to be entirely dispensed with, or whether alternative forms of distinction and even much different types of relation are required to take its place.

Is it conceivable that the proper application of these bits and pieces to themselves and to each other can lead to a reconstruction of the rationality desired?

For instance, consider the distinction between form and matter.

No matter what distinction forms the interest of the moment, there is that which discerns, draws, finds, follows, grasps, makes, sees, or seizes the distinction in question, an experimental agency that might as well be called an ''interpreter'', a ''maker'', or an ''observer'' of that distinction. With regard to any form of distinction, the agency of a ''distinguisher'' or a ''former'' is a role that seems to suit what the Greeks fairly often described, but just barely hinted at, under the name of an ''entelechy''. In general, this is a somewhat mysterious designation, stemming from a complex of terms whose various connotations are commonly translated as ''actuality'' or ''reality'', and often as ''actualization'' or ''realization'', seeming to suggest ''actualizer'' or ''realizer'' for the duty of this agent. Sticking more literally to its etymology, the function of an entelechy can be taken to mean “that which has or is its end in itself”, and thus “that which exists for its own sake” or “that which is complete as it is”. Whether the interpreter actively creates the distinction as it is drawn or passively discovers the distinction as it is traced is not yet the issue of interest, and I leave this matter to a future distinction.

The relevance of the distinction between form and matter can be traced, not solely by way of illustration, to another passage from Aristotle:

: (Aristotle)

One should not let the phrases ''tertium quid'' or ''third something'' lead one astray, going so far as to think that an additional essence, a new kind of material, or a novel substance is implied, when it might be only a third way of being, mode of existence, degree of freedom, dimension of motion, or an extra role in a relation that is actually required.

To this seminal account of interpretation the pragmatic theory of signs adds an array of general and specific elaborations, equipping it with a fully developed corpus of formal, instrumental, and material features. Since the pragmatic line of development is in some sense an alternative track to what is usually called the modern line, the naive enlightenment, or the cartesian tradition, and yet shares many aims and basic methods with this still current mode of inquiry, it is necessary to distinguish these different trends, to detect their different impacts on the present scene, and to discern their different imports for the future of inquiry. The accidental, intentional, and specific differences that the pragmatic theory of sign relations, in its currently developing form, is able to deploy over and above the ancient account, along with the differential circumstances that exist in the context of its present day applications, are taken up next, starting with the most salient augmentations and the most significant extensions of its overall lines of growth.

Some of the most important general features that mark out the pragmatic theory of sign relations from its original material are instrumental in character and arise largely due to changes in the technological base, formally speaking, between the ancient and the present times, that is, by innovations in the formal languages and the technical methods that are made available for carrying out the discussion. Three of these general instrumental features are taken up next.

# In conformity with the modern facility for thinking of relations in general in extensional terms, as collections of ordered n tuples of domain components that belong to the relation in question, current versions of the theory of signs render it easiest to think of each given sign relation as a particular collection of ordered triples. Elements of a sign relation are called ''elementary sign relations'' (ESRs), and the data of each given element of the sign relation can be represented as an ordered triple, of the form <math>(o, s, i),\!</math> that names its object, sign, and interpretant, respectively.
# Among the other props on the modern stage, the pragmatic theory of sign relations can make especially good use of the bounteous ''logics'' of relations and ''algebras'' of relative terms that are currently available, as expressed in any one of several symbolic calculi with approximately the power of predicate logic. Indeed, many of these algebras, calculi, and logics of relations received their first “modern&redquo; formulations in the work of C.S. Peirce, and in the very process of trying to deal with the problems presented by the classical theory of signs. As it happens, this coincidence of origins and this parallelism of derivations may help to account for the appearance of a quality of pre-established harmony that is presently manifested between the general subject of relations and the special subject of sign relations.
# Developments in other fields in the intervening times have caused the prevailing paradigms to shift a number of times. For starters, the lately recognized inescapability of participatory observation, and the multitude of constraints on knowledgeable action that the necessity of this contingency implies, that ought to have always been clear in marking the horizons of anthropology, economics, politics, psychology, and sociology, and the phenomenological consequences of this unavoidability that have recently forced themselves to the status of physical principles and tardily made their appearance in the symbolic rites of the attendant formalities, against all the fields of reluctance that physics can generate, and in spite of the full recalcitrance that its occasional ancillary, mathematics, can bring to heel. These cautions leave even the casual observer nowadays much more suspicious about declaring the self evident independence of diverse aspects and axes of experience, whether assuming the disentanglement of different features of experiential quality or presuming on the orthogonality of their coordinate dimensions of formal quantity, for instance, as represented by the aspects of particles versus waves, or the axes of space versus time. Features and dimensions of experience that appear as relevant or arise into salience at one level of action, exchange, or observation can disappear from the scene of relevant regard at other stages of participation and weigh imponderably on other scales of transaction. In relation to one another, aspects and axes of experience that appear unrelated just so long as they are considered at one level of interaction and perception may not preserve their appearance of indifference and independence if the scales of participation under consideration are radically shifted, whether up or down in their order of magnitude. As a result, the sort of consideration that makes a line of experience conspicuous as it falls on one plane of existence is seldom enough to draw it through every plane of being. In a related fashion, the brand of consideration whose bearing on an intermediate scale of treatment causes one to regard two features or dimensions of experience as ''moderately independent'' or as ''relatively orthogonal'' is rarely ever relevant to all levels of regard and is almost never enough to justify one's calling these aspects ''absolutely independent'' or to support one's calling these axes "perfectly orthogonal".

Next, I examine the more specific features of the pragmatic theory of sign relations, focusing on attributes that are augmented in the degrees of their development and that acquire a distinctive emphasis along with the extension and growth of this theory. These features happen to be material in character, that is, they concern the contents of individual sign relations, affecting the aspects of relational structure and the orders of relational complexity that become especially conspicuous from the pragmatic point of view. Two of these specific material features are taken up next.

# Direct relation between objects and signs.
# Irreducibly triadic sign relations.

A few more things can now be said about the conditions that are usually taken for granted in the theoretical use of sign relations. Although it is often the case that the structure of the object domain is marked for reconstruction, part for part, in the partition of the syntactic domain, as one says, in the ''divisions'' of its ''quotient structure'', or equally, in the structure of its ''semantic equivalence classes'' (SECs), which are also called its ''semantic orbits'', it is advisable not to imagine, except in the most abstractly artificial and purely formal cases, that an object is “nothing but” an orbit of signs. In every situation of concrete or practical interest, the object domain is something that has a real existence, one that is independent of the syntactic domain to some degree, and to a degree of qualified independence that can be specified, for example, as ''absolutely'', ''moderately'', or ''relatively''. That is, an object exists in a manner that is more or less independent of both the signs and the interpretants that are used to talk and to think about it, as one sooner or later discovers in any real case where one is tempted to ignore the implications of this fact.

But this explanation of the status intended for objects only serves to elevate into prominence the subordinate question: What is meant by the relation ''independent of''? Outside the realm of mathematics, where the necessity has long been recognized of declaring one's independence in the form of an explicit and public definition, one that makes clear the sense of the term that one plans to uphold, this is an issue that still manages to incite an uproarious confusion of obvious claims and often just as obvious counter claims, each of which is just as insistent in what it attributes to the terms of its relation as the actual basis for what it considers evident is kept implicit.

One thing does appear certain, at least, once the issue is addressed: Whatever it means, and however it is qualified, the relation of being ''independent of'' does not mean a relation of being ''not in relation to''. After all, did I not just call this, with all due justice, a relation? Indeed, independently of all questions of independence, the very notion of there being a relation ''not in relation to'' is a self cancelling nullity. Perhaps the closest that one can approach to conceiving of relations like ''not related to'' or ''not a relative of'', in short, perhaps the simplest analogues or approximations to such a relation that one can devise are: ''considered as not in relation to'' or ''treated as not in relation to'', prompting the questions: ''considered by whom?'' and ''treated by whom?'', all of which goes to make it manifest that a triadic relation is the minimal support needed for any such brand of speculative relation.

In trying to reach a form of relation that is minimal in a certain regard, the analysis comes to a point where it is forced to reverse its direction and to synthesize a complex relation, one that possesses a higher arity than might be expected of a structure intended as a primitive rudiment. What is ultimately suggested is a triadic relation that formulates the idea of a ''consideration'', a ''regard'', or a ''treatment''. This involves an agent that acts as the overseer of the consideration, the regard, or the treatment in question along with a couple of other entities that fall in a dyadic relation to each other under this consideration, this regard, or this treatment. The way to treat this triadic relation as sparingly as possible, in regard to the level of consideration that is assigned to it, is to let the agency of this oversight ignore as much as it possibly can about all the relations that conceivably exist between the overseen pair, the couple of agents, entities, or objects that fall within its purview. Thus, the least that the overseer can manage to do is to mark a relation between the other two parties, without codifying, conveying, recording, or retaining any information about the particular kind of relation it is. At any rate, this is the best interpretation that I find myself able to contrive at present for “<math>X\!</math> regards <math>Y\!</math> as not in relation to <math>Z\!</math>”.

The preceding analysis may appear to lead up to a trivial point, but the argument just recounted is formally identical to a demonstration that is basic to pragmatic thinking, namely, that “we have no conception of the absolutely incognizable” (Peirce, CP 5.265). Whether or not one wishes to say that there are such things as the “absolutely incognizable”, we have no conception of them. Any concept that we do have cannot truly be a concept ''of'' them, that is, it cannot be held to be ''true'' of them, since this all by itself would amount in fact to making them cognizable. The idea that a successful conception is intended by its very nature to result in a true concept is critical and crucial in this regard. If one merely wants to point out the triviality that we can have false concepts of anything we please, for example, the false concepts that are attached to the verbal formula “absolutely incognizable”, then it is easy enough to stipulate that we are likely to have false conceits and misleading concepts about very many things indeed.

Since the pragmatic theory of sign relations welcomes partial symbols and verbal formulas of every species of description as well as mental impressions, concepts, and ideas of every genus and level of generation into the same broad class of entities that it takes as signs, the idea of an object that is not the object of a sign imparts a formal impression within its material that is identical, or at least indistinguishable in the structure of the relations that it suggests, to the idea of a relation that fits the verbal formula or the specious specification ''not in relation to''. A rigorous critique of these very ideas is required in order to prevent their specious impressions from flowering into malign oppressions that obsess both the mind and the spirit. The pragmatic critique of prior philosophy and the pragmatic theory of signs are intended, in part, precisely to address this task of weeding out delusive ideas.

=====5.2.11.6. Questions of Justification=====

There is a singular misunderstanding of this pragmatic perspective that needs to have its equally singular but bad effects blunted at this point. There is a definition of good conduct that is implicit in the pragmatic ordering of the normative sciences, but it is a characterization whose true import is frequently misinterpreted by seizing too quickly on one partial formulation or another of its full intention. If the pragmatic definition of good conduct is properly considered, in light of the full circumstances of its intended application, it does not lead to the bad end often associated with the fallacy of “the ends justifying the means”. In fact, the bad effects accountable to even so facile a formulation of the pragmatic desideratum, can be seen to result, in actual practice, from a faulty application of its own stated principle, going even so far as to ignore the expressly indicated pluralities of “ends” and “means”. But that is merely a verbal scruple. In the end, it does not matter whether one speaks of “ends” or the “end”. What really matters is that the term not be interpreted in too singular a way, but only with regard to the whole conceivable effect of each contemplated form of conduct.

Accordingly, in order to counteract the brands of bad faith that arise from ignoring this holistic sense, one needs to remember that an action has many consequences. Since an action has a multitude of results, a plurality of which conceivably contribute in significant ways to a truly balanced judgment of its goodness, an action is good only in so far as all of these results are good. If an action, intended primarily for the purpose of achieving a particular good, however successful it is toward that end, nevertheless has collateral consequences that are not so good, then the action is to that degree not so good as it otherwise might be. These are moral trivialities, of course, but just as easily trifled with, and apparently as likely to slip into oblivion for all concerned as they are likely to be slighted by some. But this is the nature of singularity.

From this pragmatic point of view, it is possible to deal with many questions of justification by invoking the contexts of amenities that surround the reasoning process, outside of which it cannot be pursued and without which it makes no sense. In this frame, one can say that reason is justified by its alternative, that is to say, by unreason, but only in the peculiar sense that reason is justified by considering the properties of unreason, by contemplating the ethical consequences of acting according to its dictates, and by recognizing the aesthetic fact that one does not like these consequences. Of course, this strategy of argument does not amount to a justification of reason in any positive sense of the word ''justification''. In logical force it is tantamount to the aesthetic tautology of simply insisting that one likes what one likes, but I see nothing unreasonable about this ''form of justification'' (FOJ), at least, as it is employed in this case. The whole point of noticing the placement of logic within the concentric spheres of ethics and aesthetics is that logical arguments depend on prior considerations of ethical and aesthetic casts, so a logical argument that merely recovers and iterates this context is acting in conformity with the only objective it knows.

By way of concrete examples, FOJs of a negative character frequently arise in situations that are affected by a genuine dilemma, where it is necessary to choose just one action from a set of two or more actions, where it is impossible to do nothing and impossible to do everything, and where each action excludes all the others. At such a juncture the structure of that very situation, or a reference to it as described, is itself a sufficiently valid FOJ for choosing some action, even if not yet a full justification for any one specific choice. If an agent challenged: “Why did you do that?”, responds: “It was necessary to do something!”, then that is “just” as far as it goes, to a general not a specific extent. In summary, one finds that there are FOJs of a negative character, that proceed by the rejection of an alternative, but that are perfectly valid in their doing so. These FOJs of a negative character exist in contrast to the more familiar FOJs, at least, the more often expressed FOJs, all of which are positive and transitive in character, identical or analogous to the various forms of logical implication and logical consequence.

The dictum to the effect that ''there is no argument in matters of taste'', for what it is worth, enjoins an ''argument to'', not an ''argument from''. An aesthetic principle or judgment, that I prefer to live, for example, can have definite logical consequences, even if every justification I can expect to find for it is ultimately circular, tautologous, or logically speaking, trivial in form: “Why do I like it?” — “Just because I do!” To me, this cannot help but seem, if challenged in a case of this kind, to be a perfectly adequate and a reasonably sufficient answer. But to maintain this reason means to preserve this life, and that in its turn has decidedly logical consequences. It is likely that artisans and engineers have an easier time understanding this pragmatic principle, what it means in active practice and the wisdom it holds in general, than many varieties of logicians, mathematicians, philosophers, and scientists, although if I say that I pick my axioms with an eye to the beauty of what they can shape, in other words, that I select my logical and mathematical principles for what are essentially aesthetic reasons, then there are evidently some in these guilded ilks who already know what I mean.

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<p>Socrates not only used irony but was so dedicated to irony that he himself succumbed to it.</p>
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| align="right" | Kierkegaard, ''The Concept of Irony'', [Kier, 5]
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A person who drinks an extract of hemlock for what he says is a reason of logic either suffers from a confusion of priorities or acts according to a higher aesthetic than that of saving his own small portion of life. But a person who drains the tendered glass for lack of lighting quickly enough on a reason why not is a person who has let his extract of logic turn to a poison in its own right.

=====5.2.11.7. The Experience of Satisfaction=====

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| Die unbegreiflich hohen Werke
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| Sind herrlich wie am ersten Tag.
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| The world's unwithered countenance
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| Is bright as on the earliest day.
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| width="60%" |   || Goethe, ''Faust'', …,
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| width="60%" |   || quoted in Weyl, ''The Open World'', [Weyl, 29].
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With these considerations freshly in mind, it is possible to return to the more immediate questions:

: Why is it useful to keep a store of memory?
: How does a record of past experiences serve an agent in meeting its present goals and thereby in achieving future satisfactions?

Assuming that an agent, however accidentally, elliptically, obliquely, occasionally, partially, or transiently it may happen, ever experiences a state of satisfaction, as reflects its achievement of one of its objects or as marks its identity with one of its goals, then it is likely to be useful for that agent to try to keep track of all the incidental experiences that accompany or surround this "experience of satisfaction" (EOS). Because the questions of causal order and even of purely temporal simultaneity are difficult in general to resolve in real time, in medias res, it is advisable for the agent not to focus too fixedly on trying to sort out the precedents from the consequents, at least, at first. But why is it likely to be useful? And what does it mean to be useful? Responding to these questions requires another apparent departure, as follows.

What does it mean to be useful? It means to further the purpose of an agent's present, continued, or future satisfaction. It means to help an agent achieve a specified goal, or else, failing the possibility of that, to help an agent know the reasons why a particular goal is impossible. Taking this as a satisfactory answer on this score, for now, it leads to the question of why a record of previous experience is likely to further the purpose of future satisfactions. Looking forward to the point when these issues of justification are out of the way, I will then be able to focus on the purely technical task of showing how sign relations can be used to construct many varieties of extremely flexible memory stores, not just accumulating the images of past experience, but indexing their elements in ways that make it possible to analyze their logical imports.

If there is any consistency to experience, in other words, any form of lawful relationship between one sample of experience and other samples of experience, then it follows that almost any kind of memory structure, any facility for attention and retention that an agent can contrive to organize the interaction between transient experience and the orders of its more persistent signs, any faculty that allows an agent to note the sundry aspects of a satisfying experience or the circumstantial details of a satisfying situation, any organization of processes that permits an agent to fashion periodic or persistent notes of the tangent experiences that surround an EOS, whether these stores are internal to its initially given body of resources or external to its innate endowment, is likely to be of service in achieving future satisfactions. Properly organized for quick access, the whole index of past experience can serve as a catalyst for future achievements, in other words, it can act on the whole as the sort of sign that is conducive to actualizing its object.

One may well ask: Is there any form of lawful relationship between one sample of experience and other samples of experience? To say ''yes'' too quickly is practically vacuous, that is, it is empty of anything beyond the vaguest hopes of an implication for action, until one is willing to risk the assumption of a specific form of lawful relationship. For the sake of proposing a non trivial stake, what one needs and desires is an ''informative'' form of lawful relationship. I am not making any form of fixed assumption here, but merely contemplating the forms of hypotheses that I am able to consider as possible. I admit to occasionally having experiences that cause me to question all the more frequently exploited answers to this question, that tempt me to say ''no'' even to this modest quantum of presumption, but then I note that it is particular varieties of experience that lead me to say this and specific brands of laws that I am led to question. Then I notice that all the forms of contrariety, disagreement, discord, discrepancy, disharmony, disparity, dispersion, dissension, distribution, diversification, incongruity, and opposition that I encounter at these junctures are themselves distinctions with a difference, and each in its way renders a generic form of distinction, “which, taken at the flood, leads on to fortune”, to wit, a wealth of unsuspected approaches to the problems that positive experience poses. Without being tempted to classify or to enumerate the full diversity of logical forms by which differing samples of experience come to grate on and to grind against each other, it is possible at this point to notice their essentially differential, negative, and oppositional characters.

In this way, I arrive at the conclusion that ''forms of negation'' (FONs), or fundamentally negative logical relations, are unavoidable necessities, needed to anchor any adequate basis for stating the forms of interaction among different samples of experience. One finds, instead of a positive foundation, that irreducibly negative operations are inescapable notions, needed to support any satisfactory system of notations for detailing the collisions and the collusions that particles of experience impart to one another. In spite of the aura of negativity that chances to shade their logical aspects, to color their evidential impacts, and to weigh against their positive receptions, the counter exemplary characters and conducts of these bearings of experience on experience do, at a minimum, contrive to convey “informative forms of lawful relationship between one sample of experience and other samples of experience”.

The circumstance that absence, necessity, and privation are the mothers of invention, possibility, and plenitude is a difficult fact of logic to accustom oneself to, apparently because of the mind's innate blind spots in regard to its own nature and partly due to the mind's acquired bias in favor of positive relationships. Against these blocks and in accord with this bias, the aspects of nullity and vacuity that arise in respect of logical FONs are frequently responsible for leading the mind astray, for instance, into supposing that these FONs are:

# Purely derivative abstractions from wholly positive contents and structures of experience,
# Partially selective extractions from primary materials of experience and primitive elements of reasoning,
# Secondary, tertiary, and higher order constructions that are based on and built from basically positive forms of empirical and rational connection, and
# Wholly dependent for their practical utility and their rational justification on contents of positive experience to fill out their sparingly minimal forms.

By way of contrast, it is possible to identify a couple of dimensions along which the variety of clarification tasks can be classified into coherent associations among themselves and coordinated with each other.

<ol style="list-style-type:decimal">

<li><p>One interpretation of the clarification task fixes the object and the class of signs that figure in as implied arguments of the operation, and thus it understands the task as a process of refining the quality of significance that stems from the individual signs, that is, developing a clearer interpretant for each sign given in the input class. I refer to this brand of clarification as a ''modeling'' process, for several reasons:</p></li>

<ol style="list-style-type:lower-latin">

<li><p>Speaking of signs in the generic sense, this mode of clarification process involves the finding or the making of model signs to serve as their clarified interpretants. In other words, it takes in signs of an arbitrary quality of clarity and replaces them with their ''canonical'', ''normal'', or ''standard'' equivalents, ones that have an improved or optimal level of clarity.</p></li>

<li><p>Speaking of signs in the sense of logical expressions, this mode of clarification process involves the detection, enumeration, and organization of their logical "models", that is, their logically satisfying interpretations.</p></li>

</ol>

<p>Overall, this brand of clarification can be viewed a purely cognitive, intellectual, syntactic, or rational process, one that goes on in the absence of any interaction with the object domain beyond the initial sample of signs.</p>

<li><p>Another interpretation of the clarification task allows the object, or the information that the object avails of itself, to change over time. It is not that agents always have a lot of choice in the matter of whether changes occur or what changes take place, but only that they do have the option to envision the possibility of changes in the objects or the data, and accordingly to contrive systematic ways of tracking these changes and accounting for the developments of objects and signs through time. This rendition of the general requirement to “increase the clarity of the signs that the agent possesses about the object” comprehends the task of clarification as a matter of increasing the quantity of the agent's possessions in that regard, and it leads to a class of strategies in which agents proceed by gathering each new sign that they find of the object into the class of signs that forms their sample.</p></li>

</ol>

=====5.2.11.8. An Organizational Difficulty=====

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<p>Moreover, I did not wish to begin to reject completely any of the opinions which might have slipped earlier into my mind without having been introduced by reason, until I had first given myself enough time to make a plan of the work I was undertaking, and to seek the true method of arriving at knowledge of everything my mind was capable of grasping.</p>
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| align="right" | Rene Descartes, ''Discourse on Method'', [Des1, 39-40]
|}

At this point in my text I run into what I think of as an ''organizational difficulty'' (OD). I have already written sixty plus pages of this text and, if the stacks of notes and previous drafts that I find around me are any evidence, I am likely to write many pages more. …

=====5.2.11.9. Pragmatic Certainties=====

This brings me to the point of asking: What does ''certainty'' mean in practice, that is, what meanings can be revealed for the concept if one attempts to translate the intentions behind it into operational terms? Once one bothers to ask this eminently practical question, it becomes reasonably clear almost immediately that no brand of absolute certainty is ever in required practice. For practical purposes, only a moderate amount of certainty is demanded, just enough for a particular agent to settle on a particular course of action. A question about the kind of certainty expected or the level of certainty needed in order to make a decision is itself an interpretive issue. In other words, no matter whether its instances remain to be decided on a case by case basis, or whether a general rule can be formed to cover them, their resolution occurs in a manner that retains an irreducible degree of arbitrariness about it, since it must relate to the degrees of freedom possessed by the agents who arbitrate the matter in question. In the final analysis, this is an issue that devolves upon the nature and the constitution of the very form of agency that finds itself concerned with the question and exerts itself according to its interest in the action. Namely and solely, this form of agency can be comprised of nothing other than the particular agents and the communities of agents who are compelled or inspired to act at the moment in question.

The idea that certainty is needed to begin, whether to start thinking or to get moving in any direction of conduct, is one of the most paralyzing traps that the mind can let itself fall into. This is why the pragmatic theory of inquiry emphasizes uncertainty as the literal start of inquiry, since there is certainly no difficulty about the mind finding itself in a state of uncertainty. Thus, there is no scarcity of events to throw the mind into confusion, no trouble at all getting into trouble, and so this renders the whole fabric of one's experience rife with moments of doubt. But what is the mind to make of this dubious resource, and what is the ultimate good of noticing this abundance of ambivalence in the mind?

There is much that is curious in the picture of uncertainty that I just presented. The paradoxical phrase “certainly no difficulty”, that seemed to pop up from nowhere in my description of the situation, is evidently an artifact of reflection, that is, due to the reflective character of the description but not an element of the situation described. Specifically, it focuses on what appears at first to be a purely incidental triviality: How easy it is to find oneself in a state of uncertainty. More carefully, since to "find oneself" may be still too much to expect at such an early stage of the game, it may be said: How easy it is to be for the moment or to end up momentarily in a state of uncertainty. And yet, with yet another reflection, one is forced to ask: To what exactly, whether an aspect of the original situation, a newly introduced amendment to it, or a newly generated outlook on the situation, do all these attributions of certainty, ease, freedom from trouble, and lack of difficulty apply? Since it seems a contradiction to attribute these predicates to the problematic state of uncertainty itself, that is to say, they are not what's the matter in the original situation, they must belong either to the attitude of approach that is capable of reflection, to the ensuing state that is entered on reflection, or to the manner of viewing the whole situation.

Thus, it is possible to distinguish between two collections of properties, or, if one prefers, between two different applications of the same set of predicates:

# Those that affect the state of inquiry, its matter, and
# Those that affect the attitude toward it, the manner of regarding it, whether of absorption and irreflection or reflection and understanding.

In order to keep track of this distinction, I introduce the designations of ''lower order'' (LO) and ''higher order'' (HO) properties, attitudes, or applications of predicates.

“Aha!” you might say, here is the hidden certainty that is needed to begin the inquiry, the initial knowledge of its true motive force that makes the whole process of inquiry feasible, the unshakable faith in its prime mover that is required to rest before the rest can get started, or the ultimate security that is necessary to sustain the entire endeavor. But it is not that, not yet. The particular brand of HO certainty that arises in this situation is actually of very little use in resolving the original uncertainty. Although it can provide a modicum of security, a small peace of mind, and serve as a sop to Cerberus at critical times, the invitations to this escape can be as distracting as the delights of its certainties are in fact seductive, and the exercise of this method to the exclusion of risking the perils of real experience can just as easily become the main obstruction to the further progress of inquiry.

This business of ''being certain that one is uncertain'', the enterprises of reasoning that vie to capitalize on its purely derivative securities, the foundations of thought that try to indemnify themselves against all risk and against all hope through the guarantees of its instruments, the certificates of its stocks, and the surety of its bonds, the companies of philosophers, incorporated and limited, who leave their precious earnings so heavily invested in the specious lightness of its bearing, and all the subsidiary entertainments that are produced in pursuit of this spectacle, that grow in the absence of more penetrating lights to delight the hosts of spectators, to enrich the parasites of apparent productivity, that accrue in lieu of more genuine profits to all the participants and their silent partners who count themselves party to this form of gamble, and that provide nothing more than a nominal incentive, both for those who stake their personal fortunes on the receipts of the intervening days and those who bet their pari mutuel interests on the outcome of running along this track — this entire business, that strives to busy itself at any cost whatsoever, to protect its investment for its own sake, and to insure its continuance by means of any excuse it can arrange for itself, eventually leads to the following strategy: to institute a discipline whose rationale is precisely that of issuing warrants for unnamed and probably unnameable apprehensions, of handing down sealed indictments whose principals on principle remain as impeccable as they are obscure, and of rendering forms of certification for forms of belief that have already surrendered their original contents — this tiresome business can easily become so all consuming and global in its sphere of influence, while curiously remaining so provincial and local in its motives, that it totally derails the original train of thought.

What accounts for the fact that one way to certainty, the ostensibly “higher” order, so often gets favored to the exclusion of the other? Perhaps it is something in the nature of the one track mind that only one brand of certainty can be pursued at a time. Perhaps it is partly due to the implicit judgments of “lower” and “higher” and partially on account of the adventitious implications that cannot help but slip into their making. Although these terms originally attach themselves to the discussion as convenient labels, intended solely to mark the sides of a purely formal distinction, and in spite of all the arbitrary characters that go into their nominal conventions, the nature of the associative mind is such that these tokens are almost bound to mount up in time to the point where they come to represent judgments of value, to symbolize in intuitively suggestive or in strictly illicit manners something beyond their original intentions, and thus to connote guilt or gilt by means of their informal associations. Absurd, I know, almost as if the even more innocuous words ''left'' and ''right'' could come to represent significant value judgments. Still, it happens.

As a guard against the deleterious effects that frequently emerge from the drawing of a distinction between LO and HO attitudes of certainty, no matter whether the division concerns the attributions of properties or the applications of predicates, and that commonly arise from making the various lines of LO and HO tracks express enough to carry between them a significant import but not equal enough for both to carry their share of the moment, I must take care that the tracks laid down in the building of a RIF, and all the actions conducted on their basis, are always adequate to facilitating both levels of inquiry in parallel.

Triadic relations are a staple element of architecture that can serve the purpose of coordinating LO and HO inquiries, since a triadic relation can incorporate the dyadic relation that describes the transition from one state of inquiry to a subsequent state of inquiry, while still keeping track of its relationship to developments occurring on the other track. This allows the orders of developments taking place within each inquiry, and the sequences of states extracted from their processes, to proceed uninterrupted, but not uninterpreted, by each other's concerns, and to exhibit a partial independence, but an adequate correlation, with each other's progress.

In order to carry this discussion of certainty through with a maximum of ease, I need to find a battery of descriptive terms for the situation of uncertainty that is neutral with respect to two interpretations, that covers with equal facility the two kinds of uncertainty that one usually faces in a situation: (1) uncertainty about what is true in a situation, and (2) uncertainty about what to do in a situation. Along these lines, I describe the typical situation of uncertainty, encompassing both kinds of doubt that are fraught with peril for an agent, as ''junctures''.

For the sake of a convenient classification, I label the juncture that presents a problematic phenomenon, a surprising or unexpected state of affairs, with the generic name of a ''surprise'', and I label the juncture that presents a phenomenal problem, a demanding or unintended state of affairs, with the generic name of a ''problem''. Junctures do not always sort themselves out into cases that are clearly one or the other type, but when they do it simplifies the manners of addressing, approaching, and ultimately resolving the difficulties they present for the agent.

# If it is the aspect of a ''surprise'' that is dominant at a juncture, or the role of a spectator that is prominent for an agent, then the juncture is resolved, in its theoretical aspects, by finding an ''explanation'', a statement expressing a way of looking at the juncture that renders it less of a surprise.
# If it is the aspect of a ''problem'' that is dominant at a juncture, or the role of an actor that is prominent for an agent, then the juncture is resolved, in its theoretical aspects, by finding a ''plan of action'', a statement expressing a way of moving from the juncture that renders it less of a problem. Of course, it remains for the plan or theoretical resolution to be carried out in practice before the problem itself can disappear.

If the uncertainty that one experiences in facing a juncture reflects the complexity of the juncture that faces one there, and if these are related to the difficulty that one is likely to have in resolving the juncture, then the appropriate analysis of these complexities, difficulties, and uncertainties into several parts can serve to advance the process of their resolution.

With this picture of an agent at a juncture, appraising the uncertainties that affect the agent in that situation, indicating the complexities and the difficulties that the situation presents for the agent to resolve, sketching the forms of analysis that are called for in the process of resolution, and suggesting the relationships that obtain among these diverse ingredients of the situation, it is feasible to return to the problem of the ''cartesian step'', the one that moves from ''dubito'' to ''ergo sum'', and that simultaneously, as if perforce its very passing, creates the distinction between the LO and the HO attitudes of certainty. Can the cartesian step be viewed in this light, that is, can it be placed in a suitable way within this picture of junctures and resolutions, to be specific, posing a form of analysis that advances the cause of certainty? And if so, how does it appear when regarded in this light, that is, how well does it perform with respect to its conjectural role in reducing a fundamental uncertainty of the agent concerned?

In a sense, the cartesian step splits the agent's initial juncture into a couple of parts, or ''subjunctures''. In this attempt at resolution, there is a part identical to the initial juncture, and thus with an uncertainty of the original severity, plus a part that the agent is sure of, and thus with an uncertainty of zero. But this sort of analysis only works if it brings to light subjunctures of the initial juncture, or subsituations of the initial situation, that are actual ingredients, proper components, or non trivial constituents of it. When the HO certainty does not have an effective bearing on resolving the LO uncertainty, then the pretense of analysis is only a distraction, not a step toward a genuine resolution.

Unless the HO answer that is revealed by dint of the cartesian step has an application to the LO question that instigated the original inquiry, one that reduces the LO uncertainty that initially justified the effort, then it does not have a genuine bearing on the LO juncture that led to putting this inquiry in gear and setting its proceedings into motion, and it cannot bring to bear on the ensuing activity or the ongoing process the modicum of traction that is needed to put a brake on its continuing. But a partition of a level of uncertainty into the very same amount plus a quantity of zero is hardly a sum, however much it seems on the level, that inspires much confidence in either the practical sincerity or the ergo nomic utility of the putative sum.

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<p>When Descartes set about the reconstruction of philosophy, his first step was to (theoretically) permit scepticism and to discard the practice of the schoolmen of looking to authority as the ultimate source of truth. That done, he sought a more natural fountain of true principles, and thought he found it in the human mind; …</p>

<p>Self-consciousness was to furnish us with our fundamental truths, and to decide what was agreeable to reason. But since, evidently, not all ideas are true, he was led to note, as the first condition of infallibility, that they must be clear. The distinction between an idea seeming clear and really being so, never occurred to him.</p>
|-
| align="right" | (Peirce, CP 5.391).
|}

In the discussion that follows, I am going to use the letters <math>C, L, M\!</math> to stand for three generic features or classes of properties, yet to be fully analyzed or completely specified, that are commonly appreciated, desired, or valued as virtues of signs and expressions. For now, a list of adjectives appropriate to each class can give a sufficient indication of their intended characters, even though it is easily possible and eventually necessary to find important distinctions that exist among the items in each given list of exemplary properties.

# The class <math>C\!</math> is suggested by the adjectives ''certain'', ''cogent'', ''compelling'', or ''convincing'', and, in some of their senses, by ''apparent'', ''evident'', ''obvious'', or ''patent''.
# The class <math>L\!</math> is suggested by the adjectives ''clear'', ''lucid'', ''perspicuous'', ''plain'', ''relevant'', or ''vivid''. To the geometric imagination, these terms suggest a ''bluntness'' (of surfaces) or a ''sharpness'' (of edges).
# The class <math>M\!</math> is suggested by the adjectives ''distinct'', ''decided'', ''defined'', ''definite'', ''determinate'', ''different'', ''differentiated'', or ''discrete'', and, within a stretch of the imagination, by ''acute'', ''conspicuous'', ''eminent'', ''manifest'', ''poignant'', ''salient'', or ''striking''. To the geometric imagination, these terms suggest a ''pointedness''.

In this frame of thought, it needs to be understood that the intended sense of these last two classes excludes the common usage of words like ''clear'', ''clearly'', and so on, or ''distinct'', ''distinctly'', and so on, as elliptic figures of speech that are intended to be taken in a more literal way to mean ''clearly true'', and so on, or ''distinctly true'', and so on.

In this connection, when I mention one of these properties it is only meant as a representative of its class. Also, as they are used in this context, these terms are intended only in what is diversely called their ''impressionistic'', ''nominal'', ''subjective'', ''superficial'', or ''topical'' sense, implying the sorts of qualities that one can judge “by inspection” of the expression and its immediate situation, and without the need of a prolonged investigation. Thus, none of their intentions is damaged for this purpose by prefacing their proposal with an attitude of ''seeming''. For all one cares in these concerns, <math>{}^{\backprime\backprime} \operatorname{seems}~X {}^{\prime\prime} = {}^{\backprime\backprime} X {}^{\prime\prime},\!</math> for <math>X = C, L, M.\!</math> This makes the judgment of these qualities a matter of ''seeming syntax'' and ''seeming semantics'', involving only the sorts of decision that are commonly and easily made without carrying out complex computations or without delving into the abstruse equivalence classes of expressions.

People frequently use the adverbs ''immediately'' or ''intuitively'' to get this sense across, and even though these terms have technical meanings that prevent me from using them in this way in anything but a casual setting, they can do for the moment. Still, when I use ''immediately'' in this sense it is meant in contrast only to ''ultimately'', and more or less synonymous to ''mediately'', suggesting that which holds in the meantime. In a pinch, a determination of seeming certainty or seeming clarity is enough to put an inquiry on hold for a time being, but the distinction between ''seeming so to me, for now'' and ''seeming so to all, forever'' still holds, with only the latter deserving the title of ''being so''.

These observations on im/mediate, intuitive, or meantime determinations of certainty, clarity, and distinctness have a bearing on the styles of mathematical formulation and the modes of computational implementation that are candidates for mediating a natural style of inquiry, in other words, the sort of inquiry that a human being can relate to. Because a decision that a sign or expression has one of the virtues <math>C, L, M,\!</math> even to a mediate, a moderate, or a modest degree, is often enough to end an inquiry on a temporary basis, it becomes necessary to recognize a form of recursive foundation that also rests on a temporal basis. And yet, because these modes of judgment are all the while fallible and subject to change, it is possible that deeper foundations remain to be found.

What does this mean for the topic of reflection? Well, reflection is precisely that mode of thinking that is capable of beginning with the axioms and working backward, that is, of searching out the more basic forms that conceivably underlie one's received formulations.

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<p>I thereby concluded that I was a substance, of which the whole essence or nature consists in thinking, and which, in order to exist, needs no place and depends on no material thing; so that this “I”, that is to say, the mind, by which I am what I am, is entirely distinct from the body, and even that it is easier to know than the body, and moreover, that even if the body were not, it would not cease to be all that it is.</p>
|-
| align="right" | Rene Descartes, ''Discourse on Method'', [Des1, 54]
|}

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<p>And voila, I have, finally, spontaneously returned to there where I wanted to be. For because it now be known to me that bodies themselves are properly perceived not by the senses or by the faculty of imagining, but rather by the intellect alone, and that bodies are perceived not from thence that they would be touched or seen, but rather from thence only that they were to be understood, I cognize overtly that nothing can be perceived by me more easily or more evidently than my mind.</p>
|-
| align="right" | Rene Descartes, ''Meditations on First Philosophy'', [Des2, 117]
|}

On reflection, the observation that appeared just before these last questions arose can be seen to make a very broad claim about a certain class of properties affecting expressions, namely, all those properties that can be analogous to the ordered measures of expressive quality. For future reference, let me call this the ''monotone assumption'' (MA). This generatrix of so many future and specious assumptions takes for granted a sweeping claim about the ways that an order of analysis of expressions translates into an order of comparison of their measures under one of these properties. But this entire and previously unstated assumption is itself just another manner of working hypothesis for the mental procedure or the process of inquiry that makes use of it, and its proper understanding is perhaps better served if it is rephrased as a question: Can the <math>X\!</math> of a claim or a concept be greater than the <math>X\!</math> of the subordinate claims and concepts that it calls on, where <math>{}^{\backprime\backprime} X {}^{\prime\prime}\!</math> stands for ''certainty'', ''clarity'', or any one of the corresponding class of measures, orders, properties, qualities, or virtues?

Rather than taking this claim for granted, suppose I go looking for any properties, that might be similar to certainty or clarity, for which the measure of a whole expression is capable of exceeding the measure of its parts. Is there an order property that is dependent on the constitution of the whole expression and a function of its analytic constituents but not necessarily tied down to monotonically conservative relationships like the sum, the average, or the lowest common denominator of the measures affecting its syntactic elements? Once I take the trouble to formulate the question in explicit terms, any number of familiar examples are free to come to mind as fitting its requirements. Indeed, since the notions of dependency and independence that accompany the use of mathematical functions and mathematical forms of decomposition do not by themselves implicate the more constrained types of dependency and the more radical types of independence that arise in relation and in reaction to the MA, it is rather easy to think of many that will do.

=====5.2.11.10. Problems and Methods=====

The relationship between a “problem” and a “method” needs to be given another look, in view of the discussion that has transpired since the initial steps of this proposal. …

To approach the distinction between problem and method in the present setting, in the light of the discussion that has transpired since I naively assumed a distinction between them, …

What is the nature of the relationship between a problem and a method? What is the distinction between them, and what sort of difference is it? These questions are made especially acute in view of the fact that the present inquiry nominates “inquiry” to both of these roles, proposing to take “inquiry” as naming both a problem and a method. If it makes any sense to do this, and if there is anything to the distinction between a problem and a method, then it must be a distinction of relational roles rather than a distinction of absolute essences. Trying to make sense of this requires me to ask: What manner of common, generic, indifferent, or shared existence do a problem and a method both possess, logically prior to taking on their distinctive roles in relation to each other?

In approaching the distinction between a problem and a method, I use a piece of advice that is helpful in approaching any important distinction, especially a distinction that is naively taken as given or a distinction that has been taken for granted for too long a spell of time, like the distinction between the problem and the method, the work and the tool, or the object and the sign. This recommends that one stand back from a full involvement in the drawing of the distinction under review, to partially withdraw one's commitment to having it drawn the way it is, and to contemplate how it came to be drawn that way in the first place, in other words, to consider the process that initially draws it and that keeps on drawing it in just the way that it presently appears.

If this is done, then one realizes that the problem and the method are both constituted in part by the way their distinction is drawn, by the sort of distinction that one takes it to be, whether a sign of the roles that entities take up in relation to each other or a mark of the natures that entities have in and of themselves, and by the items that one takes as instances on either side of the distinction. In this way, every form of distinction, with respect to the contents of its counterposed sides, plays the role of a mediator in their mutual constitution of each other.

Standing back from the picture a little further, one can see that the distinction between a problem and a method is itself a tool of method, and one that is not ordinarily considered to be a problem. To see this, notice that ''distinction'' is an ''-ionized'' term, and thus denotes both a process and a result, the process being the drawing of the distinction and the product being the distinction drawn, so any form of distinction is available for consideration in the light of its instrumental meaning. In this regard, the distinction between a problem and a method is itself an instrumentality of reasoning, a procedural means to an end, in short, a method or a tool. This particular distinction, between a problem and a method, falls among those of a very basic order, the kind that one takes as given without hesitation or reflection, uses to construe almost every situation that one finds oneself in, and does not usually question the utility of, until, as presently, some special attention is drawn to it. In summary, the distinction that is drawn between the problem seen and the method used, the conventional form of designation that says what is the work and what is the tool, is itself an artifice, a construct, an invention of the mind, or an intervention of the thinking process whose correspondence with anything else in reality and whose constitution as an enduring reality in itself is something that demands to be tested.

If the terms ''problem'' and ''method'' refer to phenomena and activities that take place in the world, then that is one mode of existence they have in common. If the “world” is further circumscribed to the kinds of phenomena that have effective descriptions, that is, computational models, and the kinds of activities that have effective prescriptions, that is, computational implementations, then the mode of existence one commonly denotes by means of programs, codes, effective procedures, or other practical recipes is another domain that is capable of providing instances that fill both the roles of a problem or a method. Taking a clue from this interpretation, I can shift my approach to the question and consider the medium of signs that is used to address the things in comparison. By starting with the syntactic side of the issue I avail myself of ready made handles on the question, even if the mechanism of these conventional modes quickly becomes a difficulty in its own right, blocking further progress and demanding to be tackled in terms of the influential biases and the instrumental characters it brings to bear.

===5.3. Reflection on Reflection===

====5.3.1. Looking Back====

Let me review the developments that bring me to this point. I began by describing my present inquiry, <math>y_0,\!</math> as an inquiry into inquiry, <math>y \cdot y.\!</math> Then I focused on the activities of discussion, <math>d,\!</math> and formalization, <math>f,\!</math> as two components of the faculty or the process of inquiry, <math>y >\!\!= \{ d , f \}.\!</math> This led me to the present discussion of formalization, <math>f \cdot d.\!</math> Considered as classes of activities, the collective instances of formalization, <math>F,\!</math> appeared to be encompassed by the collective instances of discussion, <math>D,\!</math> thereby yielding the relationship <math>F \subseteq D.\!</math>

I initially characterized discussion and formalization, in regard to each other, as being an “actively instrumental” versus a “passively objective” aspect, component, or “face” of the inquiry <math>y >\!\!= \{ d , f \}.\!</math> In casting them this way I clearly traded on the ambiguity of “-ionized” terms to force the issue a bit. In other words, I used the flexibility that is freely available within their “-ionic” construals, as processes or as products, to cast discussion and formalization into sundry molds, drawing out the patent energies that are manifested by the active process of discussion and placing them in contrast with the latent inertias that are immanent in the dormant product of formalization. In this partially arbitrary way, I decided on the one hand to treat discussion in respect of its ongoing process, the only thing that it has any assurance of accomplishing, but I decided on the other hand to treat formalization in respect of its end product, the abstract image or the formal model that constitutes its chief qualification and thus becomes the mark of what it is.

By casting inquiry into the form <math>y >\!\!= \{ d , f \},\!</math> I made it more likely that my development of its self application <math>y \cdot y >\!\!= \{ d , f \}\{ d , f \}\!</math> would first take up the application of discussion to formalization, <math>f \cdot d,\!</math> and only later get around to the application of formalization to discussion, <math>d \cdot f,\!</math> that brings the active side of the formalization process into a greater prominence. But the bias that I exploited in these readings does not seem at present to be a property of the incipient algebra that would determine the sense of the applications and the decompositions envisioned here. Thus, if I initially saw a difference between the two presentations <math>\{ d , f \}\!</math> and <math>\{ f , d \},\!</math> then it must have been a purely interpretive and not a substantial one, and the task of giving explicit notice to these interpretive distinctions and working out their algebra or calculus yet remains to be carried out in any sort of convincing fashion.

Still, the casting of discussion and formalization as active and inert, respectively, was not entirely out of character with their distinctive natures, since a process that has an end is more naturally suited to be represented by its result than a process that conceivably never ends. And whereas a ''discussion'' was allowed to be a form of discourse that does not need to have an end, with the possible exception of itself, a ''formalization'' was sensed to be a form of discourse that has, needs, seeks, or wants a distinct end, not just any end but a form of product that is preferred to satisfy a general description, and one that most likely resides outside the form of a vacuous vanity that simply refers, in a reflexive but hollow echo, to the entirety of its own proceedings.

In this merely penultimate analysis, and to the extent that the question of ends has been analyzed up to the present, it needs to be noted that more than a bit of ambiguity yet remains. When one speaks of a form of discourse each of whose instances necessarily has an end, does one mean that the definition of the form requires each instance to have an end, and does one then mean that each valid instance actually achieves its end, or does one only mean that each instance of some empirically given class of discourses actually reaches some end or another?

The word “reflection” first entered this discussion in what seemed like a purely incidental and instrumental way, as a part of the definition of a “meditation” as “a discourse intended to express its author's reflections or to guide others in contemplation” (Webster's). I converted this term to my own use as a name for a particular class of activities, describing the class of ''meditations'', <math>M,\!</math> as a brand of ''measured'' and ''motivated'' discussions that can serve to mediate formalizations within the realm of discussions at large. Thus, I borrowed the term for no better reason than that of interposing a middle term between formalized discussions and discussions in general, thereby yielding the relationship <math>F \subseteq M \subseteq D.\!</math>

In this respect, it seems to be instructive that the issue of reflection first arrived on the present scene, quietly enough, under the aegis of a borrowed term, imported without deliberate design among the components and the connotations of its associated sample of discourse, and involved in a process that seeks to negotiate the conflicting claims that arise between formal and casual discourse. In the simplest sense of the word, an activity of reflection implies only that an agent thinks quietly and calmly about a matter, the etymology of the word suggesting the actions of bending, bounding, casting, folding, giving, turning, throwing, or yielding back again, and hence a pause, a return, or a review. In this regard, the word "reflection" barely alludes to the idea that what the agent turns back to is something that involves itself, its own patterns of activity, and thus the word only hints as yet at the complicities of self reference and self application that are involved in an agent turning back to view its past, its present, or its ongoing forms of conduct.

Whereas I dragged the topics of discussion and formalization somewhat arbitrarily and forcibly into the arena of inquiry, the issue of reflection appeared to develop naturally on its own, without significant foresight on my part, and without deliberation or design gradually forced itself on my attention, as if rebounding from contingent obstacles that exist just beyond my current horizon, reflecting the necessary constraints of a natural law, or echoing the bounds of an inherent capacity that I yet know not of.

When one crosses a critical threshold or a threshold of decision, ...

A notion of reflection, in a more authentically reflexive sense, was implicitly involved in the application of inquiry to itself, <math>y_0 = y \cdot y,\!</math> and was eventually encountered on a recurring basis in the application of each newly recognized component of inquiry to itself: <math>y \cdot y >\!\!= d \cdot d, f \cdot f.\!</math> In a more substantial role, the option of a capacity for reflection was already noticed as a significant parameter in the constitution of an IF.

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<p>Often, an IF is founded and persists in operation long before any participant is able to reflect on its structure or to post a note of its character to the constituting members of the framework.</p>
|-
| align="right" | (§ 1.3.3.5, page 14)
|}

More substantially, a notion of reflection was invoked as something necessary …

In other contexts, something called “reflection” was seen as necessary to avoid certain types of unfortunate outcomes, …

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<p>To avoid the types of cul de sac (cultist act) encountered above, I am taking some pains to ensure a reflective capacity for the interpretive frameworks I develop in this project.</p>
|-
| align="right" | (§ 1.3.3.5, page 15)
|}

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<p>A radical form of analysis &hellip requires interpreters … to reflect on their own motives and motifs for construing and employing objects in the ways they do, and to deconstruct how their own aims and biases enter into the form and use of objects.</p>
|-
| align="right" | (§ 1.3.4.12, page 36)
|}

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<p>Thus, the radical project in all of these directions demands forms of interpretation, analysis, synthesis that can reflect a measure of light on the initially unstated assumptions of their prospective agents.</p>
|-
| align="right" | (§ 1.3.4.12, page 36)
|}

The relationships among the activities and faculties of discussion, contemplation, formalization, meditation, and reflection need to be explored in more detail. In particular, the relationship between formalization and reflection is especially relevant to the task of constructing a RIF.

Unlike a discussion of discussion, <math>d \cdot d,\!</math> which is easy to start and hard to put an end to once it gets going, it is difficult for a reflection on reflection, <math>r \cdot r,\!</math> to get itself going with nothing to reflect on but itself. I have just illustrated one way of doing this, namely, by leading a text to reflect on itself, as long as you understand this figure of speech to mean that it leads its interpreters, its writer and reader, without whose agency there would be no reflection at all, to reflect on how it reflects on itself. But I obviously need other ways than this to demonstrate the functions and properties of reflection in anything like their full variety.

Toward this end, it can also help to illustrate the action of reflection if I find it some material besides itself to reflect off of, in other words, if I supply it with an independently generated and concretely finished text as an argument to exercise its powers of reflection on. Accordingly, in the next part of this discussion I will interleave my text with …

The preview that follows takes up the first stirrings of many subjects that cannot be meaningfully engaged, much less fully formalized, until much later in the investigation. In many cases the ongoing discussion can afford to pause only long enough to toss a provisional name in the direction of a prospective topic, both the name and the place of which promise an eventual return and revision, if not a recurring visitation. If I were forced to give a formal title to my sketch in reconnaissance of this area, I would call it the “phenomenology of inquiry”, but this fine label is already too heavy an emblem for my effort to bear at this point.

This subsection presents a broad overview of the questions raised and the conceptual needs occasioned by the prospects of constructing a RIF. It makes no attempt to completely cover the topics it identifies and does not try to answer the questions it raises or to fill the needs it notices, but it merely points out a selective sample of the most salient concerns that need to be addressed.

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<p>The long route which I propose also aspires to carry reflection to the level of an ontology, but it will do so by degrees, following successive investigations into semantics and reflection.</p>
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| align="right" | Paul Ricoeur, ''The Conflict of Interpretations'', [Ric, 6]
|}

The subject of reflection is first approached in a topical, paraphrastic, or even periphrastic manner. The notion of a capacity for reflection is submitted to a tentative analysis, as treated in a variety of different modes of conception, by listing the duties that are typically demanded of a reflective agent, by compiling and reconciling the properties that are commonly ascribed to the process of reflection, and by contemplating the responsibilities and the results that the faculty of reflection is supposed to have in the many contexts where it is expected to serve.

“Reflection” is a word that is used with a wide variety of meanings in both ordinary language and technical contexts. Some of these uses have little to do with the sorts of inquiry being pursued here. Other uses, though related to inquiry, refer to processes that are fully as complex as inquiry itself. Neither of these extremes of meaning falls within the present focus of discussion.

The task for this project is to identify a coherent set of operations: (1) that fall within the scope of conceptual analysis and computational modeling, (2) that bear a recognizable and illuminating relationship to what is commonly called “reflection”, (3) that make an operational contribution to inquiry, and (4) that constitute simpler components of the operation of inquiry.

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<p>I define “symbol” as any structure of signification in which a direct, primary, literal meaning designates, in addition, another meaning which is indirect, secondary, and figurative and which can be apprehended only through the first.</p>
|-
| align="right" | Paul Ricoeur, ''The Conflict of Interpretations'', [Ric, 12]
|}

The types of reflective operations of interest in this work are best approached and analyzed within a sign theoretic setting. …

Presented in the guise of an allegorical figure, reflection appears as the bailiff or the sergeant at arms who escorts an agent, faculty, or process from the office of power to the dockets of observation, examination, and potential revision. But still, putting all allegory aside, this mode of transport goes nowhere at all, nor travels through any space, but turns on a mere change of views for all that it ushers in. One is perfectly capable of using a power of inquiry of which no account is yet given. But reflection is a part of the inquiry into conduct that gives the conduct in question a description.

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<p>He paused and nibbled absentmindedly on his branch for a while, as if gathering his thoughts.</p>
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| align="right" | Daniel Quinn, ''Ishmael'', [DQ, 11]
|}

If inquiry is viewed as a process of reasoning, then it takes place in thought, and so in the signs that are the public expression of thought. This means that an inquiry into inquiry has the duty to concern itself with thoughts, signs, and the relationship between them. To do this in an appropriate setting, it has to consider thoughts and signs as things taken apart from, but placed in relation to, their own particular objects. Indeed, the task that distinguishes an inquiry into inquiry, the specific difference that sets it apart, both from all of its object inquiries and again from inquiry in general, is the question of how thought is to be conducted if the goals of inquiry are to be met.

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<p>In such places (he went on at last), where animals are simply penned up, they are almost always more thoughtful than their cousins in the wild.</p>
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| align="right" | Daniel Quinn, ''Ishmael'', [DQ, 11]
|}

If one considers the formula that characterizes an inquiry into inquiry, <math>y_0 = y \cdot y,\!</math> and examines the term <math>y \cdot y\!</math> that factors <math>y_0\!</math> along the lines of an ostensible self-application, it is evident that any power invoked on the right is instantly echoed on the left and so required to survive the application or else be revoked. If the use of a given power of inquiry, working from the right and serving in the role of an operator, leads to a prospective description of inquiry, worked on the left from the role of an operand to the role of a result, and if the proffered characterization of inquiry is found to be out of accord with significant instances of its actual practice, then either the depiction of inquiry, as it is mediately improvised in progress, or the performance of inquiry, as it is actually conducted in practice, can turn out to be at fault.

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<p>This is because even the dimmest of them cannot help but sense that something is very wrong with this style of living.</p>
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| align="right" | Daniel Quinn, ''Ishmael'', [DQ, 11]
|}

The philosophical point of view called “pragmatism” takes a particular position on the relation of thoughts to signs, and this determines a particular method of approach to the nature of thinking. To preview here what is presented in detail later, the pragmatic point of view involves: (1) an assertion that thoughts are a special case of signs, (2) a theoretical definition of signs in terms of sign relations, and (3) a corresponding approach to the nature of thought as “praxis”, in other words, of thinking as a process, or inquiry as a form of conduct.

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<p>When I say that they are more thoughtful, I don't mean to imply that they acquire powers of ratiocination.</p>
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| align="right" | Daniel Quinn, ''Ishmael'', [DQ, 11]
|}

The way that reflection turns action into depiction and description, how it gives a sign to the act and provides a code for its future conduct, is the chief mystery of the whole process of reflection. Expressed in the substantive fashion that the “-ionized” character of “reflection” permits, this riddle arises from wondering how a reflection on the action can be transubstantiated as a sign of the action and resurrect itself in a code of the conduct. In other words, how does a signification calling for an interpretation arise from the very interruption of its full transmission, the comical section of its secular extension, the transient abdication of its permanent tradition, …, or the discrete truncation of a continuous conduction?

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<p>But the tiger you see madly pacing its cage is nevertheless preoccupied with something that a human would certainly recognize as a thought.</p>
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| align="right" | Daniel Quinn, ''Ishmael'', [DQ, 11]
|}

The way that inquiry obstructs itself, or that each inquiry interferes with every other, is a phenomenon that constitutes a principal target of this investigation. At times it seems as if the present construction is always the main obstruction to the intentions that are embodied in it, forestalling every chance of change, every hope of growth, and every possibility of progress into the future. How do the very processes of analysis, inquiry, and reflection come to have their aims so distorted, their airs so polluted, and their purposes so perverted that the very bodies of their own past effects become the blocks to their moving on?

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<p>And this thought is a question: ''Why?''</p>
|-
| align="right" | Daniel Quinn, ''Ishmael'', [DQ, 11]
|}

An inquiry into inquiry can start out so close to the start of its subject that it finds its subject inquiry in a state of inchoate incipience, one where no hint of reflection is yet to be discerned. But once the action of reflection is suspected to lie beneath the manifest activity of the subject inquiry, from the effects of reflection that are discovered to issue from the inquiry in question, the analysis of this reflection can take its charge to such outrageous extremes, breaking down the actual process of reflection to its most inert levels of miniscule details, breaking up the living continuity that is appropriate to a realistic depiction, incrementally revealing the fragments of reflection that are revealed by this analysis, omitting every sense of connection and surgically excising every ligament of extenuating context, or that are incrementally and inspecting revealed by this analysis with such a myopic scope, that no trace of the overriding imagination appears among the pieces that remain, the original aspect of reflection is lost among the residual debris.

When it comes to understanding a living activity, like reflection, or inquiry, or analysis itself for that matter, there are forms of analysis that go too far in their favored or particular directions while failing to take into account the workings of significant “ligaments”, or connective factors, that help to constitute and coordinate both their subjects and themselves, and on which the integrity of both depends. In particular, the purely syntactic analysis of a reflective narrative is liable to be carried to the bounds of such extremities that no trace of reflection is evident within the functional structures of the parts that are obtained or appears beneath the few points of light that the analysis yet throws. At this point, the instrument of reflection is broken so utterly, into so many pieces, and of such a small size that neither a close nor a distant inspection of any fraction of their number reveals any longer the aspect of reflection for which their matter was originally prized and pried into. When this state of analysis is reached, the medium of reflection, though it is still reflective in a certain sense, scatters the ambient light of nature that remains to it and disperses its sense through an evanescent void that presents nothing more illuminating to the imagination than the shimmering opacity of an opalescent haze.

The way that reflection, in adjunction to conduct, leads that conduct to a description of itself, not only in the sense of begetting an image but also in the sense of encountering a design, needs itself to have a name. I dub this turn of reflection, through which it converts the conduct of experience into an experience of conduct, by the name “metamorphism”. The way that a reflection of the action is a sign of the action needs to be investigated further, and is pursued through the rest of this work. An apology is due for continuing to harp on this point, but it remains a crucial point for the whole method of reflection, if it is to be a method.

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<p>“Why, why, why, why, why, why?” the tiger asks itself hour after hour, day after day, year after year, as it treads its endless path behind the bars of its cage.</p>
|-
| align="right" | Daniel Quinn, ''Ishmael'', [DQ, 11]
|}

Returning to the formula of an inquiry into inquiry, <math>y_0 = y \cdot y,\!</math> it is possible to derive a few of its consequences for the character of the operation that is to be called “reflection”. In general, a formula like <math>f = g \cdot h\!</math> constitutes a movement of conceptual reorganization, one whose resultant syntactic structure may or may not reflect an objective form of being, that is, an aspect of structure in the being that constitutes its object. If there is a similarity of structure to be found between the formula and the object, then one has what is called an ''iconic formula'', but this is not always the case, and even this special situation requires the proper interpretation to tell in exactly what respect the form of the sign and the form of the object are alike. Whatever the case, the role of the formula as a sign should not be confused with the role of the object in reality, no matter how similar their forms may be.

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<p>It cannot analyze the question or elaborate on it.</p>
|-
| align="right" | Daniel Quinn, ''Ishmael'', [DQ, 11]
|}

If one reads the form <math>y \cdot y\!</math> according to the convention adopted, where a latently but actively instrumentalized inquiry on the right applies to a patently but patiently objectified inquiry on the left, almost as if they were two distinct agencies, faculties, or processes, then it is clear that an inquiry into inquiry can begin with little more than a nominal object, taking the name of “inquiry” in its sights to yield a clue in name only, while it can reserve all the power of an established capacity for inquiry to conduct its review, of which no account, no prescribed code, nor any catalog of procedure has to be given at the outset of its investigation.

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<p>If you were somehow able to ask the creature, “Why ''what''?” it would be unable to answer you.</p>
|-
| align="right" | Daniel Quinn, ''Ishmael'', [DQ, 11]
|}

But it is important to remember that the full intention of this factious formulation is more analogous to an interpretive doubling of vision, an amplification of resolving power and a coordination of perspectives, than it is to an objective division of being, a substantial disconnection of essentials or a disintegration of being. Even when the factions of the term <math>y \cdot y\!</math> are conceived in practice to be implemented by substantially different parts of the same agency, constitutionally they embody but a single power.

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<p>Before long I too began to ask myself ''why''.</p>
|-
| align="right" | Daniel Quinn, ''Ishmael'', [DQ, 11]
|}

The form of inquiry into inquiry, <math>y \cdot y,\!</math> requires that any power assumed on the part of the right is open to be indicted on the part of the left. This entails that any power arrogated for the ends of inquiry has to be given a name, not only under which it is invoked as an executive power, but also by which it is entered on the agenda of issues to inquire into, and finally through which it is indicted for submission to all the powers of inquiry that be. This combination of ''appellation'' and ''supplication'', or ''nomination for'' and ''submission to'' the jurisdiction of a reflexive application, makes up a large part of what is usually called ''reflection''.

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<p>Being neurologically far in advance of the tiger, I was able to examine what I meant by the question, at least in a rudimentary way.</p>
|-
| align="right" | Daniel Quinn, ''Ishmael'', [DQ, 11]
|}

Many times, otherwise unrelated uses of the word ''reflection'' in the physical sciences can be suggestive. For example, reflection is one of the ways that a continuous appearing phenomenon can be brought into a form of interaction with itself and thereby to exhibit its nature as a pattern of activity with definite features and discrete characteristics. But metaphors like these can be kept from spinning out misleading clues only if the keys to understanding them as analogies can be found.

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<p>I remembered a different sort of life, which was, for those who lived it, interesting and pleasant.</p>
|-
| align="right" | Daniel Quinn, ''Ishmael'', [DQ, 11–12]
|}

With respect to the present project, that peers into its own appearances for the sake of seeing what logic lies in the rawness of experience, the analogous question is: How can the continuation of experience be conducted in such a way as to reveal within experience itself the conditions that connect its unconducted to its conducted course?

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<p>By contrast, this life was agonizingly boring and never pleasant.</p>
|-
| align="right" | Daniel Quinn, ''Ishmael'', [DQ, 12]
|}

This motion, the kind of movement just put in question or the form of development just wondered about, occurs throughout the whole field of inquiry, making it convenient to find a name for it. Because it steps across a boundary in a state space while keeping its eye on both sides of the displayed distinction, it can be called a ''circumspect transition'' or, more dramatically, a ''peripeteic strophe''. At times it appears in lights that earn it the titles of the ''yoke of knowing existence'' (YOKE), the ''conduct of subjectivity'' (COS), or just the ''junctive element'' (JE). It is pertinent in this connection that I am treating the condition of subjectivity as a special case of pragmatic objectivity, as it appears in this instance, one that involves an element of reflection.

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<p>Thus, in asking ''why'', I was trying to puzzle out why life should be divided in this way, half of it interesting and pleasant and half of it boring and unpleasant.</p>
|-
| align="right" | Daniel Quinn, ''Ishmael'', [DQ, 12]
|}

So let me dispense for the moment, even if this dispensation has to be carried out for now in the flippant manner and the fugal mannerisms of the foregoing counterpoint of discussion, with what I earnestly desire, all in good time and by means of well tempered arguments, to dispense with once and for all: The notion that individual words, sentences, paragraphs, articles, and so on up the scale can be anything like the final arbiters of thought and anything approaching the units of thought of ultimate interest to inquiry. But I want to do this without giving up the spirit of analysis altogether, merely by looking for forms of analysis that are fit to address the ''text of inquiry'' (TOI) in its full integrity.

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<p>It was in puzzling out such small matters as these that my interior life began — quite unnoticed.</p>
|-
| align="right" | Daniel Quinn, ''Ishmael'', [DQ, 12]
|}

In the process of elucidating the pragmatic point of view and applying it to the present array of problems, it soon becomes useful to examine the very notion of a ''point of view'' (POV). As reflections on the idea of a POV gradually mount up, the analysis of the general conception of a POV begins not only to take on a definite shape for itself, that is, with respect to the conditions it needs for its continued progress, but also to give a determinate, if slightly schematic, form to its object. Eventually, a sufficient confidence in the accumulated developments of the concept of a POV can lead this analysis to the point of suggesting provisional definitions and even to the point of attempting a specific formalization of POVs in general. This series of developments occurs concurrently throughout the construction of the current or any RIF, the necessity of which ought to be clear from the fact that the sequence of POVs from which one reflects is critical to the result and the success of any process of reflection.

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<p>A child hears it said that the stove is hot. But it is not, he says; and, indeed, that central body is not touching it, and only what that touches is hot or cold. But he touches it, and finds the testimony confirmed in a striking way. Thus, he becomes aware of ignorance, and it is necessary to suppose a ''self'' in which this ignorance can inhere. …</p>

<p>In short, ''error'' appears, and it can be explained only by supposing a ''self'' which is fallible.</p>

<p>Ignorance and error are all that distinguish our private selves from the absolute ''ego'' of pure apperception.</p>
|-
| align="right" | (Peirce, CP 5.233–235)
|}

Peirce makes the point that one's first awareness of a personal existence arises in reaction to the brute impact of experience and is ultimately compounded by way of reflection on its imports. Taking this to echo the exchange between Brutus and Cassius, I have the points that I need to stake out and to sound out a significant portion of the RIF that I intend to discuss.

Before passing over the subject of the JE, in order to continue with a sketch of reflection, it is useful to notice a couple of features that affect this style of movement, as recapitulated in this strophe, and that determine the prospective turns of events, as predicated on its action. In one sense the motion is reversible, since the boundary in the basic state space, once crossed, is easily crossed again. In another sense the transition is irreversible, since what is learned from a single crossing, assuming the agent is capable of acquiring knowledge in the process, is neither so quickly forgotten nor so easily lost.

It is the usual thing, in contemplating the forms of development that are epitomized by the figure of the JE, to find the images of winding gyres and helical ascents coming to mind, in other words, the trajectories of open curves in an extended space that project onto closed curves in a more basic space. In anticipation of later developments, I propose to attribute the ''basic'' and the ''extended'' aspects of this strophic segue to its ''dynamic'' and its ''symbolic'' components, respectively.

If it is important, at first sight, to recognize the JE as an irreducible primitive, inscribing its expression of its own being in the self signed tokens of a uniquely traced but perfectly typical autograph, all along personalizing its form of possession with an irrepressible panache, and leaving its legacy in an otherwise irreproducible style of paraph, it is just as important, on second thought, to try various schemes of analysis on it, with the aim, however artificially, of articulating, approximating, or explaining its form.

A number of questions arise at this point, concerning the justification of these moves, not just to justify the initial JE but to rationalize all of the ensuing action that is predicated on it. Just to name a few:

# What justifies a particular way of leading experience to reflect on itself?
# What justifies a particular way of causing reflection to comment on itself?

If it is asked, with respect to the legitimacy of all such questions, what is the justification for imposing extraneous ventures and superimposing foreign notions on the otherwise natural course of things, the answer has to be that, otherwise, it could not be articulated at all, and that, once reflected, the naturalness of expression that affected the original intention cannot be recovered without some risk of artificiality.

The object example of a strophic transition is the action of inquiry itself. For instance, if an inquiry is something that strives to move from a state of uncertainty toward a state of certainty about its object, then an inquiry into inquiry is something that proposes to move from a state of uncertainty toward a state of certainty about the identifiable elements and features of inquiry itself, including the natures of doubt and certainty, and of the distinction or the relationship between them. Distributing the terms a bit, this means that an inquiry into inquiry tries to move from a state of doubt about doubt toward a state of knowing about doubt.

To deal with the relationship between the dynamic and the symbolic aspects of the JE, I try a couple of strategies, ranging in character from the casual to the formal.

# To start, I adapt an informal distinction between the ''matter'' of a thought and the ''manner'' of thinking it. For example, in the effort to think about uncertainty one hopes to develop a certain concept of it. So, even though one continues to think about uncertainty, one hopes to become fairly certain about it. Here, the matter or content of one's thinking is fixed on uncertainty while the manner or conduct of one's thinking is hoped to change from the dubious to the certain.
# Eventually, it is necessary to develop a formal concept and even a mathematical model of this relationship. To do this, I adopt the intuitive notions of a ''point of view'' (POV) and its ''point of development'' (POD), gradually turning them into formal concepts of a very general character. This requires distinguishing between two kinds of propositions that are associated with POVs and PODs, namely: (a) the propositions that are ''attached to'' or ''contained in'' them, and (b) the propositions that are ''applied to'' or ''maintained about'' them.

Just to give a rough idea of how these two distinctions relate to each other, the ''matter'' of a thought corresponds to an ''attached'' proposition, ''in'' a POV or ''at'' a POD, while the ''manner'' of a thought corresponds to an ''applied'' proposition, ''on'' a POV or ''about'' a POD. Employing this language to describe the case of an inquiry successfully self-applied, one can say the following things. An agent of inquiry has a POV that changes from one POD to the next in a series of developments, and this can be a POV that concerns itself with the question of inquiry, among other things, and thus with the topics of uncertainty and certainty, or doubt and belief. In such a case, as the POV moves from an initial POD to a terminal POD, a part of its matter stays fixed on ''doubt'', while its whole manner is transformed from one of ''doubt'' toward one of ''belief''.

It is not always necessary to distinguish a POV from each of its PODs, except when one needs to emphasize the dynamic aspect of these ideas, especially the fact that a single POV can pass through or incorporate many different PODs in the course of its development. It is legitimate to say that the POV is present at each of its PODs, or that the PODs are incorporated in their overall POV. Accordingly, it is not always necessary to lose sight of the successive PODs in a series, so long as they are amenable to being incorporated in the last POD, or final POV.

When one says that a POV is associated with a particular proposition, whether containing it or instancing it, one always means a POV as it exists at a particular POD, or through a particular range of its PODs. For example, if I say <math>{}^{\backprime\backprime}J ~\text{thinks}~ K ~\text{is smarter than}~ L{}^{\prime\prime},\!</math> then I am implicating a POV that <math>J\!</math> has at a particular POD, assumed to be capable of specification. Moreover, I am relying on the specific information inherent in this POD to index the particular persons <math>K\!</math> and <math>L\!</math> that I am assuming <math>J\!</math> has in mind at that POD. In technical terms, this requires the “intentional context” that is signaled by the verb ''thinks'', normally “opaque” to all distributions of contextual information from any point outside its frame, to be treated as “transparent” to the packet of information that is assumed to be represented by the POD in question.

In the application of mediate interest to this project, a POV corresponds to a computational system, while a POD corresponds to one of its states. It is desirable to have a way of referring to the system as a whole, but in ways that are implicitly quantified by the relevant classes of states. For example, I want to have a system of interpretation in place where it is possible to write <math>{}^{\backprime\backprime}j : x = y{}^{\prime\prime}\!</math> to mean that <math>{}^{\backprime\backprime}j ~\text{sets}~ x ~\text{equal to}~ y{}^{\prime\prime},\!</math> to read this as a statement about a system <math>j\!</math> and two of its stores <math>x\!</math> and <math>y,\!</math> and to understand this as a statement that implicitly refers to a set of states that makes it true. Further, I want to recognize this statement as the active voice, attributed account, or authorized version of the more familiar, but passive, anonymous, or unavowed species of assignment statement <math>{}^{\backprime\backprime}x := y{}^{\prime\prime}.\!</math>

The rudimentary parallels between these different distinctions should not be treated too rigidly, as a number of finer points about their true relationship remain to be sorted out. The next few remarks are given just to provide a hint of what is involved.

The distinction between matter and manner of thought is correlated with the distinction between object and sign, so long as these are recognized as distinctions of momentary use and not as distinctions of fixed nature. In the beginning, these distinctions also coincide with the distinction between dynamic and symbolic aspects of inquiry. A typical instance of this is when a change of physical configuration is needed to carry out an experiment, while what is learned as a result of that experience is tantamount to a change in the organization of one's thought, and where this includes changes in the signs, texts, and other sorts of records that form one's knowledge base as well as changes in the ideas, or signs in the mind, that make up one's mental configuration.

Nevertheless, it is important to appreciate that all of these distinctions can become increasingly and divergently relativized as higher orders of reflection on the initial domain of objects turn wider circles of signs around it and heap higher towers of ideas upon it. When this happens, any initial portion of the objective and lower order syntactic domains can form the matter of a thought, while any final portion of the higher order syntactic domains can embody the manner of a thought.

Here is the critical point. A conceptual distinction is not absolute, but relative to the POV that sees it, makes it, draws it, or uses it. Indeed, one of the reasons for introducing the concept of a POV is to formalize this general insight and thereby to permit reflection on specific POVs. But the intention of this move is to include any distinction that can be made in the process of an inquiry and found essential to its progress. Accordingly, the aim of this insight marks an intention to comprehend, not just the distinctions between previously identified predicates, like the attributes ''dubious'' and ''certain'' already mentioned, but also any future distinctions that might be discovered as necessary to inquiry. Almost immediately, for instance, the distinction just made by way of formalizing the concept of a POV, between the ''propositions at'' and the ''propositions about'' a POV, falls into the category of distinctions in question, at least, put under examination for the purposes of a review.

{| align="center" cellpadding="0" cellspacing="0" width="90%"
|
<p>Consciousness is a movement which continually annihilates its starting point and can guarantee itself only at the end. In other words, it is something that has meaning only in later figures, since the meaning of a given figure is deferred until the appearance of a new figure.</p>
|-
| align="right" | Paul Ricoeur, ''The Conflict of Interpretations'', [Ric, 113]
|}

{| align="center" cellpadding="0" cellspacing="0" width="90%"
|
<p>In a sense, knowledge shrinks as wisdom grows: for details are swallowed up in principles. The details of knowledge which are important will be picked up ad hoc in each avocation of life, but the habit of the active utilisation of well-understood principles is the final possession of wisdom.</p>
|-
| align="right" |
<p>Alfred North Whitehead, ''The Aims of Education'', [ANW, 46]<br>
cited in Stephen R. Covey, ''First Things First'', [CMM, 71]</p>
|}

If I reflect on my own POV, it becomes evident that the critical focus and main interest of this inquiry is on the kinds of inquiry that cross a certain threshold, that of becoming deliberately conducted and critically controlled. But in order to remain critical and reflective I have to be interested in both sides of this distinction. This is the usual pattern of the circumspect transition that I just personified in the form of the JE. But here it seems to lead to the disconcerting conclusion that reason is founded on unreason, and that this reason is justified by the end that it leads to, not by the state that it starts in. This sounds ominous and dangerous, but I think that the difficulties it raises are partly verbal, the fault of ambiguities that are quick to right themselves on reflection, but more seriously, partly due to misleading theory of what constitutes a foundation, a theory that tends to be applied on the modern scene as an uncritical reflex of that very same and everyday modern POV.

The general problem encountered here can be expressed in the question: How can reason rationally address its other, the real irrationality of unreason that it finds in its experience, not only in and of the world but at the beginnings of its own being? If reason addresses unreason as its founder, then it seems to founder in its justification of itself. This is how the question and its puzzing echo first present themselves, but in order to approach a genuine answer it seems more likely that a transformation of the question is demanded, perhaps recasting it so: How can one form of reason rationally address another, whether it is wholly and radically other or complementary to and continuous with its own being? Other versions of this question, and other attempts to respond to it, appear throughout the remainder of this work.

One thing is apparent: If reflective inquiry, based on the rationality of the intellectual share of reason, addresses instinctive inquiry, based on the sensibility of the affective, emotional, and motivational portion, as something wholly other, indeed, as its logical opposite, then it obstructs the eventuality and even precludes the possibility of discovering their compatibilities and continuities, and it interferes with the chances of seeing how each form of inquiry completes and extends the other.

====5.3.2. The Light in the Clearing====

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<p>Thus we can understand that “the entire sensible world and all the beings with which we have dealings sometimes appear to us as a text to be deciphered”.</p>
|-
| align="right" |
<p>Paul Ricoeur, ''The Conflict of Interpretations'', [Ric, 222]<br>
quoting Jean Nabert, ''Elements for an Ethic'', [Nab, 77]</p>
|}

Before this discussion can proceed any further I need to introduce a technical vocabulary that is specifically designed to articulate the relation of thought to action and the relation of conduct to purpose. This terminology makes use of a classical distinction between ''action'', as simply taken, and ''conduct'', as fully considered in the light of its means, its ways, and its ends. To the extent that affects, motivations, and purposes are bound up with one another, the objects that lie within the reach of this language that are able to be grasped by means of its concepts provide a form of cognitive handle on the complex arrays of affective impulsions and the unruly masses of emotional obstructions that serve both to drive and to block the effective performance of inquiry.

Once the differentiation between sheer activity and deliberate conduct is comprehended on informal grounds and motivated by intuitive illustrations, the formal capabilities of their logical distinction can be sharpened up and turned to instrumental advantage in accomplishing two further aims:

# To elucidate the precise nature of the relation between action and conduct.
# To facilitate a study of the whole variety of contingent relations that are possible and maintained between action and conduct.

When the relations among these categories are described and analyzed in greater detail, it becomes possible forge their separate links together, and thus to integrate their several lines of information into a fuller comprehension of the relations among thought, the purposes of thought, and the purposes of action in general.

It is possible to introduce the needed vocabulary, while at the same time advancing a number of concurrent goals of this project, by resorting to the following strategy. I inject into this discussion a selected set of passages from the work of C.S. Peirce, chosen with a certain multiplicity of aims in mind.

# These excerpts are taken from Peirce's most thoughtful definitions and discussions of pragmatism. Thus, the general tenor of their advice is pertinent to the long-term guidance of this project.
# With regard to the target vocabulary, these texts are especially acute in their ability to make all the right distinctions in all the right places, and so they serve to illustrate the requisite concepts in the context of their most appropriate uses.
# Aside from their content being crucial to the scope of the present inquiry, their form, manner, sequence, and interrelations supply the kind of material needed to illustrate an important array of issues involved in the topic of reflection.
# Finally, my reflections on these passages are designed to illustrate the variety of relations that occur between the POV of a writer, especially as it develops through time, and the POV of a reader, in the light of the ways that it deflects its own echoes through a text in order to detect the POV of the writer that led to its being formed in that manner.

The first excerpt appears in the form of a dictionary entry, intended as a definition of ''pragmatism''.

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<p>'''Pragmatism.''' The opinion that metaphysics is to be largely cleared up by the application of the following maxim for attaining clearness of apprehension: "Consider what effects, that might conceivably have practical bearings, we conceive the object of our conception to have. Then, our conception of these effects is the whole of our conception of the object."</p>
|-
| align="right" | (Peirce, CP 5.2, 1878/1902).
|}

The second excerpt presents another version of the ''pragmatic maxim'', a recommendation about a way of clarifying meaning that can be taken to stake out the general POV of pragmatism.

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<p>Pragmaticism was originally enounced in the form of a maxim, as follows: Consider what effects that might conceivably have practical bearings you conceive the objects of your conception to have. Then, your conception of those effects is the whole of your conception of the object.</p>
|-
| align="right" | (Peirce, CP 5.438, 1878/1905).
|}

Over time, Peirce tried to express the basic idea contained in the ''pragmatic maxim'' (PM) in numerous different ways. In the remainder of this work, the gist of the pragmatic maxim, the logical content that appropriates its general intention over a variety of particular contexts, the common denominator of all of its versionary approximations, can be referred to with maximal simplicity as “PM”. Otherwise, subscripts can be used in contexts where it is necessary to mention a particular form, for instance, referring to the versions just given as “PM<sub>1</sub>” and “PM<sub>2</sub>”, respectively.

Considered side by side like this, any perceptible differences between PM<sub>1</sub> and PM<sub>2</sub> appear to be trivial and insignificant, lacking in every conceivable practical consequence, as indeed would be the case if both statements were properly understood. One would like to say that both variants belong to the same ''pragmatic equivalence class'' (PEC), where all of the peculiarities of their individual expressions are absorbed into the effective synonymy of a single operational maxim of conduct. Unfortunately, no matter how well this represents the ideal, it does not describe the present state of understanding with respect to the pragmatic maxim, and this is the situation that my work is given to address.

I am taking the trouble to recite both of these very close variants of the pragmatic maxim because I want to examine how their subsequent interpretations have tended to diverge over time and to analyze why the traditions of interpretation that stem from them are likely to develop in such a way that they eventually come to be at cross-purposes to each other.

There is a version of the pragmatic maxim, more commonly cited, that uses ''we'' and ''our'' instead of ''you'' and ''your''. At first sight, this appears to confer a number of clear advantages on the expression of the maxim. The second person is ambiguous with regard to number, and it can be read as both singular and plural, since the …

Unfortunately, people have a tendency to translate ''our concept of the object'' into ''the meaning of a concept''. This displacement of the genuine article from ''the object'' to ''the meaning'' obliterates the contingently indefinite commonality of ''our'' manner of thinking and replaces it with the absolutely definite pretension to ''the'' unique truth of the matter // changing the emphasis from common conception to unique intention. This apparently causes them to read ''the whole of our conception'' as ''the whole meaning of a conception'' … // from ''thee'' and ''thy'' to ''the'' and ''our'' //

The pragmatic maxim, taking the form of an injunctive prescription, a piece of advice, or a practical recommendation, provides an operational description of a certain philosophical outlook or ''frame of reference''. This is the general POV that is called ''pragmatism'', or ''pragmaticism'', as Peirce later renamed it when he wanted more pointedly to emphasize the principles that distinguish his own particular POV from the general run of its appropriations, interpretations, and common misconstruals. Thus the pragmatic maxim, in a way that is deliberately consistent with the principles of the POV to which it leads, enunciates a practical idea and provides a truly pragmatic definition of that very same POV.

I am quoting a version of the pragmatic maxim whose form of address to the reader exemplifies a second person POV on the part of the writer. In spite of the fact that this particular variation does not appear in print until a later date, my own sense of the matter leads me to think that it actually recaptures the original form of the pragmatic insight. My reasons for believing this are connected with Peirce's early notion of ''tuity'', the second person character of the mind's dialogue with nature and with other minds, and a topic to be addressed in detail at a later point in this discussion.

By way of a piece of evidence for this impression, one that is internal to the texts, both versions begin with the second person POV that is implied by their imperative mood.

Just as the sign in a sign relation addresses the interpretant intended in the mind of its interpreter, PM<sub>2</sub> is addressed to an interpretant or effect intended in the mind of its reader.

The third excerpt puts a gloss on the meaning of a ''practical bearing'' and provides an alternative statement of the pragmatic maxim (PM<sub>3</sub>).

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<p>Such reasonings and all reasonings turn upon the idea that if one exerts certain kinds of volition, one will undergo in return certain compulsory perceptions. Now this sort of consideration, namely, that certain lines of conduct will entail certain kinds of inevitable experiences is what is called a "practical consideration". Hence is justified the maxim, belief in which constitutes pragmatism; namely,</p>

<p>In order to ascertain the meaning of an intellectual conception one should consider what practical consequences might conceivably result by necessity from the truth of that conception; and the sum of these consequences will constitute the entire meaning of the conception.</p>
|-
| align="right" | (Peirce, CP 5.9, 1905).
|}

The fourth excerpt illustrates one of Peirce's many attempts to get the sense of the pragmatic POV across by rephrasing the pragmatic maxim in an alternative way (PM<sub>4</sub>). In introducing this version, he addresses an order of prospective critics who do not deem a simple heuristic maxim, much less one that concerns itself with a routine matter of logical procedure, as forming a sufficient basis for a whole philosophy.

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<p>On their side, one of the faults that I think they might find with me is that I make pragmatism to be a mere maxim of logic instead of a sublime principle of speculative philosophy. In order to be admitted to better philosophical standing I have endeavored to put pragmatism as I understand it into the same form of a philosophical theorem. I have not succeeded any better than this:</p>

<p>Pragmatism is the principle that every theoretical judgment expressible in a sentence in the indicative mood is a confused form of thought whose only meaning, if it has any, lies in its tendency to enforce a corresponding practical maxim expressible as a conditional sentence having its apodosis in the imperative mood.</p>
|-
| align="right" | (Peirce, CP 5.18, 1903).
|}

I am including Peirce's preamble to his restatement of the principle because I think that the note of irony and the foreshadowing of comedy intimated by it are important to understanding the gist of what follows. In this rendition the statement of the principle of pragmatism is recast in a partially self-referent fashion, and since it is itself delivered as a "theoretical judgment expressible in a sentence in the indicative mood" the full content of its own deeper meaning is something that remains to be unwrapped, precisely through a self-application to its own expression of the very principle it expresses. To wit, this statement, the form of whose phrasing is forced by conventional biases to take on the style of a declarative judgment, describes itself as a "confused form of thought", in need of being amended, converted, and translated into its operational interpretant, that is to say, its viable pragmatic equivalent.

The fifth excerpt, PM<sub>5</sub>, is useful by way of additional clarification, and was aimed to correct a variety of historical misunderstandings that arose over time with regard to the intended meaning of the pragmatic POV.

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<p>The doctrine appears to assume that the end of man is action — a stoical axiom which, to the present writer at the age of sixty, does not recommend itself so forcibly as it did at thirty. If it be admitted, on the contrary, that action wants an end, and that that end must be something of a general description, then the spirit of the maxim itself, which is that we must look to the upshot of our concepts in order rightly to apprehend them, would direct us towards something different from practical facts, namely, to general ideas, as the true interpreters of our thought.</p>
|-
| align="right" | (Peirce, CP 5.3, 1902).
|}

If anyone thinks that an explanation on this order, whatever degree of directness and explicitness one perceives it to have, ought to be enough to correct any amount of residual confusion, then one is failing to take into consideration the persistence of a ''particulate'' interpretation, that is, a favored, isolated, and partial interpretation, once it has taken or mistaken its moment.

A sixth excerpt, PM<sub>6</sub>, is useful in stating the bearing of the pragmatic maxim on the topic of reflection, namely, that it makes all of pragmatism boil down to nothing more or less than a method of reflection.

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<p>The study of philosophy consists, therefore, in reflexion, and pragmatism is that method of reflexion which is guided by constantly holding in view its purpose and the purpose of the ideas it analyzes, whether these ends be of the nature and uses of action or of thought. …</p>

<p>It will be seen that pragmatism is not a Weltanschauung but is a method of reflexion having for its purpose to render ideas clear.</p>
|-
| align="right" | (Peirce, CP 5.13 note 1, 1902).
|}

The seventh excerpt is a late reflection on the reception of pragmatism. With a sense of exasperation that is almost palpable, this comment tries to justify the maxim of pragmatism and to reconstruct its misreadings by pinpointing a number of false impressions that the intervening years have piled on it, and it attempts once more to correct the deleterious effects of these mistakes. Recalling the very conception and birth of pragmatism, it reviews its initial promise and its intended lot in the light of its subsequent vicissitudes and its apparent fate. Adopting the style of a ''post mortem'' analysis, it presents a veritable autopsy of the ways that the main truth of pragmatism, for all its practicality, can be murdered by a host of misdissecting disciplinarians, by its most devoted followers. This doleful but dutiful undertaking is presented next.

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<p>This employment five times over of derivates of ''concipere'' must then have had a purpose. In point of fact it had two. One was to show that I was speaking of meaning in no other sense than that of intellectual purport. The other was to avoid all danger of being understood as attempting to explain a concept by percepts, images, schemata, or by anything but concepts. I did not, therefore, mean to say that acts, which are more strictly singular than anything, could constitute the purport, or adequate proper interpretation, of any symbol. I compared action to the finale of the symphony of thought, belief being a demicadence. Nobody conceives that the few bars at the end of a musical movement are the purpose of the movement. They may be called its upshot.</p>
|-
| align="right" | (Peirce, CP 5.402 note 3, 1906).
|}

There are notes of emotion ranging from apology to pique to be detected in this eulogy of pragmatism, and all the manner of a pensive elegy that affects the tone of its contemplation. It recounts the various ways that the good of the best among our maxims is "oft interrèd with their bones", how the aim of the pragmatic maxim to clarify thought gets clouded over with the dust of recalcitrant prepossessions, drowned in the drift of antediluvian predilections, lost in the clamor of prevailing trends and the shuffle of assorted novelties, and even buried with the fractious contentions that it can tend on occasion to inspire. It details the evils that are apt to be done in the name of this précis of pragmatism if ever it is construed beyond its ambition, and sought to be elevated from a working POV to the imperial status of a Weltanshauung.

The next three elaborations of this POV are bound to sound mysterious at this point, but they are necessary to the integrity of the whole work. In any case, it is a good thing to assemble all these pieces in one place, for future reference if nothing else.

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<p>When we come to study the great principle of continuity and see how all is fluid and every point directly partakes the being of every other, it will appear that individualism and falsity are one and the same. Meantime, we know that man is not whole as long as he is single, that he is essentially a possible member of society. Especially, one man's experience is nothing, if it stands alone. If he sees what others cannot, we call it hallucination. It is not "my" experience, but "our" experience that has to be thought of; and this "us" has indefinite possibilities.</p>
|-
| align="right" | (Peirce, CP 5.402 note 2, 1893).
|}

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<p>Nevertheless, the maxim has approved itself to the writer, after many years of trial, as of great utility in leading to a relatively high grade of clearness of thought. He would venture to suggest that it should always be put into practice with conscientious thoroughness, but that, when that has been done, and not before, a still higher grade of clearness of thought can be attained by remembering that the only ultimate good which the practical facts to which it directs attention can subserve is to further the development of concrete reasonableness; so that the meaning of the concept does not lie in any individual reactions at all, but in the manner in which those reactions contribute to that development. …</p>

<p>Almost everybody will now agree that the ultimate good lies in the evolutionary process in some way. If so, it is not in individual reactions in their segregation, but in something general or continuous. Synechism is founded on the notion that the coalescence, the becoming continuous, the becoming governed by laws, the becoming instinct with general ideas, are but phases of one and the same process of the growth of reasonableness.</p>
|-
| align="right" | (Peirce, CP 5.3, 1902).
|}

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<p>No doubt, Pragmaticism makes thought ultimately apply to action exclusively — to conceived action. But between admitting that and either saying that it makes thought, in the sense of the purport of symbols, to consist in acts, or saying that the true ultimate purpose of thinking is action, there is much the same difference as there is between saying that the artist-painter's living art is applied to dabbing paint upon canvas, and saying that that art-life consists in dabbing paint, or that its ultimate aim is dabbing paint. Pragmaticism makes thinking to consist in the living inferential metaboly of symbols whose purport lies in conditional general resolutions to act.</p>
|-
| align="right" | (Peirce, CP 5.402 note 3, 1906).
|}

The final excerpt touches on a what can appear as a quibbling triviality or a significant problem, depending on one's POV. It mostly arises when sophisticated mentalities make a point of trying to apply the pragmatic maxim in the most absurd possible ways they can think of. I apologize for quoting such a long passage, but the full impact of Peirce's point only develops over an extended argument.

{| align="center" cellpadding="0" cellspacing="0" width="90%"
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<p>There can, of course, be no question that a man will act in accordance with his belief so far as his belief has any practical consequences. The only doubt is whether this is all that belief is, whether belief is a mere nullity so far as it does not influence conduct. What possible effect upon conduct can it have, for example, to believe that the diagonal of a square is incommensurable with the side? …</p>

<p>The proposition that the diagonal is incommensurable has stood in the textbooks from time immemorial without ever being assailed and I am sure that the most modern type of mathematician holds to it most decidedly. Yet it seems quite absurd to say that there is any objective practical difference between commensurable and incommensurable.</p>

<p>Of course you can say if you like that the act of expressing a quantity as a rational fraction is a piece of conduct and that it is in itself a practical difference that one kind of quantity can be so expressed and the other not. But a thinker must be shallow indeed if he does not see that to admit a species of practicality that consists in one's conduct about words and modes of expression is at once to break down all the bars against the nonsense that pragmatism is designed to exclude.</p>

<p>What the pragmatist has his pragmatism for is to be able to say: here is a definition and it does not differ at all from your confusedly apprehended conception because there is no practical difference. But what is to prevent his opponent from replying that there is a practical difference which consists in his recognizing one as his conception and not the other? That is, one is expressible in a way in which the other is not expressible.</p>

<p>Pragmatism is completely volatilized if you admit that sort of practicality.</p>
|-
| align="right" | (Peirce, CP 5.32–33, 1903).
|}

Let me just state what I think are the three main issues at stake in this passage, leaving a fuller consideration of their implications to a later stage of this work.

#<li value="1">Reflective agents, as a price for their extra powers of reflection, fall prey to a new class of errors and liabilities, any one of which might be diagnosed as a ''reflective illusion'' or a ''delusion of reflection'' (DOR). There is one type of DOR that is especially easy for reflective agents to fall into, and they must constantly monitor its swings in order to guard the integrity of their reflective processes against the variety of false images that it admits and the diversity of misleading pathways that it leads onto. This DOR turns on thinking that objects of a nature to be reflected on by an agent must have a nature that is identical to the nature of the agent that reflects on them.</li>

An agent acts under many different kinds of constraints, whether by choice of method, compulsion of nature, or the mere chance of looking outward in a given direction and henceforth taking up a fixed outlook. The fact that one is constrained to reason in a particular manner, whether one is predisposed to cognitive, computational, conceptual, or creative terms, and whether one is restrained to finitary, imaginary, rational, or transcendental expressions, does not mean that one is bound to consider only the sorts of objects that fall into the corresponding lot. It only forces the issue of just how literally or figuratively one is able to grasp the matter in view.

To imagine that the nature of the object is bound to be the same as the nature of the sign, or to think that the law that determines the object's matter has to be the same as the rule that codifies the agent's manner, are tantamount to special cases of those reflective illusions whose form of diagnosis I just outlined. For example, it is the delusion of a purely cognitive and rational psychology, on seeing the necessity of proceeding in a cognitive and rational manner, to imagine that its subject is also purely cognitive and rational, and to think that this abstraction of the matter has any kind of coherence when considered against the integrity of its object.

#<li value="2">The general rule of pragmatism to seek the difference that makes a difference has its corollaries in numerous principles of indifference. Not every difference in the meantime makes a difference in the end. That is, not every difference of circumstance that momentarily impacts on the trajectory of a system nor every difference of eventuality that transiently develops within its course makes a difference in its ultimate result, and this is true no matter whether one considers the history of intertwined conduct and experience that belongs to a single agent or whether it pertains to a whole community of agents. Furthermore, not every difference makes a difference of consequence with respect to every conception or purpose that seeks to include it under its "sum". Finally, not every difference makes the same sort of difference with regard to each of the intellectual concepts or purported outcomes that it has a bearing on.</li>

To express the issue in a modern idiom, this is the question of whether a concept has a definition that is ''path-dependent'' or ''path-invariant'', that is, when the essence of that abstract conception is reduced to a construct that employs only operational terms. It is because of this issue that most notions of much import, like mass, meaning, momentum, and number, are defined in terms of the appropriate equivalence classes and operationalized relative to their proper frames of reference.

#<li value="3">The persistent application of the pragmatic maxim, especially in mathematics, eventually brings it to bear on one rather ancient question. The issue is over the reality of conceptual objects, including mathematical "objects" and Platonic "forms" or "ideas". In this context, the adjective "real" means nothing other than "having properties", but the import of this "having" has to be grasped in the same moment of understanding that this old schematic of thought loads the verb "to have" with one of its strongest connotations, namely, that nothing has a property in the proper sense of the word unless it has that property in its own right, without regard to what anybody thinks about it. In other words, to say that an object has a property is to say that it has that property independently, if not of necessity exclusively, of what anybody may think about the matter. But what can it mean for one to say that a mathematical object is "real", that it has the properties that it has independently of what anybody thinks of it, when all that one has of this object are but signs of it, and when the only access that one has to this object is by means of thinking, a process of shuffling, sifting, and sorting through nothing more real or more ideal than signs in the mind?</li>

The acuteness of this question can be made clear if one pursues the accountability of the pragmatic maxim into higher orders of infinity. Consider the number of "effects" that form the "whole" of a conception in PM<sub>1</sub>, or else the number of "consequences" that fall under the "sum" in PM<sub>2</sub>. What happens when it is possible to conceive of an infinity of practical consequences as falling among the consequential effects or the effective consequences of an intellectual conception? The point of this question is not to require that all of the items of practical bearing be surveyed in a single glance, that all of these effects and consequences be enumerated at once, but only that the cardinal number of conceivable practical bearings, or effects and consequences, be infinite.

Recognizing the fact that "conception" is an "-ionized" term, and so can denote an ongoing process as well as a finished result, it is possible to ask the cardinal question of conceptual accountability in another way:

What is one's conception of the practical consequences that result by necessity from a case where the "conception" of practical consequences that result by necessity from the truth of a conception constitutes an infinite process, that is, from a case where the conceptual process of generating these consequences is capable of exceeding any finite bound that one can conceive?

It is may be helpful to append at this point a few additional comments that Peirce made with respect to the concept of reality in general.

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|
<p>And what do we mean by the real? It is a conception which we must first have had when we discovered that there was an unreal, an illusion; that is, when we first corrected ourselves. Now the distinction for which alone this fact logically called, was between an ''ens'' relative to private inward determinations, to the negations belonging to idiosyncrasy, and an ''ens'' such as would stand in the long run. The real, then, is that which, sooner or later, information and reasoning would finally result in, and which is therefore independent of the vagaries of me and you. Thus, the very origin of the conception of reality shows that this conception essentially involves the notion of a COMMUNITY, without definite limits, and capable of a definite increase of knowledge.</p>
|-
| align="right" | (Peirce, CP 5.311, 1868).
|}

{| align="center" cellpadding="0" cellspacing="0" width="90%"
|
<p>The real is that which is not whatever we happen to think it, but is unaffected by what we may think of it.</p>
|-
| align="right" | (Peirce, CE 2:467, 1871).
|}

{| align="center" cellpadding="0" cellspacing="0" width="90%"
|
<p>Thus we may define the real as that whose characters are independent of what anybody may think them to be.</p>
|-
| align="right" | (Peirce, CP 5.405, 1878).
|}

Having read these exhibits into evidence, if not yet to the point of self-evidence, and considered them to some degree for the individual lights they throw on the subject, let me now examine the relationships that can be found among them.

These excerpts are significant not only for what they say, but for how they say it. What they say, their matter, is crucial to the whole course the present inquiry. How they say it, their manner, is itself the matter of numerous further discussions, a few of which, carried out by Peirce himself, are already included in the sample presented.

Depending on the reader's POV, this sequence of excerpts can appear to reflect anything from a radical change and a serious correction of the underlying POV to a mere clarification and a natural development of it, all maintaining the very same spirit as the original expression of it. Whatever the case, let these three groups of excerpts be recognized as forming three successive ''levels of reflection'' (LORs) on the series of POVs in question, regardless of whether one sees them as disconnected, as ostensibly related, or else as inherently the very same POV in spirit.

From my own POV, that strives to share this spirit in some measure, it appears that the whole variety of statements, no matter what their dates of original composition, initial publication, or subsequent revision, only serve to illustrate different LOR's on what is essentially and practically a single and coherent POV, one that can be drawn on as a unified frame of reference and henceforward referred to as the ''pragmatic'' POV or as just plain ''pragmatism''.

There is a case to be made for the ultimate inseparability of all of the issues that are brought up in the foregoing sample of excerpts, but an interval of time and a tide of text are likely to come and go before there can be any sense of an end to the period of questioning, before all of the issues that these texts betide can begin to be settled, before there can be a due measure of conviction on what they charge inquiry with, and before the repercussions of the whole sequence of reflections they lead into can be brought to a point of closure.

If one accepts the idea that all of these excerpts are expressions of one and the same POV, but considered at different points of development, as enunciated, as reviewed, and as revised over an interval of many years, then they can be taken to illustrate the diverse kinds of changes that occur in the formulation, the development, and the clarification of a continuing POV.

====5.3.3. The Face in the Mirror====

{| align="center" cellpadding="0" cellspacing="0" width="90%"
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<p>We cannot proceed in the question of the who without introducing the problem of everyday life, self-knowledge, the problem of the relation to the other — and, ultimately, the relation to death.</p>
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| align="right" | Paul Ricoeur, ''The Conflict of Interpretations'', [Ric, 232]
|}

It is the business of this preview to take up the initial stirrings of many subject matters that cannot hope to be completely developed, even clearly engaged, and much less fully formalized at the present stage of discussion. At many points I can afford to pause only long enough to toss a provisional name in the direction of a subject I plan to address more fully later on, and at times I can merely indicate in cursory terms a topic that I intend to return to at several later points in this work.

There are aspects of inquiry that are difficult to address from a purely cognitive point of view, that is, from a standpoint that expects agents of inquiry to behave in a strictly rational fashion. At times it appears that there are complexities in the phenomenology of inquiry that are impossible to compass, comprehend, deal with, fathom, grasp, or tackle by means of a purely cognitive or strictly rational method of approach. Nevertheless, it is not that the manner of proceeding ought to be anything other than what it aims to be, but merely that it cannot expect its subject matter to mirror nothing beyond its own manners of limitations, principles, and scruples.

The reason that these issues have to be discussed at this particular point of this particular inquiry is simply that it cannot proceed beyond this point without facing up to them and facing them down again, that is, without addressing them in some measure and reducing the force of their obstruction. In short, these issues present a real obstacle to the progress of this inquiry.

It is such a rarity for this theme of "internal opposition" to find a clear expression in the light of day that the latest and best descriptions one sees of it are not all that far removed in their basic characters and their degrees of clarity from the very first inklings one can find of it, namely, those intimations of its nature that remain wrapped up in the perennial figures of perceptive allegory and the ancient images of insightful mythology that are passed down from the foundings of every civilized culture. Thus, I find I cannot discuss this issue, even dimly, without first invoking this class of symbols, no matter what difficulties of interpretation they bring in their train. I can only promise to try to clarify their empirical and rational meanings as the discussion continues.

In general, one can treat the dramatic images presented by literature and mythology as not essentially different from scientific images of action, behavior, and conduct. They all attempt to capture something about human aims and human actions in a compact, exact, and memorable image, to catch a glimmer of human behavior in a singular but still moving form, and to cache it in the coin of a common exchange so that it embodies not just a commemorative but a communerative inheritance, one that anyone with access to the store can recurrently cash in as warranted by the occasion and the development of a particular situation, that all can repeatedly bank on to their individual and common profit. No matter what genre it is cast within, a successful image records a moment's insight into an aspect of human nature, preserving it in such a way that it can reveal itself at relevant moments in future experience, and thus avail itself in such a form that it makes this particular insight into this particular aspect available for the use of many other points of view.

An image is useful if it codifies significant insights about character and conduct. To do this it needs to indicate, descriptively or normatively, important aspects of behavior, involving the necessary structures or the contingent potentials of conduct. An image is especially useful if it is indexed to come to mind at the times when it is appropriate, called for, and helpful. An image begins to make itself useful in a more scientific, or a deliberately controlled way, if it can serve flexibly and not just fixedly in the act of reflecting on a present scene or a real situation. This means that it can make itself available as a suggestive hypothesis about a situation in general and can serve as a constructive critique of the ways to proceed from that point forward, but that it does not force itself on any particular situation, does not take the role of an active ingredient in the objective problem to be resolved, and does not elect itself as the primary constituent of that very predicament.

To be useful in critical reflection, an image has to remain transparently suggestive of the potentials that reside in a real and present situation. To continue its utility in a viable process of reflection, an image cannot allow itself to be so cast in stone that it blocks contemplation of all that demands attention in a real situation, and it cannot become so blindingly opaque in its own right that it obscures and occludes any concern that does not fit within its pre-cut scene. In order to preserve its uses in feasible and viable conditions, an image cannot be allowed to coerce the satisfaction of its own application, as in the style of a self-fulfilling prophecy or the spell-binding picture of a prescience that predestines its own end, descriptively and prescriptively determining the form and the fact of its own fate.

In this connection, it is not just in story and fable that one finds the images of self-fulfilling prophecies. The scientific picture of human behavior embodies an equal number of self-satisfying limitations and self-predicting reductions, analogous to the image of a person who is looking under the lamppost for something lost elsewhere simply because the light is better there, or the image of a person with no tool but a hammer who is determined by this utility to view everything as a nail.

These possibilities highlight and point up another source of perils that one risks in trying to craft a serviceable RIF. One of the main reasons for seeking a RIF is to make reflective and critical images of a chosen framework available within the scope of that very same framework itself. But the original aim of the framework, that orients its agent toward an objective reality, should not be lost in the process of adjoining these reflections on its own form. So the next predicament to be solved, once the inclusion of self-images is provisionally made possible and after the license of self-reference is tentatively permitted, is how to prevent the extended framework from being swamped by nothing but images of itself, with nothing on the order of an objective nature and nothing of the other that it originally sought.

The particular problematic I am trying to capture here seems to demand a symbol of a sufficient power, and this forces me to raid the reserves of archetype and mythology. So let me stake out this issue under the rubric of "Cerberus". It deserves this name for at least three reasons. First, there is the way that it blocks one's way to underlying sources of insight, the way it closes off one's access to a potential wealth of deeper understandings of oneself. Second, there is the irresistible challenge presented by external forms of resistance to one's own ideals, the instigation that this very resistance represents to one's own drives against it, and the incitement to collective effort and deliberate trials that it causes to mount, thereby harnessing one's motive instincts and promoting their expression in particular directions. Third, there is the snarling dogmatism that it betrays, of a kind that one encounters most acutely in one's dealings with one's counterparts in the community, but that one can tackle only within oneself, as if all along the doubts and the difficulties that the other comes to represent for oneself are merely the reflections, in the mirror afforded by the external world, of one's own internal opposition.

In practice, the obstructions that an effort of inquiry is bound to meet up with in the world outside its state of intention would not be able to knock it off its target if it were not for the opposing tensions that it maintains within its own intentional bearing. Sometimes, these take the form of intrinsic or inherent oppositions, but more often than not they are "introjected" obstacles, the sort that get themselves internalized at a particular point in time. This brings the mythological portion of this tale around to the figure of "Oedipus", portraying the predicament of an inquiry that deliberately blinds itself, not just for what it has seen, but for what it might continue to see.

Given the license to temporarily invoke such powerful cultural symbols, and since the issues they try to capture appear already bound together, I can organize the manifestations of this problematic under three heads:

# The "negative onus" (NO) of inquiry.
# The "affective mood" (AM) of inquiry.
# The "existential subjective tone" (EST) of inquiry.

The pragmatic description of inquiry is notorious for the constellation of problematic and negative features that its account the matter points out. In practice, this complex of negative aspects recurrently presents itself as the most difficult to steadily face up to without flinching and the most troublesome to squarely address without blanching, blinking, or otherwise to break off the approach and even to abandon the question. By way of deriving the minimal service that the simplest name affords, I refer to this problematic aspect as the "negative onus" (NO) of inquiry.

The natural instinct of the mind brings it into frequent encounters with this NO, at least, its short term need for adventure and exploration is often brought up short by it. But there are other tendencies at work in the meantime, and independent dispositions can be found at play on each larger scale of the process of inquiry. For instance, there is evidently a concurrent tendency of the mind to run away from this NO, since its desire for security in the intermediate term is threatened by the practical consequences of affronting and so bearing the brunt of it. In any case, this is how the matter typically develops after the initial fascination with a surprising phenomenon or the original compulsion to resolve a problematic situation have faded, and this is usually sometime before a reasonable explanation or a suitable plan of action can appear. And so one finds that successful inquiries often require a second stage, or even more, that their incipient motivations are seldom enough to get them past the first signs of trouble, that it takes renewable resources of persistence to countenance the negative aspects of inquiry, and that it takes a higher order of dedication to keep from running away in the face of a genuine question's more troubling demeanor.

The NO of inquiry arises at a couple of distinct moments in its process. It arises naturally enough at the outset of inquiry, due to the negative, problematic, and troubling aspects of doubt and uncertainty. But again, no sooner does an inquiry get started in some hopeful direction than it runs into a host of distortions, diversions, obstacles, and resistances that act to impede its progress. Some of these obstructions can be seen as derivative expressions of the original uncertainty that occasioned a specific inquiry, but others go deeper than the issues that stem from a particular topic. Other obstacles, harder to address, have to do with a deficient motivation for a specific inquiry, a deficit in competence at inquiry in general, a constitutional incapacity to change beliefs once their status is fixed, or a lack of desire to make the necessary changes in belief and in conduct that are always at risk in an authentic inquiry. Still other obstacles abide at another order entirely, arising from the necessary properties and structures of inquiry itself.

The fact that the pragmatic point of view recognizes uncertainty as the beginning of inquiry means that the dynamic start of its process and the rational foundation of its method are energetically and logically sundered from each other, if not forever split apart, then at least until the inquiry in question is itself resolved.

All attempts to found inquiry on positive intellectual powers and virtues, or to chart its progress through purely rational methods and principles, always come to grief and founder on the fact that one cannot squarely address the obstructions to inquiry without facing up to the negative, troubling, and unruly aspects of truly problematic uncertainties, nor without being willing to look at the affective investments that people have in their own pet notions and their favorite group's fondest ideas. It is neither possible nor necessary to deal with this problem, in its entirety, within the present context. However, it is both necessary and possible to articulate the ways that it impacts on the present concerns. In this regard, a measure of "affective investment" in a customary idea or interpretive conduct is the persistence of the custom or the strength of the habit that associates itself with that particular conduit of ideas. This translates the notion of "affective investment" into a concept with dynamic import, giving it an operational significance and a quantifiable meaning for the current approach to a "dynamic symbolics".

Depending on the accidents and the inevitabilities of their historical reception, the trains of consequences that accompany them, the general drift of intervening events, and the finer shade of the mood from which the totality of this sweep is reviewed, a retrospective LOR can take on all the character of any pure choice or any mixture of options from a whole spectrum of characters. Just by way of example, these can range from apology, casuistry, and attempts at exculpation or explanation by way of mitigating circumstances, through elegy, encomium, and eulogy, then all the way to lamentation, melancholy, panegyric, and requiem.

Is the lack of a mirror really lamented? And if it is literally so, then what is the source of remorseful affect that abides in this impression? From whence does the vast but vaporous plague of advisory injunctions, censorious restrictions, circumspect suspicions, consensual admonitions, delinquent proscriptions, imminent suspensions, monitory animadversions, mundane cautions, and tardy but truant afterthoughts arise, along with the vague but vampirous unease of something forgotten, lost, or forlorn? If one arrives at the point of asking in what sort of space does a point of view reside, and what manner of a multitude does it share this manifold with, then the sense of this whole, collective, irksome host begins to appear.

One clue arises from the circumstance that the very purpose of affects, emotions, and motives is to change a point of view, to alter a state of being or a position in space, to get through a fit of pique or a point of impasse, to break out of a fixed opinion in the labyrinth of purely intellectual and speculative convictions. And yet this presents nothing but problems for the effort to see the formal skeleton within the living body of inquiry.

Although emotion remains a pervasive factor in the informal realm, it is difficult to see, at least, at first, how its forces could be transmitted into and through the formal arena, even if, in spite of all appearances, there were found to be forms of good judgment that permitted it to try. The only hint of an answer that I can see at this point is the insight that is stored up in the very etymology of "form" itself, whose often forgotten connotation is literally "beauty".

The final consequence that I want to derive from these excerpts is their bearing on the differential relationships that exist between a conduct, as it distinguishes itself in action from an action, and the respective ends of this conduct and this action, all in all, which may amount to the very same object in the end. In particular, I want to point out a remarkable implication of this approach that I call the "paradox of conduct" (POC).

: Considered as action, the end of life is death, but
: considered as conduct, the end of life is life itself.

I am attempting no definition of life here, but merely noting one of its most exemplary and elegant formal properties. In game theoretic terms, if life is a game then its aimed for value is a robustly recursive goal. In generic terms, the pay off, intended target, or desired value of the whole game of life is just the whole game back again. In species terms, the aim of a form of life is to continue its life in the same form, or in what is recognizable to itself as being in its original spirit ("anima"). The content that abides in a form of life is not a satisfaction with the continual repetition of precisely the same state of being or the unending replication of perfectly identical copies of itself, as these by themselves are just different forms of death, but it takes an ability to recognize and regenerate the spirit within the letter, to promote the true content its own essential form, not merely to reproduce the shadow of its shape.

In this sense, and purely with regard to the topographies of their fields and courts of play, life is less like soccer, where the goals fall at the extreme ends of the field, than it is like tennis, where the net measure of its results lies in the midst of the ongoing play. Altogether, this makes life a system of organization that is topologically "open" at both ends, both with regard to its foundations, its "arche", and also with respect to its goals, its "telos".

To draw the conclusion: To say that death is the end of life, in the sense of a goal, is obviously going a bit too far. Death is merely the contingent end of life as form of action, not the intentional end of life as a form of conduct, and all the rest of the confusion is merely verbal equivocation around and about these two senses of an end. In the light of the distinction between action and conduct it is easy to see that death is just a bit beyond the true end of life.

{| align="center" cellpadding="0" cellspacing="0" width="90%"
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<p>In the same way as it is not sufficient, before beginning to rebuild the house in which one lives, only to pull it down and to provide material and architects, or oneself to try one's hand at architecture, and moreover, to have drawn the plan carefully, but one must also provide oneself with some other accommodation in which to be lodged conveniently while the work is going on, so, also, in order that I might not remain irresolute in my actions during the time that my reason would oblige me to be so in my judgements, and so that I would not cease to live from that time forward as happily as I could, I formed a provisional moral code which consisted of only three or four maxims, which I am willing to disclose.</p>
|-
| align="right" | Rene Descartes, ''Discourse on Method'', [Des, 45]
|}

The advisability of justifying one's actions as much as one can is clear, but the problems that arise in trying to do so are not trivial, …

In view of these problems, it is necessary to examine the formation of the JE and to consider its import for the ''justification of inquiry'' (JOI). It is useful to begin with a traditional idea or a received sense of what a JOI must be. The default justification, that almost everyone seems to fall back on, even when deliberately trying to be critical and reflective, arises from the common notion of a “foundation” as something that forms a necessary prerequisite to all attempts at reasoning.

After giving a critical account of the standard model in the light of a few additional reflections, I review the question of what a real JOI must be like, at least, if it is to allow for inquiry into inquiry and to account for the other features of inquiry already observed. At last, I renew the quest for those JOIs that befit a pragmatic perspective and that can be found within its survey.

There is a standard sort of proposal to justify inquiry that attempts to place its foundations at the beginning of its process and that insists on making them out as certain. I think it is fitting to describe this as a variety of ''fundamentalism''. If this form of understanding is submitted to reflection, it begins to look inconsistent, or at least hypocritical, since it promises a distinct JOI, but one that it can just as easily tell, by the right reflection at the outset, is not forthcoming by these means. In essence, it claims to have a different sort of justification for itself than every other claim to one's allegiance, but a careful examination of its more finely printed disclaimers begins to turn up the evidence that this, too, is ultimately on a par with every other belief system, with the technical exception that it demands unquestioning faith at the level of a method rather than on the grounds of a doctrine. Even here, it leaves one wondering how to discern these levels in practice, or whether they can be distinguished in principle.

On this fundamental model of inquiry, any appearance of a passage from doubt to certainty has its authenticity placed in doubt, and begins to have its pretensions of creatively discovering new knowledge fall into question, looking more like the illusions of a derivative performance. Indeed, every semblance of a genuine inquiry is parasitic on the host of axioms and methods already taken for granted, and it creates no greater capital of knowledge than the fund of certainty already established in a prior method of inquiry. In effect, this prior method is taken on faith, since it begs to be imitated in a ritual fashion and to have its formulas, while invoked without question, to be invested with blind forms of trust.

In fine, the default manner of approaching the question of foundations makes inquiry into inquiry a moot question, an otiose endeavor that is neither possible to pursue in a bona fide way nor necessary to venture. Given the fundamentalist understanding of inquiry, the application of inquiry to itself can neither accomplish any real purpose nor achieve any goal that is actually at risk. The pretense of establishing the integrity of inquiry under a self-application of its principles always results in something of a put up job, a kangaroo court, or a show trial.

Under these conditions, the proceedings that declaim themselves to be engaged in honest inquiry are nothing more than a hypocritical display. They imitate the exterior form of a due process, but their judgment is fixed in advance and their conclusion but extravagantly reconstructs a previously settled system of belief, one that is never really doubted or put in question. The outer inquiry in the self-application is not a live inquiry but a ''frame'' that is prefabricated to isolate the object inquiry. Whether expertly or inertly, it is designed ahead of time to contain and to delimit a picture of inquiry that may or may not already be painted.

Notice that this is not a question of whether the original inquiry is genuine or not. The object inquiry, typically ignited by an external phenomenon, is commonly taken up in good faith, that is, with honest doubts at stake. But when there is never any doubt about what method to use, or about how to use it, or about the chances of its leading to a satisfactory end of the doubts inflamed in the first place, then there is never any need for inquiry into inquiry, and all show of it is vanity.

As a result, the fundamental JOI renders the hallowed method of inquiry just another doctrine among others, equal in its manner of justification, its final appeal, and its ultimate justice to every other belief system. But this is not the criticism that finally condemns it. Being just the same as other systems of belief is not the fatal flaw. That only makes all systems of belief equal under the law, if no longer a law of inquiry but a law of compromising positions and convenient resolutions. Still, there would not necessarily have been anything wrong with this, if it were not for the self-imposed burden that inquiry brings down on itself via the dishonesty or the self-deception of promising something else.

<br>

{| align="center" border="0" cellpadding="0" cellspacing="0" style="text-align:left; width:90%"
| colspan="2" | Could great men thunder
|-
| colspan="2" | As Jove himself does, Jove would never be quiet,
|-
| colspan="2" | For every pelting petty officer
|-
| colspan="2" | Would use his heaven for thunder, nothing but thunder.
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| colspan="2" | Merciful heaven,
|-
| colspan="2" | Thou rather with thy sharp and sulphurous bolt
|-
| colspan="2" | Split'st the unwedgeable and gnarled oak
|-
| colspan="2" | Than the soft myrtle. But man, proud man,
|-
| colspan="2" | Dressed in a little brief authority,
|-
| colspan="2" | Most ignorant of what he's most assured,
|-
| colspan="2" | His glassy essence, like an angry ape
|-
| colspan="2" | Plays such fantastic tricks before high heaven
|-
| colspan="2" | As makes the angels weep, who, with our spleens,
|-
| colspan="2" | Would all themselves laugh mortal.
|-
| width="50%" |  
| ''Measure for Measure'', 2.2.113–126
|}

<br>

It is probably wise to stress this point. It is not being claimed that an authority based system of belief, simply by building itself on traditional foundations, is necessarily hypocritical or inconsistent in its own right. It can be as accurate, authentic, and honest in what it says and tries to say as any other belief system or knowledge base. In fact, a modicum of reliance on one source of authority or another is not only prudent but most likely to be found inescapable. Authority based systems, in form so analogous to axiom systems, if not in the context of their use, simply have the specific properties and the generic limitations that they can be observed to have.

At this point, let authority based systems and axiom systems be lumped together into the same class, at least temporarily, on the basis of the forms of derivation that they allow, and without regard for the different ways that they are initially brought to light or subsequently put to use. Further, let this whole class be described as ''founded systems'', for the moment ignoring the distinction between informal and formal systems, or regarding all prospectively, in anticipation of their formalizations.

Every project of a founded system voluntarily risks certain limitations. But there is one limitation that appears to be a genuine defect from the standpoint of this inquiry, amounting to the chief source of worry that this inquiry has about the whole class of founded systems. This is the fact that whatever acuteness of reverence or accuracy of reference to their objects they do in fact achieve is a matter of grace or luck, and not something that can be subjected to change, criticism, or correction. This puts it outside the sphere of inquiry, as I understand it, even if its formulations are suggested by data within the sphere of experience.

In effect, there is no amplification of intelligence, no leverage of reason, in short, no instrumental gain or “mechanical advantage” to be acquired from the use of a founded system. It can transmit the force of reason, in a conservative way at best, from premisses to conclusions, but the effective output of the system achieves no greater level of certainty as it bears on any question than the level of authority it can grant itself on input or justly claim for itself at the outset. If I can continue to use the image of a lever, while delaying the examination of its exactness until a later point of this work, it is as if the lack of leverage in a founded system can be traced back to one of several defects:

# The ''fulcrum'' of a founded system, the fixed point of its critique, is the ''examen'' of its critical powers, the tongue of its balance, and this has to be placed so evenly between the objective domain, whereof its ignorance is writ so large, and the fund of applied information, wherein its share of accumulated knowledge resides, that no gain in the effective intelligence of the actions thus founded can be derived.
# A founded system is forced to be a ''grounded system'', that is, one that requires a moderately strong emplacement on grounds already settled and a preponderance of certainty on the side of the applied intelligence.
# In effect, a ''founded or grounded'' (FOG) system requires absolute certainty with respect to some of its points, the points on which it is said to rest. It is as if these fixed points put it in contact with an infinite source of knowledge or connect it to an infinite sink for uncertainties. Of course, a FOG system that casts itself as a beacon of enlightenment and sells itself under the label of “science” can never admit to seeing itself in this image, since the very act of making the claim explicit already puts its grant in jeopardy. But that is what it amounts to, nevertheless.

Another way to see the over-constrained nature of these FOG conditions, for the certainty of foundations, is by expressing them in terms of the ''boundary conditions'' that a given system of belief is assumed to have. In this regard, it helps to make the following definition. An ''open'' system of belief is one that has each of its points ''mediated'' by the system itself, in other words, surrounded by, apprehended within, and evidentially or argumentatively justified by a neighborhood of similar points that falls entirely within the system in question.

When it is considered in the light of this definition of ''openness'', a FOG system is clearly seen to constitute a ''non-open'' system of belief. In short, not all of its axioms, points, or tenets are mediated within the system itself, but have their motives, reasons, and supports lying in points ulterior to it. In hopes of serving both the understanding and the memory, let me try to express this situation in a couple of striking, if slightly ludicrous, metaphors, a pair of judicial, if not entirely judicious, figures of speech:

# The ''corpus delicti'', the body of material evidence and substantial fact that is necessary to justify the institution of the system and the initiation of its every process, is always found to lie in such a disposition that it rests partially beyond the system in question.
# The ''habeas corpus'', the body of probable causes and sufficient reasons that is tendered to justify the holding of certain points, is always deposed in such a demeanor that its true warrant either stays unwrit or is writ largely outside the system in question.

Whether it is verifiably jurisprudent or merely a fantastic simile, whether it is really a conspiracy of their natural bents or purely a coincidence of their accustomed distortions, the parody of a judicial process that one constantly sees being carried on in the name of this or that FOG system, and always apparently up to the limits of their several FOG boundaries, makes a mockery of the spirit of inquiry, and of all its pretensions to a critical reflection, since it places not only the first apprehension but the final justice of such a system beyond all question of executive examination, judicial review, and constitutional amendment. The whole matter is even more deceptive that it appears at first sight, precisely because a FOG system, as lit within, or according to its own lights, often takes on all the appearance of being open. But this is only because the boundaries of its viability and the outlines of the external obstacles that represent a threat to the illusions of its omni pervasiveness are actively being obscured by the limitations inherent in its unreflective nature.

This is just the kind of situation that one would expect in the purely deductive or demonstrative sections of science, for instance, in logics and mathematics of the “purer” and less “applied” sorts. In these more abstract traces and more refined extracts of a fully scientific method, the authority of the conclusions, or the level of certainty achieved on output, is no greater than the authority of the premisses, or the level of certainty possessed on input. Thus, the work of reasoning in such a case is purely ''expliative'', that is, wholly expository or explicational.

But a truly synthetic or ''ampliative'' analysis should be able to reduce a complex induction to simple inductions, meanwhile gaining a measure of certainty in the process, and all without losing the power to reconstruct the complex from the simple. The perceived gain of practical certainty that develops in this analysis can be explained in the following manner. A complex induction, prior to analysis, is likely to be a very uncertain induction, but is likely to have its certainty shored up if the analysis to simple inductions is successful.

This is a pretty sorry picture, especially in view of all the bright promises of enlightenment through inquiry that inquiry makes, to be a veritable system of belief for constituting systems of veritable belief. But the promise of inquiry to be better than all that, to be an advance over other systems of belief, not just another dogma in the management of uncertainty but a unique way of life, holds out hopes that are still tempting and that deserve to be pursued further. So it is time to ask: If not by means of these foundations, then what form of constitution can provide the sought for JOI?

Fortunately, there is another JOI, arising from the pragmatic critique of even the most enlightened fundamentalism. If the fundamental approach is viewed as a project to conjoin three positive features — ''founding'', ''beginning'', and ''certain'' — in a single point of conceptual architecture, then the pragmatic critique of this plan can be understood as objecting that this point is overloaded. There are ways to preserve this triarchic association, but not without protracting other angles of approach to the juncture and not without compassing other senses of the terms than the meanings originally intended. It is perhaps easier just to abrogate one of the terms, either rescinding its constraint or trading it in for its logical negation.

The pragmatic approach to the foundations of inquiry, more precisely, its approach to the hoped for JOI, whether or not this leaves room in the end for a notion of secure foundations, suggests that reason does begin with unreason, but only in the sense that inquiry starts from a state of uncertainty. If one objects that this doubt is not radical, because many things in the meantime are never in fact doubted at all, then this is correct, but only in the sense that these things are not doubted because they are never even consciously questioned. If that sort of lack of doubt is the type one plans to found their reason on, then I think it is a very fond notion indeed.

<br>

{| align="center" border="0" cellpadding="0" cellspacing="0" style="text-align:left; width:90%"
| colspan="2" | There's a double meaning in that.
|-
| width="50%" |  
| ''Much Ado About Nothing'', 2.3.246
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<br>

(Yes, there is a subtext. (There is always a subtext.) A reader who has access to the subtext, who can read it in the face of the pretext, and who remains both sensitive to and sensible about its connotations, is already beginning to suspect that what I intend to argue in the end is exactly that the chief justification of inquiry is nothing less and nothing more than the pure joy of it. But the moment that I depend on this subtext to carry the logical argument, to go beyond supporting the intuition and encouraging the effort of reasoning, is the moment that I utterly fail in my intention. This bears on the matter of a harmonious balance between rhetoric and logic, where the former appreciates and is bound to consider the affective and the impressionable nature of the interpreter, and takes into account the need for reason's ponderous beacon to be buoyed over the deep by incidental glosses and light exhortations.)

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<p>Self-awareness is our capacity to stand apart from ourselves and examine our thinking, our motives, our history, our scripts, our actions, and our habits and tendencies. It enables us to take off our “glasses” and look at them as well as through them. It makes it possible for us to become aware of the social and psychic history of the programs that are in us and to enlarge the separation between stimulus and response.</p>
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| align="right" | Covey, Merrill, and Merrill, ''First Things First'', [CMM, 59]
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How is it possible for one to use an organization of thought in order to think about that same organization of thought, or indeed, about others? How is it possible to draw distinctions, even the most basic distinctions necessary to thought, in such a way that they can be redrawn and even withdrawn when necessary? In other words, what are the conditions for having a ''critical reflection of inquiry'' (CROI), a system of assumptions and methods that acts continuously and self-correctively to constitute a critically reflective belief system? This would be tantamount to a POV where no assumption is forced to be taken for granted, even if at any given moment many assumptions are contingently being acted on just as if they were true. For instance, if a distinction between dynamic and symbolic systems, or aspects of systems, is a part of one's present POV, to what extent can one reflect on that fact, and thus be able to think about alternative POVs or to think about changing one's current POV?

This ends my preview of the kinds of issues that the pragmatic theory of sign relations and their reflective extensions is intended to comprehend.

In the sequel I propose a particular way of approaching these problems. I introduce a simplified model of the general situation to be addressed, but one with sufficient structure to embody analogous versions of many of the problems and phenomena of ultimate interest. By exploring the issues that develop in this miniature model, and by looking for ways of resolving them that work on this scale, I hope to gain insight into ways of dealing with the corresponding issues in the larger study of inquiry.

To be specific, I restrict my discussion at first to ''propositional'' or ''sentential'' models of POVs, and I examine a particular type of logical strategy that allows agents operating within this framework to describe the constitutions of a broad class of POVs. If this strategy turns out to be flexible enough, it can permit agents to reflect on the bases and the biases of their POVs and those of others, at least, to some degree.

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<p>This circumscription of expressions with a double meaning properly constitutes the hermeneutic field.</p>
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| align="right" | Paul Ricoeur, ''The Conflict of Interpretations'', [Ric, 13]
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Even with its meaning duly circumscribed, “reflection” retains the dual senses of an “-ionized” word, referring to both a process and a result. As such, it is already on its way to becoming a highly charged term in this investigation. Even though a complete analysis of inquiry, from the top to the bottom of its putative hierarchy, is yet to be made available, the tendency to invoke “reflection” at every step and stage of inquiry is already apparent. This is clear from the fragmentary and scattered, but steadily mounting evidence of the word's textual circumstances that is currently piling up at the level of inquiry's most primitive details.

In other words, the constant invocation of “reflection” as an auxiliary to inquiry is apparent from the elementary syntactic fact that the charge of “reflection” is found in the mission statements of so many processes that are already noted to be involved in inquiry. In this connection, any time one senses the need to add the adjective “reflective” to the title of an agent, process, or faculty, then it speaks to the suspicion that the simple carrying out of actions and the perfunctory execution of procedures is not enough for the sake of composing intelligent conduct, but that there is an obligation to adjoin a component of reflection to whatever else is going on.

The elliptic nature of the discussion in this subsection, touching on topics that must be left in the forms of questions, raising issues that cannot be answered or even fully addressed until later sections of this project, lighting on a range of promontories in a field of problematic icebergs, and glancing up against problems that stay largely submerged and keep barely connected only through a medium of chance associations constantly in flux — all of this makes it advisable for the writer to come up with a device for continually warning the reader of the text's approaching discontinuities.

In view of these requirements, the text proceeds by highlighting a number of thematic points that find themselves to be reinforced in prior stages of its own construction, not all of which stages survive erasure enough to be explicitly marked in the text, and all in all continuing to develop as if by a pattern of constructive and destructive interference. It is hoped that this can reveal significant aspects, however partial and confounded, of its subject, its medium, and the forms that shape them.

A few words need to be spent in advance on the status of these points. Most of them are no longer controversial from my current POV, indeed, they partially constitute that POV. However, I recognize that some of them are likely to be controversial from the perspective of other POVs. Thus, these points are not intended to be taken as self-evident axioms, the kinds of logistical supports on the basis of which one customarily and confidently marches forward to the conquest of ever more powerful theorems. It is true that one of the best ways of testing these points is to take them up as premisses and to reason forward from them as far as one can. But the main reason for pointing them out in an explicit form of expression is so that their meanings, their logical implications, and their practical consequences can be examined in a circumspect light.

In short, none of the points to be staked out here is taken as evident or proven, and nothing of final certainty can be proved from them, but a demonstration can be made from them in the sense of an illustration, showing and testing their strength, trustworthiness, and utility for organizing an otherwise overwhelming complexity and depth of material. This process of examination and clarification, just as often as it has to reason forward, in the direction of the contingent theorems, also has to reason backward, to interrogate the mediately obvious principals and to ask whether more basic points can be discerned, as if lurking within the points already noted and secretly required to shore them up.

Out of this material I need to develop a method of inquiry, one that is extensible to its self-application. As an adjunct, or in adjutant fashion, I need to develop a justification of this method that can lend support to the justification of inquiry in general, and in its turn help to justify the application of inquiry to itself. Accordingly, the prospective aim to be sighted through the series of points ahead, and the line of survey to be projected through the elliptic text that charts it, are directed toward an effective theory of sign relations, one that is capable of resolving some of the subtleties it discerns in discourse, on occasions when a resolution is what is called for.

====5.3.4. Points Forward====

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<p>If exegesis raised a hermeneutic problem, that is, a problem of interpretation, it is because every reading of a text always takes place within a community, a tradition, or a living current of thought, all of which display presuppositions and exigencies — regardless of how closely a reading may be tied to the ''quid'', to “that in view of which” the text was written.</p>
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| align="right" |Paul Ricoeur, ''The Conflict of Interpretations'', [Ric, 3]
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'''Point 1.'''     Thought takes place in signs.

This makes a sign relation the setting of thought, where thought occurs. In particular, the connotative plane of a sign relation is the medium of thought proper, and the denotative plane of a sign relation embodies the lines of thought's orientation toward its objects.

This point is one that may be thought controversial, until it is realized that the meaning of the term “sign” is being extended to cover anything that might conceivably occur in thought. Far from intending to restrict thought to a circumscribed domain of signs, it expands the definition of “sign” to encompass anything that might enter into thought, so long as this entrance into thought is understood not in the sense of being its object but as something that lends a place to it. Properly taken, this point is tantamount to an empirical definition of the term “sign”, more like an indication of where in experience a ready supply of examples can be found. It says that if you seek signs then look to your thoughts.

'''Point 2.'''     Thinking is a form of conduct.

Conduct is action with a purpose. Synonymous with the term “purpose”, as used in this statement, are “aim”, “end”, “goal”, or “object”. The object domain of a sign relation is the place where these objects are envisioned to be, and thinking is the action that is carried out with a view to these ends.

Rightly taken, this point, too, is purely definitional. It classifies thinking as a species of action that has, or is meant to have, a purpose. In particular, thinking is the kind of action that passes from sign to interpretant sign in relation to an object. If one wishes to object that not all that passes for thinking has any assignable purpose, and if one desires to maintain an alternative POV that recognizes forms of aimless thinking, then it is nothing more than a technical problem to translate between the two ways of thinking, reclassifying unconducive thinking as a “degenerate form” from the standpoint of the pragmatic POV.

'''Point 3.'''     Reflection on thinking is reflection on conduct.

Even though it can appear too evident, too immediate, and too obvious to bear pointing out, there are several good reasons to make a point of noticing this simple corollary of the previous point, namely, that if thinking is a special case of conduct then reflection on thinking is a special case of reflection on conduct.

First of all, it means that reflection on thinking and reflection on conduct have a reciprocal bearing on each other, the way that special cases and general types always do. Reflection on thinking can tell us something about reflection on conduct in general. This is because the special case informs the general type and can be used inductively to discover its possible properties. Reflection on conduct in general can tell us something about reflection on thinking. This is because the general type constrains the special case and can be used deductively to derive its necessary properties.

(Bearing on the order of the normative sciences : logic < ethics < aesthetics?)

There is more to this point than first meets the eye, especially when it is considered in the light of its abstract form. Aside from its present application to the matters of reflection, thinking, and conduct, one can see in this instance the form of a distributive law, that distributes an operation (“reflection”) across a relation (“implication” or “inclusion”), and where this order of dyadic relation is the very one that constitutes the ordering of special cases under general forms. The point of this is that the general intention of this dyadic relation, in its full extension, must be to capture the relation of a special application of any principle (say, a distributive law) to its own general formulation. For instance, therefore, reflection on a special kind of distribution is a special kind of reflection on distribution in general.

In light of these relations between the specialization of thinking and the general capacity for conduct, I can now turn to a logical analysis of the concept of conduct for the light it reflects on the nature of thought.

'''Point 4.'''     Conduct = (Act, End) = (State<sub>1</sub>, State<sub>2</sub>, State<sub>3</sub>).

One can say that a conduct is a pair comprised of an act and an end. In this formula, the act can be anything from a complex activity to an extended action and the end can be anywhere among a vast diversity of destinations that are found to be encompassed by a general description. If it is recognized that the data needed to specify a minimum of action, a mere transition, is an ordered pair of states, and if it is remembered that the data appropriate to specifying a singular end is a single state, then an element of conduct, at its minimum, can be conceived to consist of an ordered triple of states.

'''Point 5.'''     Reflection, joined to conduct, generates an image of it.

Reflection on conduct produces an image of that conduct. In relation to the active nature of the conduct the image is just what its etymology says it is, an inactive sign or an inert icon of the action. The image of a conduct presents itself as a hypothesis about it, a tentative description that may or may not be accurate out of the starting blocks and may or may not continue to be useful in the long run.

'''Point 6.'''     There is a type of reflection that only reproduces the images produced by previous reflections.

The images produced by this kind of reflection, affected by an imitative or nearly identical character, can be referred to as “reproductions”, “stereotypes”, or “simple copies”. A reproductive reflection has the option of attaching additional marks to distinguish the reproduced copy from the original image. If it does add a distinguishing mark or a distinctive notation to identify the source, then one has the type of reproduction that can safely be regarded as a reflective “quotation”.

'''Point 7.'''     There is a type of reflection that captures an extended sequence of events in a single image.

The images produced by this kind of reflection, affected by a creative, critical, reductive, selective, or truly imaginative character, using manners of plastic representation that can condense, edit, summarize, and transform, all at the risk of serious distortions that go beyond simple errors in the transmission, can be referred to as “adaptations”, “redactions”, “renditions”, “versions”, or “transformed interpretations”.

These effect of reflection, when it is efficient, is to do just this, to produce a single image that captures a poignant, salient, or relevant aspect of an entire dramatic sequence.

'''Point 8.'''     Inquiry, if deliberate and critical, involves reflection.

The capacity for reflection is necessary to carry out the deliberately conducted and critically controlled varieties of inquiry that make up the principal interest of this work, and especially to entertain any form of inquiry into inquiry.

The pragmatic theory of signs sets the stage for a broad definition of inquiry. It includes under “inquiry” all the fortuitous and instinctive processes that agents exploit to escape from states of uncertainty, to soothe the “irritation of doubt”, in Peirce's phrase, along with all the deliberate and intelligent procedures that enable communities of agents to deal in systematic ways with the surprises and the problems that they encounter in their several and common experiences. At one end of this spectrum, the more incidental, instinctive, and casually intuitive forms of inquiry can be carried on without the interruptions of critical reflection. But an intelligent inquiry is necessarily a reflective inquiry.

'''Point 9.'''     The need for a capacity of reflection is the reflection of a certain incapacity to see certain things without it.

This point has a bearing on the capacity that one has to recognize one's own character as an objective form of being and to realize it within an active pattern of conduct.

'''Point 10.'''     At this point, the circumstances bearing on the previous few points interact in such a way as to produce a series of further points.

Expressed in abstract fashion, the injunction of a reflective capacity and the injunction of a capacity limitation are recognized to impinge on each other in a way that brings to light a number of additional issues. Expressed in more concrete detail, the experiential instances that lead to the formation of these two points in the first place, as organizing poles of topics explicitly noticed, and that continue to surround their particular arrangements, …

'''Point 11.'''     Computational models of intelligent agents are limited to the consideration of “finitely informed constructions and computations”, or as I more affectionately call them, ''finitely informed creatures'' (FICs).

This point arises as a specialization of the point about capacity limits, where the discussion is restricted to the kinds of interpretive agents and the models of interpretive faculties that are available in a computational framework.

Something is a FIC to the extent that it falls into any of the following sorts:
# Anything that exists in the form of a finite number of bits,
# Anything whose objective being can be described in terms of a finite number of bits,
# Anything whose moment to moment activity can be specified by means of a finite number of bits.

Notice that this depiction makes being a FIC a term of description, and thus of possible approximation, not of necessity an exact definition of the thing's essential substance. An objective being or a real activity, even one that escapes all bounds of finite description, can be usefully represented “as” or “by means of” a FIC precisely to the extent that a particular description of it in this form succeeds in helping the agent concerned to orient toward its underlying reality and to deal with its ultimate consequences.

'''Point 12.'''     Reflection involves higher orders of sign relations.

As a minimum requirement, a capacity for reflection implies an ability to generate names for the elements, processes, and principles of thought. Assuming the tenet of pragmatism that all thought takes place in signs, this is tantamount to having signs for signs, signs for sign processes, and signs for sign relations. Further, each higher order sign that is generated in a process of reflection is required to take its place and to find its meaning within a correspondingly higher order sign relation.

In this connection, the designation ''higher order'' (HO) can be used as a generic adjective to describe a sign of any object whose nature it is to involve signs as a part of its being. The use of this adjective is subject to extension in natural ways to describe not only entire classes of signs but also the kinds of sign relations that involve them.

In order to reflect on signs themselves, it is necessary to have signs for signs, a necessary supply of which can be generated by quotation. But reflection on sign processes requires a much larger supply of signs. Initially, it requires a HO sign for each sign transition that actually occurs, that is, a name for each ordered pair of signs that is observed. Eventually, it requires a HO sign for each sign sequence that actually appears in experience, that is, a name for each <math>k\!</math>-tuple of signs seen. And reflection on sign relations requires an even larger stock of signs. It requires, initially, a HO sign for each sign transaction of the form <math>(o, s, i)\!</math> that is observed in experience and, ultimately, a HO sign for each sign relation that is encountered in experience or contemplated in a hypothetical situation.

If reflection is to constitute more than a transient form of observation, then provision needs to be made for permanently recording its HO signs. Under these conditions the capacity for instituting and maintaining an order of reflection is just a capacity for creating and storing HO signs.

This gives a brief glimpse of the issues involved in the effort toward reflection and the roughest possible estimate of the kinds of growth rates in the population of HO signs that are engendered by the need to provide a durable and stable medium for reflection. Further discussion of these topics can be put off to a later point. At this point it only needs to be clear that the injunction of a reflective capacity and the injunction of capacity limitations have an acute bearing on each other.

The combinatorial explosion engendered by reflection impinges on the capacity limitations of a FIC with such an impact that neither the standpoints of “naive empiricism” or “naive intuitionism” can continue to support viable forms of inquiry.

This is what makes the mediation of a ''higher order hypothesis'' (HOH), a hypothesis about the qualifications of a hypothesis, or a hypothesis about what can count as a hypothesis, so essential to the life of a FIC.

The process of generating signs that refer to things already signs is incited by a syntactic operation that is commonly called a “quotation”. Strictly speaking, the descriptive term “quotation” refers to generic class of syntactic functions, each of which maps one order of signs into the next higher order of signs. A proper form of quotation function is required to map signs in a one to one or “injective” fashion, and thus associates each element of its source domain with a HO sign that denotes it and it alone. In short, a quotation produces a unique “name” or a distinctive “number” to index each piece its source material.

Some sort of quotation operation has to be made available as a standard mechanism to support almost any level of theoretical discussion about syntax. In computational settings, various types of quotation operation need to be implemented as computable functions and provided among the basic resources for almost any adequate system of symbolic computation. Conceived as a stock device of computation, and supplied with domains of arguments already well established as signs, quotation is relatively easy to implement.

Given a well defined domain of signs as the initial material, it is not difficult to contemplate the generation of successively higher orders of signs that stem from the examples of the founding domain.

But a level of genuine reflection on sign processes and sign relations exceeds the generative capacity of mere quotation.

'''Point 13.'''     A ''finitely informed creature'' (FIC), if it is reflective up to the point that it reflects on its own nature as such, crosses a singular threshold of reflection, whereupon it not only obeys its own capacity limitations, as it instinctively and necessarily must, but also observes and reflects on their character.

'''Point 14.'''     Higher order sign relations tax the pragmatic resources of an interpretive agent to such a severe extent that they impinge on the practical limits of its representational capacity and computational ability.

When it is necessary to be precise, I use the term “matriculation” to refer to the first permanent recording of a sign by an agent, the one that marks in a relatively indelible fashion the initial recognition, original declaration, or principal registration of a sign by an agent, on which every subsequent use of that sign by that agent depends, and to which every later usage of that sign by that agent implicitly or explicitly refers.

The feature of matriculation that is important to the present argument is that it uses up memory capacity in a monotonic way. It is an economical strategy of memory usage to matriculate only the first token of each sign type observed and to let the observation of each subsequent token generate only a derivative reference to the primary registration. However, the present argument does not depend on the hypothesis of such a model actually being used, since this standard is only proposed to establish a lower bound on memory usage.

If quotation were the only mechanism for introducing HO signs, then each new round of HO signs would require for its primary registration only the same constant amount of memory capacity. The laying down of each new order of signs over the original foundation would take up an increment of memory equal to that used by the initial domain of signs.

But tagging sign processes and sign relations with signs that actually stick to them requires an agent to catch them first. In other words, the generation of signs for sign processes and signs for sign relations demands that an agent be able to perceive them or conceive them amidst the flow of an ongoing sign process that is itself governed by the law of a prevailing sign relation. This involves a contribution from the higher faculties of reasoning, in particular, taking steps of synthetic inference to introduce or invent the necessary signs. Since it resorts to the processes of inductive and abductive reasoning, this is naturally much more difficult to achieve.

In order to reflect on sign processes it is necessary to have signs for sign processes. One needs to start with signs for sign transitions, that is, signs for ordered pairs of signs, and work up to signs for arbitrary sequences of signs. As an empirical matter, every transition between signs that actually appears in experience is worth noting. By extension, it is useful to note as many sequences of transitions from sign to sign as actually occur, so long as one can spare the capacity to record them. If one also attends to the objects with regard to which these transitions occur, then one has the material of an empirical sign relation.

But this is a tricky matter, much less obvious than it seems at first. Pragmatic objects are more than just the physically compacted objects that happen to be present in a given situation, at, during, or in causal relation to a particular transition. In general, a pragmatic object is a hypothetical object, one whose presence in a situation, relevance to a transition, or association with a system of interpretation has to be hypothesized. But a hypothesis incurs a risk of error that goes beyond the elementary faults of observation and recording. The hypothesis has to be tested in subsequent experience and corrected by future inquiry. What this all comes down to is the circumstance that not even the raw empirical matter of a theory of signs can be panned from the pure stream of consciousness without a good admixture of speculation.

To amplify this point, in many cases the objects of a sign relation cannot be pointed out with any sense of clarity or resolve until the semantic equivalence classes are fairly and adequately sampled and the semantic partition that mirrors the structure of the object domain is at least partially reconstructed in the experience of an interpretive agent.

In order to reflect on sign relations, it is necessary to have signs for sign relations. Failing this, the laws or principles that sign processes follow, even if fleetingly half intuited, remain forever semi conscious, and thus they continue to rule in a subcritical state of representation.

At this point it becomes clear that the ideals of a naive empiricism must be left behind. The combinatorial explosion set off by the need to contemplate HO sign relations …

If it becomes necessary to entertain hypotheses about sign transitions, then the space of HO signs that has to be matriculated is potentially as large as the space of all ordered pairs of signs from the initial domain. If it becomes necessary to hypothesize about sign processes in general, then the space of HO signs that has to be matriculated grows like the union of the spaces of <math>k\!</math>-tuples of signs from the initial domain, where <math>k = 1, 2, 3, \ldots,\!</math> and possibly increases with no limit in principal.

The acuteness of this point, if taken in its full generality, brings the discussion to an appreciation of the next point.

'''Point 15.'''     Pragmatic incapacities have practical consequences.

A limitation of an agent's capacity along a pragmatic dimension …

'''Point 16.'''     Reflection involves a sense of context, and this involves a notion of community.

The capacity for reflection involves an ability to view one's own conduct in a context of other conceivable actions, and this implies viewing one's choices not just in a context of other possible actions for oneself, but also in a context of other conceivable actors, ones that are comparable to but characteristically distinct from oneself.

Remarkably, the capacities for criticism and creativity that are needed for reflection spring from a common source, namely, from the sense of possibility that can regard every process as occurring within a context of alternative actions. An inquiry, to be intelligent and innovative, critical and creative, has to be reflective, with the capacity to regard itself as one inquiry among others. In this “regard” is implied the ability of an interpretive agent to reference and to evaluate its own progress in inquiry, to observe it more dispassionately in subsequent reflections as the conduct of one inquirer among a host of many others, choosing one way of doing inquiry from the array of others conceivable. Accordingly, solely out of these reflections is developed the notion of a virtual or a potential community, quite independently of the empirical matter of how any actual or present community is constituted or realized at the moment.

'''Point 17.'''     Recalling the proposed application <math>{}^{\backprime\backprime}y \cdot y{}^{\prime\prime}\!</math> once again, it needs to be pointed out that an action cannot really act on an action, but only on its signs.

In technical terms, an action can act only on certain signs that exist in association with another or the same action, signs that are often called the “images” of the action to be affected.

'''Point 18.'''     The images, depictions, or descriptions of conduct generated by reflection, as records of experience, can be accumulated into theories and compiled into models of the corresponding conduct.

The collected images of conduct serve as “codes”, in both the senses of a descriptive datum and a prescriptive emblem. Both types of code fall subject to being tested in future experience, for their trustworthiness as bodies of observation or recommendation, respectively, with regard to their objects or intentions, as the case may be. Reviving an old term with just this spectrum of meanings, an encyclopedic corpus of received code can be called a “pandect”.

'''Point 19.'''     The power of reflection involves a risk of distortion.

The quality that separates reflection from introspection is its admission of fallibility. Although it is often troublesome to undo its distortions, the very fact that it can be in error, can miss its mark, or is by nature defeasible and falsifiable is exactly what makes a reflective image useful as a hypothesis, as an approximation to an infinitely subtler reality and as a simplification of an infinitely more complex and detailed truth, and yet one that retains a sufficient measure of realistic truth to be useful in the meaner times of a mortal existence.

The capacity of reflection to create an image in description of an action incurs a liability toward corruption in the image, both before and after its initial form is cast. The way that an image produced by reflection is designed to act as a sign of the action or permitted to behave as a code of the conduct is bound to be an imperfect device, due in large part to limitations of the media and affected in unaccounted measures by flaws in the mechanisms of reflection. How these distortions can be undone with repeated reflections, and how this clarification can be achieved without waylaying the conduct that reflection is meant to describe and control, is one of the main technical problems for empirical inquiry.

The power of reflection involves a capacity to project false images and thereby to generate distorting perspectives. The possibilities include the following:

# Views in which small things seem large and large things seem small,
# Value systems in which the apparent imports of things are reversed in relation to their actual imports,
# Forms of representation in which the places of contents interior and exterior to the surfaces of reflection are exchanged, reversed, or transposed.

There is a positive spin on the fallibility of the reflective imagination. In terms of its practical bearings on continued experience, the fallibility of reflection involves an ability, not only to make its errors over again in the form of their consequences for experience, but eventually to find its faults recognized as such in a finite order of subsequent reflections.

The way that reflection, in adjunction to conduct, leads conduct to yield a description of itself, thus creating a relation between action and sign, behavior and code, that is open to be traced in either direction, is the principal mystery of its operation. How reflection can be led to do this in a way that positively reinforces the intention of the conduct and that constructively criticizes its ongoing performance, controlling its desire to control the action so that it does not destructively interfere with the completion of the task, is the critical practical question of the work.

In sum, the feature of reflection that seems to render it most defective, its fallibility, which involves its ability to be recognized as false in the future of reflection, is the main trait that allows it to play a part in the staging of empirical inquiry.

'''Point 20.'''     The capacity for reflection involves an ability to question one's working assumptions, especially when there is occasion to suspect that they are no longer working as well as they once did.

Whenever one operates on a particular assumption, whether knowingly or otherwise, one tends to see certain patterns of features in perception and to miss others, but until one reflects on the operative assumption, makes it explicit, considers its alternatives, and thereby is empowered to put it in question, then one lacks a fundamental insight into how these figures are generated in perception, failing to see how one's own sensitivities and dispositions are biased toward allowing them to arise.

To act on the basis of a certain assumption, as though the assumption were already certain, is to act in abstraction of the total situation. When a feature or a pattern of features is abstracted from a situation, there is always something left behind, the grounds from which a feature or pattern originally rises and against which it subsequently becomes a figure of importance to the moment. There needs to be a name for this actively recessed background, suggesting the potential complement of alternative features and elliptic patterns that it contains within its share of the total configuration. But it is important to remember that this is not just the ground that comes to complement a figure in the present situation but the ground that is dynamically pushed into the past so that the current configuration can come to be formed as it is.

Exactly what it is that abstraction leaves out is something that seems currently to escape description, failing to be pinned down by any name I can think of in common or in technical use. The abstraction itself, as the process whose result is signaled by its “-ionized” designation, acts toward the end of constellating a figure that is relevant to the moment. But the concurrent and complementary process that results in a residual plurality is one that lacks a common denomination. For the sake of a harmonious balance between the syntactic expressions of these actions, it would be good if the process that recesses the background were also to be assigned an “-ionized” term.

'''Point 21.'''     There appears to be a large variety of ways that the process of reflection can go wrong.

One of the jobs of an inquiry into inquiry is to classify this variety, compassing the diversity of incidental errors and systematic distortions that are likely to occur in reflection.

One dimension of variation that runs through this variety of pathologies characterizes the degree of fixity or persistence that is invested in the images of conduct. The range of variation conceivable can be suggested by marking the prototypical figures that fall at its two extremes.

# At one extreme there is the character of a stolid fixity that can be adumbrated in terms of a mythological or a psychological archetype, appearing to be ruled by the image of Narcissus. This identifies the kind of regressive and fixed ideation that leads one to seize on a single image of one's characteristic conduct, to fix it in mind as a static ideal, and to resist at all costs letting go of its hold on the imagination.
# At the other extreme there is the character of an insipid volatility that corresponds to the complementary archetype, answering a bit dully to the name of Echo. This identifies the kind of digressive and fluid skepticism that leaves one in a permanently fugitive state. Paradoxically enough, it is typically pursuant to a precocious but transient condition of dedication, one that marks its earliest forms of conscious recognition. If it follows the usual course, it can start from being too soon fixed on the initial object of attention or the original ideal of conduct, but it eventually falls into a compensatory, defensive, and reactionary pattern. Soon it withers away into little more than the afterimage of a reflexive reaction, an account due to the ensuing trauma of disappointment, and a record commemorating a final disillusionment with its distant illusions. Whatever the initial case, the issue is such that it makes one reluctant to commit to any future image of behavior or ideal of conduct, at least, readily enough to try its utility in action or steadily enough to test it out in practice. Instead, it disposes one merely to keep repeating in an automatic, derivative, imitative, involuntary, reflexive, stereotypical, and tautologous manner any impression of the outside world that seems to inform the moment.

'''Point 22.'''     Intelligent inquiry involves inquiry into inquiry.

In view of the previous points, it appears that intelligent inquiry is necessarily reflective inquiry, seeing itself as one inquiry among others and evaluating its own progress in a setting of comparable alternatives. This means that intelligent inquiry into any subject whatever is forced to embody a component of self-study, of inquiry into inquiry. Thus, the general capacity for successfully conducting inquiry both relies on and bears on a specialized kernel of talent for doing inquiry into inquiry.

'''Point 23.'''     Inquiry into inquiry involves integrating independent inquiries.

One way of gathering data that is relevant to the task of self-study is to conduct a multiplicity of independent studies, each of which tries to track what the others are most likely to miss. This requires a monitor, a moderator, or a non-parallel but mutually concurred upon medium of comparison for overseeing and reconciling the mosaic of disparate and scattered results that can derive from a multitude of isolated studies.

Finally, this project of self study demands a comprehensive method for integrating the divergent and fragmentary imports of individual studies into a unified form, constituting the resultant bearing that they are meant to have on the main inquiry. Toward this end, it is a frequent stratagem of intelligent inquiry to maintain a form of “outrigger”, an attached but esthetically distant study that serves to steady the main course of study by embodying a full program of peripheral perspectives and exploratory investigations. In this way, the global aims of even a specialized inquiry can be achieved more robustly by keeping a studied eye out for its own systematic alternatives, often involving precisely those “outliers” that are ignored by the more focal styles of inquiry.

In times of shifting paradigms the outrigger of an established inquiry can take on a signal purpose as the forerunner of a new investigation, and can with added reinforcements even take over the role of the main. With nothing more than a few spare kernels of aptitude for reflective inquiry, that is, with a minimal but germinal talent for inquiry into inquiry to serve as a catalyst, the outriding projections and their deponent objections, testifying all along in what seems like a purely negative fashion to the mounting accumulations of anomalous evidence, can find themselves converted, refitted, and positively reconditioned. Transformed in this way, the original outrigger, with its outrageous hypotheses and its crew of motley anomalies, are ready to become the new hull, the mainstays, and the supporting constituency of a renewed constitution for inquiry, one that can sustain its overall course but more significantly its overriding cause through another day.

'''Point 24.'''     Reflective projects being partial, their refractory parts are likely to remain partial to their outward projections.

An ''unreflective framework'' (UF), if it does not devolve into a condition of total confusion, and thus deserves to be called a framework at all, ordinarily maintains a clear separation between the objective and the interpretive parts of its organization. This pragmatic division of labor coincides with a substantive distinction that is ordained to exist between the object system that is subject to observation or interpretation and the agent system that observes or interprets it.

But the goal of reflection is to make one's own conduct an object among other objects, something that can be critically evaluated as one choice among many and subsequently amended if found wanting. In this aim a realistic project of reflection never sees more than partial success. There is always a refractory residue of ongoing conduct that resists analysis and remains unreflected in any clear form of representation. Thus, the actual effect of a reflective project is to represent only a part of one's interpretive conduct as a part of one's objective regard, in other words, to reconfigure a part of one's IF as a part of one's OF.

'''Point 25.'''

The purpose of constructing a RIF is to demonstrate how it might be possible for interpretive agents to reflect on their own processes of interpretation, to critically evaluate the interpretive choices they make, and to choose from alternate interpretations based on the results of this reflection and evaluation. These are the abilities that interpreters need to carry out inquiry, and especially to pursue an inquiry into inquiry.

It seems that human beings do have the ability to reflect on their own interpretive processes, at least, to the extent that they can observe the obvious aspects of the interpretive experience and control the overt features of the interpretive activity, and insofar as these aspects and features of the experimental activity are manifested at the phenomenal surfaces of its underlying processes. Moreover, it seems that people do know how to interrogate their own judgments, turning again and again to investigate the traces of their past reflections and pausing in anticipation to examine the balance of their next evaluation.

Consequently, it must be possible to explain these apparent abilities in just one of two ways: either to account for the faculties of reflection and selection by presenting a logical model of the processes involved, or else to dispel the illusion of each performance by showing what goes on in its place. In either case, an inquiry into the virtues of critically reflective phenomena is called on to provide a plausible model for what is happening beneath the semblances of reflective and critical thought. Whether the resulting resolution of a particular phenomenon preserves or dissolves its appearances is a matter that depends on the details of the case, and perhaps to a degree on personal taste.

'''Point 26.'''     This marks a branch point. I tentatively assume that the apparent power of reflection is really more or less as it appears to be, at least, in the same spirit as it appears to be, and not some radically insidious self-deception arising on the part of its apparent agents.

Setting out initially on the positive track, I begin with the assumption that a RIF is a real possibility. In order to conceive of a RIF being possible it is necessary to set aside a host of set theoretic difficulties that might be imagined to afflict any invocation of self referent themes. No matter whether interpretation is presented in terms of a framework, a faculty, a process, a trajectory, or a hypostatic agent that is assumed to carry out its procedures, there is a problem about how anything so fleeting and so sweeping as an ongoing interpretation can refer to itself as a situated form of activity, in other words, as an objective system of interpretation that rests within a context of alternative interpretations.

There is a piece of terminology that is often useful in this connection. In set-theoretic contexts, either one of the phrases ''X collects Y'' or ''X encases Y'' can be used to mean the same thing as ''Y'' ∈ ''X''. These formulations can be taken as abbreviated ways of saying that ''X'' enumerates ''Y'' among its cases. Thus, they express the converse of the membership relation but manage to avoid the ambiguity of the phrase ''X contains Y'', a form that would otherwise have to be qualified on each occasion of its use by specifying whether one means ''contains as an element'' or ''contains as a subset'', as the case may be.

To wrap up the development of this reflective project in a single line: When the mind's original effort to catch itself at work seizes on the inventions of set theory to encapsulate its speculations, the ensuing breed of self reification that comes from mingling an unbridled capacity for self referent expressions with an unchecked propensity for creating abstract objects gives rise to the generation of set theoretic paradoxes. As a result, it is incumbent on me to show how the concretely limited kinds of constructions that I have in mind can avoid a similar excess and steer clear of the corresponding difficulties.

If formalized, a RIF would be an IF that can properly, if only partially, refer to itself as an OF. Thus, as formalized, a RIF amounts to both a reflexive and a recursive SOI, one that can refer to itself as an object, to the extent that any formal system can. As a reflexive SOI, a RIF has a sign that refers to itself. As a recursive SOI, a RIF has a character that can be determined by invoking the record of signs that it uses to refer to simpler versions and earlier developments of itself.

But more than all this, in order to be genuinely reflective a RIF's consideration of itself as a situated form of activity must extend to the consideration of alternative selves. This means that a RIF must have references to other SOIs, not only those that are continuous with the space of its own potential conduct and correlated to the course of its own form of activity, but also those that are discontinuous from and independent of its own way of being.

In keeping with the spirit of a discussion based on concrete examples, the RIF to be improvised here is restrained to the scale of a minimal IF that can reflect on the scene of A and B, in this case, synthesizing a portion of the OFs and IFs suggested by the sign relations <math>A\!</math> and <math>B\!</math> into an integrated SOI. While I do not plan to specify the additional constraints that would be needed to determine this RIF uniquely, even to say whether it is finite or infinite, it forms a convenient reference point for the rest of this section to designate the purported ideal as ''the RIF generated by <math>A\!</math> and <math>B\!</math>'' and to notate it as <math>\operatorname{RIF}(A, B).\!</math>

In accord with the customary figure of speech, a RIF can be personified in the agency of a “reflective interpreter” that possesses the faculties to carry out its actions, and this agent is in turn characterized as the localized representative of a suitably reflective and situated process of interpretation.

A reflective interpreter needs a capacity for referring to its own role in the process of interpretation, for conceptualizing each transition from sign to interpretant sign as occurring within a context of alternatives, and for noticing that each option has a potentially distinctive value with respect to a prevailing object or objective. ''Capacity'', as used in this connection, is a word with both structural and functional connotations. It implies the structural capacity that is required to articulate, record, and maintain data about observable forms of interpretive conduct, and it involves the functional capacity that is demanded to create and exploit this data, in effect, constituting a higher order of interpretive activity.

If one tries to understand the conduct of a reflective interpreter as a process of interpretation there are a number of questions that arise. How can anything so ongoing as a process of interpretation refer to an object, and how can anything so fleeting as a process of interpretation be referred to as an object?

A process that refers to itself is not like a set that collects itself, or a collection that would enroll itself among its own elements, even if some attempts to process the reference and to lay it out in a literal account do try to dissect and explain it as such. A sign that is elemental to a universe, perhaps by means of which one seeks to explain the universe, does not in fact collect, dominate, or encase the entire universe simply by referring to it, even if some interpretive interloper, at the risk of vitiating the whole account, is tempted to explain the elementary part in terms of the complex totality.

One reason for introducing the distinction between OFs and IFs into the present discussion is to keep track of the complex relationships between object domains and sign domains, between the constitutions of objects and the constitutions of signs. It is a frequent practice in mathematics to blur this distinction, often saying that an object is constituted as a set of further objects when one really means that the sign or information one has about the object is constituted as a set of further signs or further informations about the object, all of which can refer to further objects, but not always the sorts of objects that are literally intended as elementary constituents of the original object. Furthermore, each use of the directive ''further'' in this description marks a place where a suitably reflective interpreter ought to ask whether ''further'' implies ''simpler'' or merely ''other'', and in turn whether ''other'' means essentially other or only otherwise appearing.

But the distinction between object and sign, however important, is still a pragmatic distinction, involving a thing's use in a particular role, and not an essential distinction, fixing a thing's prior and eternal nature. Of course, it can turn out that some objects will never serve as signs and that some signs will never be observed as objects, but these types of eventuality involve empirical questions and contingent facts, and their actualization depends on the kinds of circumstances that have to be discovered after the fact rather than dictated ''a priori''.

The construction of a RIF forces the discussion to a point where the OFs and IFs and the relationships between them suddenly become much more complex, and where confusion can arise precisely from the fact that the purpose of a RIF is to convert an IF into the sort of thing that can be referred to and reflected on as an object. Developments like these make it all the more necessary to understand the exact character of the distinction between OFs and IFs. In a complex IF signs do participate in constitutional relationships, with complex signs being constructed out of simpler signs. But the relations involved in denotation and connotation are not limited to constitutional linkages of this sort, and thus they cannot be expected to generate by themselves the necessary sorts of analytic and synthetic hierarchies.

All in all, a RIF involves the close coordination of an OF and an IF, plus mechanisms for carrying out the so called ''reflective operations'' (ROs) that go to negotiate between the objective and the interpretive realms. The work of ROing permits processes of interpretation, initially taking place largely in the IF and impinging on the OF only at isolated points, to be formalized and objectified, thereby becoming segments of the OF. Taken over time the cumulative effect of this ROing motion gradually turns more and more of the IF into new sectors and layers of the OF.

'''Point 27.'''

There is a portion of reasoning that consists in drawing distinctions, signifying the features thereby distinguished by means of logical terms, recognizing constraints on the conjoint occurrences of these features, expressing these constraints in the form of logical premisses, and then drawing the implications of these premisses as the occasion warrants. This part of logic, in its formalizable aspects, is generally referred to as ''propositional calculus'' (PropC), ''sentential logic'' (SL), or sometimes as ''zeroth order logic'' (ZOL).

With any system of logic, at least, that does not propose a purely syntactic rationale for itself, it is necessary to draw a distinction between the logical object that is denoted, expressed, or represented in thinking and the logical sign that denotes, expresses, or represents it. Often one uses the contrast between ''proposition'' and ''expression'' or the shade of difference between ''statement'' and ''sentence'' to convey the distinction between the logical object signified and the syntactic assemblage that signifies it. Another option is to let the division lie between a ''position'' and a ''proposition'', with the suggestion being that the function of a symbolic proposition is to indicate indifferently a plurality of logical positions. In accord with my personal preference, I use the term ''proposition'' ambiguously, expecting context to resolve the question, and resorting to the term ''expression'' when it does not.

'''Point 28.'''

Adequate reasoning about the propositional constitution or the sentential representation of POVs and PODs requires a logical system that can work with ''higher order propositions'' (HOPs).

'''Point 29.'''

Finally, interlaced with the structures of the OF and the IF, there is a need for a structure that I call a ''dynamic evaluative framework'' (DEF). This is intended to isolate the twin aspects of process and purpose that are observable on either side of the objective interpretive divide and to assist in formalizing the graded notions of directed change that are able to be actualized in the medium of a RIF.

----
<div align="center">
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems|Contents]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 1|Part 1]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 2|Part 2]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 3|Part 3]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 4|Part 4]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 5|Part 5]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6|Part 6]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 7|Part 7]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 8|Part 8]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Appendices|Appendices]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : References|References]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Document History|Document History]]
•
</div>
----

[[Category:Artificial Intelligence]]
[[Category:Critical Thinking]]
[[Category:Cybernetics]]
[[Category:Education]]
[[Category:Hermeneutics]]
[[Category:Information Systems]]
[[Category:Inquiry]]
[[Category:Intelligence Amplification]]
[[Category:Learning Organizations]]
[[Category:Knowledge Representation]]
[[Category:Logic]]
[[Category:Philosophy]]
[[Category:Pragmatics]]
[[Category:Semantics]]
[[Category:Semiotics]]
[[Category:Systems Science]]

Differential Logic and Dynamic Systems • Part 3

2026-02-09T17:48:04Z

Jon Awbrey: update

<b>Author: [[User:Jon Awbrey|Jon Awbrey]]</b>

----
<div align="center">
• [[Differential Logic and Dynamic Systems • Overview|Overview]]
• [[Differential Logic and Dynamic Systems • Part 1|Part 1]]
• [[Differential Logic and Dynamic Systems • Part 2|Part 2]]
• [[Differential Logic and Dynamic Systems • Part 3|Part 3]]
• [[Differential Logic and Dynamic Systems • Part 4|Part 4]]
• [[Differential Logic and Dynamic Systems • Part 5|Part 5]]
• [[Differential Logic and Dynamic Systems • Appendices|Appendices]]
• [[Differential Logic and Dynamic Systems • References|References]]
• [[Differential Logic and Dynamic Systems • Document History|Document History]]
•
</div>
----

==Transformations of Discourse==

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
It is understandable that an engineer should be completely absorbed in his speciality, instead of pouring himself out into the freedom and vastness of the world of thought, even though his machines are being sent off to the ends of the earth;  for he no more needs to be capable of applying to his own personal soul what is daring and new in the soul of his subject than a machine is in fact capable of applying to itself the differential calculus on which it is based.  The same thing cannot, however, be said about mathematics;  for here we have the new method of thought, pure intellect, the very well‑spring of the times, the <i>fons et origo</i> of an unfathomable transformation.
| width="4%" |  
|-
| align="right" colspan="3" | — Robert Musil, ''The Man Without Qualities'', [Mus, 39]
|}

Here we take up the general study of <i>logical transformations</i>, or maps relating one universe of discourse to another.  In many ways, and especially as applied to the subject of intelligent dynamic systems, the argument will develop the antithesis of the statement just quoted.  Along the way, if incidental to my ends, I hope the present essay can pose a fittingly irenic epitaph to the frankly ironic epigraph inscribed at its head.

The goal is to answer a single question:  <i>What is a propositional tangent functor?</i>  In other words, the aim is to develop a clear conception of what manner of thing would pass in the logical realm for a genuine analogue of the tangent functor, an object conceived to generalize as far as possible in the abstract terms of category theory the ordinary notions of functional differentiation and the all too familiar operations of taking derivatives.

As a first step we examine the types of transformations we already know as <i>extensions</i> and <i>projections</i> and we use their special cases to illustrate several styles of logical and visual representation which figure in the sequel.

===Foreshadowing Transformations : Extensions and Projections of Discourse===

{| width="100%" cellpadding="0" cellspacing="0"
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| width="92%" |
And, despite the care which she took to look behind her at every moment, she failed to see a shadow which followed her like her own shadow, which stopped when she stopped, which started again when she did, and which made no more noise than a well‑conducted shadow should.
| width="4%" |  
|-
| align="right" colspan="3" | — Gaston Leroux, ''The Phantom of the Opera'', [Ler, 126]
|}

Many times in our discussion we have occasion to place one universe of discourse in the context of a larger universe of discourse.  An embedding of the type <math>[\mathcal{X}] \to [\mathcal{Y}]</math> is implied any time we make use of one basis <math>\mathcal{X}</math> which happens to be included in another basis <math>\mathcal{Y}.</math>  When discussing differential relations we usually have in mind the extended alphabet <math>\mathfrak{Y}</math> has a special construction or a specific lexical relation with respect to the initial alphabet <math>\mathfrak{X},</math> one which is marked by characteristic types of accents, indices, or inflected forms.

====Extension from 1 to 2 Dimensions====

Figure 18-a lays out the ''angular form'' of venn diagram for universes of 1 and 2 dimensions, indicating the embedding map of type <math>\mathbb{B}^1 \to \mathbb{B}^2</math> and detailing the coordinates that are associated with individual cells.  Because all points, cells, or logical interpretations are represented as connected geometric areas, we can say these pictures provide us with an ''areal view'' of each universe of discourse.

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[File:Diff Log Dyn Sys -- Figure 18-a -- Extension from 1 to 2 Dimensions.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 18-a.} ~~ \text{Extension from 1 to 2 Dimensions : Areal}</math>
|}

Figure 18-b shows the differential extension from <math>X^\bullet = [x]</math> to <math>\mathrm{E}X^\bullet = [x, \mathrm{d}x]</math> in a ''bundle of boxes'' form of venn diagram. As awkward as it may seem at first, this type of picture is often the most natural and the most easily available representation when we want to conceptualize the localized information or momentary knowledge of an intelligent dynamic system. It gives a ready picture of a ''proposition at a point'', in the present instance, of a proposition about changing states which is itself associated with a particular dynamic state of a system. It is easy to see how this application might be extended to conceive of more general types of instantaneous knowledge that are possessed by a system.

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[File:Diff Log Dyn Sys -- Figure 18-b -- Extension from 1 to 2 Dimensions.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 18-b.} ~~ \text{Extension from 1 to 2 Dimensions : Bundle}</math>
|}

Figure 18-c shows the same extension in a ''compact'' style of venn diagram, where the differential features at each position are represented by arrows extending from that position that cross or do not cross, as the case may be, the corresponding feature boundaries.

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[File:Diff Log Dyn Sys -- Figure 18-c -- Extension from 1 to 2 Dimensions.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 18-c.} ~~ \text{Extension from 1 to 2 Dimensions : Compact}</math>
|}

Figure 18-d compresses the picture of the differential extension even further, yielding a directed graph or ''digraph'' form of representation. (Notice that my definition of a digraph allows for loops or ''slings'' at individual points, in addition to arcs or ''arrows'' between the points.)

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[File:Diff Log Dyn Sys -- Figure 18-d -- Extension from 1 to 2 Dimensions.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 18-d.} ~~ \text{Extension from 1 to 2 Dimensions : Digraph}</math>
|}

====Extension from 2 to 4 Dimensions====

Figure 19-a lays out the ''areal view'' or the ''angular form'' of venn diagram for universes of 2 and 4 dimensions, indicating the embedding map of type <math>\mathbb{B}^2 \to \mathbb{B}^4.</math> In many ways these pictures are the best kind there is, giving full canvass to an ideal vista. Their style allows the clearest, the fairest, and the plainest view that we can form of a universe of discourse, affording equal representation to all dispositions and maintaining a balance with respect to ordinary and differential features. If only we could extend this view! Unluckily, an obvious difficulty beclouds this prospect, and that is how precipitately we run into the limits of our plane and visual intuitions. Even within the scope of the spare few dimensions that we have scanned up to this point subtle discrepancies have crept in already. The circumstances that bind us and the frameworks that block us, the flat distortion of the planar projection and the inevitable ineffability that precludes us from wrapping its rhomb figure into rings around a torus, all of these factors disguise the underlying but true connectivity of the universe of discourse.

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[File:Diff Log Dyn Sys -- Figure 19-a -- Extension from 2 to 4 Dimensions.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 19-a.} ~~ \text{Extension from 2 to 4 Dimensions : Areal}</math>
|}

Figure 19-b shows the differential extension from <math>U^\bullet = [u, v]</math> to <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v]</math> in the ''bundle of boxes'' form of venn diagram.

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[File:Diff Log Dyn Sys -- Figure 19-b -- Extension from 2 to 4 Dimensions.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 19-b.} ~~ \text{Extension from 2 to 4 Dimensions : Bundle}</math>
|}

As dimensions increase, this factorization of the extended universe along the lines that are marked out by the bundle picture begins to look more and more like a practical necessity. But whenever we use a propositional model to address a real situation in the context of nature we need to remain aware that this articulation into factors, affecting our description, may be wholly artificial in nature and cleave to nothing, no joint in nature, nor any juncture in time to be in or out of joint.

Figure 19-c illustrates the extension from 2 to 4 dimensions in the ''compact'' style of venn diagram. Here, just the changes with respect to the center cell are shown.

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[File:Diff Log Dyn Sys -- Figure 19-c -- Extension from 2 to 4 Dimensions.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 19-c.} ~~ \text{Extension from 2 to 4 Dimensions : Compact}</math>
|}

Figure 19-d gives the ''digraph'' form of representation for the differential extension <math>U^\bullet \to \mathrm{E}U^\bullet,</math> where the 4 nodes marked with a circle <math>{}^{\bigcirc}</math> are the cells <math>uv,\, u \texttt{(} v \texttt{)},\, \texttt{(} u \texttt{)} v,\, \texttt{(} u \texttt{)(} v \texttt{)},</math> respectively, and where a 2-headed arc counts as 2 arcs of the differential digraph.

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[File:Diff Log Dyn Sys -- Figure 19-d -- Extension from 2 to 4 Dimensions.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 19-d.} ~~ \text{Extension from 2 to 4 Dimensions : Digraph}</math>
|}

===Thematization of Functions : And a Declaration of Independence for Variables===

{| width="100%"
| align="left" |
''And as imagination bodies forth''<br>
''The forms of things unknown, the poet's pen''<br>
''Turns them to shapes, and gives to airy nothing''<br>
''A local habitation and a name.''
| align="right" valign="bottom" | A Midsummer Night's Dream, 5.1.18
|}

In the representation of propositions as functions it is possible to notice different degrees of explicitness in the way their functional character is symbolized. To indicate what I mean by this, the next series of Figures illustrates a set of graphic conventions that will be put to frequent use in the remainder of this discussion, both to mark the relevant distinctions and to help us convert between related expressions at different levels of explicitness in their functionality.

====Thematization : Venn Diagrams====

{| width="100%" cellpadding="0" cellspacing="0"
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The known universe has one complete lover and that is the greatest poet. He consumes an eternal passion and is indifferent which chance happens and which possible contingency of fortune or misfortune and persuades daily and hourly his delicious pay.
| width="4%" |  
|-
| align="right" colspan="3" | — Walt Whitman, ''Leaves of Grass'', [Whi, 11–12]
|}

Figure 20-i traces the first couple of steps in this order of ''thematic'' progression, that will gradually run the gamut through a complete series of degrees of functional explicitness in the expression of logical propositions. The first venn diagram represents a situation where the function is indicated by a shaded figure and a logical expression. At this stage one may be thinking of the proposition only as expressed by a formula in a particular language and its content only as a subset of the universe of discourse, as when considering the proposition <math>u\!\cdot\!v</math> in the universe <math>[u, v].</math> The second venn diagram depicts a situation in which two significant steps have been taken. First, one has taken the trouble to give the proposition <math>u\!\cdot\!v</math> a distinctive functional name <math>{}^{\backprime\backprime} J {}^{\prime\prime}.</math> Second, one has come to think explicitly about the target domain that contains the functional values of <math>J,</math> as when writing <math>J : \langle u, v \rangle \to \mathbb{B}.</math>

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[File:Diff Log Dyn Sys -- Figure 20-i -- Thematization of Conjunction (Stage 1).gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 20-i.} ~~ \text{Thematization of Conjunction (Stage 1)}</math>
|}

In Figure 20-ii the proposition <math>J</math> is viewed explicitly as a transformation from one universe of discourse to another.

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[File:Diff Log Dyn Sys -- Figure 20-ii -- Thematization of Conjunction (Stage 2).gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 20-ii.} ~~ \text{Thematization of Conjunction (Stage 2)}</math>
|}

{| align="center" border="0" cellpadding="10"
|
<pre>
o-------------------------------o o-------------------------------o
| | | |
| o-----o o-----o | | o-----o o-----o |
| / \ / \ | | / \ / \ |
| / o \ | | / o \ |
| / /`\ \ | | / /`\ \ |
| o o```o o | | o o```o o |
| | u |```| v | | | | u |```| v | |
| o o```o o | | o o```o o |
| \ \`/ / | | \ \`/ / |
| \ o / | | \ o / |
| \ / \ / | | \ / \ / |
| o-----o o-----o | | o-----o o-----o |
| | | |
o-------------------------------o o-------------------------------o
\ / \ /
\ / \ /
\ / \ J /
\ / \ /
\ / \ /
o----------\---------/----------o o----------\---------/----------o
| \ / | | \ / |
| \ / | | \ / |
| o-----@-----o | | o-----@-----o |
| /`````````````\ | | /`````````````\ |
| /```````````````\ | | /```````````````\ |
| /`````````````````\ | | /`````````````````\ |
| o```````````````````o | | o```````````````````o |
| |```````````````````| | | |```````````````````| |
| |```````` J ````````| | | |```````` x ````````| |
| |```````````````````| | | |```````````````````| |
| o```````````````````o | | o```````````````````o |
| \`````````````````/ | | \`````````````````/ |
| \```````````````/ | | \```````````````/ |
| \`````````````/ | | \`````````````/ |
| o-----------o | | o-----------o |
| | | |
| | | |
o-------------------------------o o-------------------------------o
J = u v x = J<u, v>

Figure 20-ii. Thematization of Conjunction (Stage 2)
</pre>
|}

In the first venn diagram the name that is assigned to a composite proposition, function, or region in the source universe is delegated to a simple feature in the target universe. This can result in a single character or term exceeding the responsibilities it can carry off well. Allowing the name of a function <math>J : \langle u, v \rangle \to \mathbb{B}</math> to serve as the name of its dependent variable <math>J : \mathbb{B}</math> does not mean that one has to confuse a function with any of its values, but it does put one at risk for a number of obvious problems, and we should not be surprised, on numerous and limiting occasions, when quibbling arises from the attempts of a too original syntax to serve these two masters.

The second venn diagram circumvents these difficulties by introducing a new variable name for each basic feature of the target universe, as when writing <math>J : \langle u, v \rangle \to \langle x \rangle,</math> and thereby assigns a concrete type <math>\langle x \rangle</math> to the abstract codomain <math>\mathbb{B}.</math> To make this induction of variables more formal one can append subscripts, as in <math>x_J,</math> to indicate the origin or derivation of the new characters. Or we may use a lexical modifier to convert function names into variable names, for example, associating the function name <math>J</math> with the variable name <math>\check{J}.</math> Thus we may think of <math>x = x_J = \check{J}</math> as the ''cache variable'' corresponding to the function <math>J</math> or the symbol <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> considered as a contingent variable.

In Figure 20-iii we arrive at a stage where the functional equations <math>J = u\!\cdot\!v</math> and <math>x = u\!\cdot\!v</math> are regarded as propositions in their own right, reigning in and ruling over the 3-feature universes of discourse <math>[u, v, J]</math> and <math>[u, v, x],</math> respectively. Subject to the cautions already noted, the function name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> can be reinterpreted as the name of a feature <math>\check{J}</math> and the equation <math>J = u\!\cdot\!v</math> can be read as the logical equivalence <math>\texttt{((} J, u ~ v \texttt{))}.</math> To give it a generic name let us call this newly expressed, collateral proposition the ''thematization'' or the ''thematic extension'' of the original proposition <math>J.</math>

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[File:Diff Log Dyn Sys -- Figure 20-iii -- Thematization of Conjunction (Stage 3).gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 20-iii.} ~~ \text{Thematization of Conjunction (Stage 3)}</math>
|}

The first venn diagram represents the thematization of the conjunction <math>J</math> with shading in the appropriate regions of the universe <math>[u, v, J].</math> Also, it illustrates a quick way of constructing a thematic extension. First, draw a line, in practice or the imagination, that bisects every cell of the original universe, placing half of each cell under the aegis of the thematized proposition and the other half under its antithesis. Next, on the scene where the theme applies leave the shade wherever it lies, and off the stage, where it plays otherwise, stagger the pattern in a harlequin guise.

In the final venn diagram of this sequence the thematic progression comes full circle and completes one round of its development. The ambiguities that were occasioned by the changing role of the name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> are resolved by introducing a new variable name <math>{}^{\backprime\backprime} x {}^{\prime\prime}</math> to take the place of <math>\check{J},</math> and the region that represents this fresh featured <math>x</math> is circumscribed in a more conventional symmetry of form and placement. Just as we once gave the name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> to the proposition <math>u\!\cdot\!v,</math> we now give the name <math>{}^{\backprime\backprime} \iota {}^{\prime\prime}</math> to its thematization <math>\texttt{((} x, u ~ v \texttt{))}.</math> Already, again, at this culminating stage of reflection, we begin to think of the newly named proposition as a distinctive individual, a particular function <math>\iota : \langle u, v, x \rangle \to \mathbb{B}.</math>

From now on, the terms ''thematic extension'' and ''thematization'' will be used to describe both the process and degree of explication that progresses through this series of pictures, both the operation of increasingly explicit symbolization and the dimension of variation that is swept out by it. To speak of this change in general, that takes us in our current example from <math>J</math> to <math>\iota,</math> we introduce a class of operators symbolized by the Greek letter <math>\theta,</math> writing <math>\iota = \theta J</math> in the present instance. The operator <math>\theta,</math> in the present situation bearing the type <math>\theta : [u, v] \to [u, v, x],</math> provides us with a convenient way of recapitulating and summarizing the complete cycle of thematic developments.

Figure 21 shows how the thematic extension operator <math>\theta</math> acts on two further examples, the disjunction <math>\texttt{((} u \texttt{)(} v \texttt{))}</math> and the equality <math>\texttt{((} u, v \texttt{))}.</math> Referring to the disjunction as <math>f(u, v)</math> and the equality as <math>f(u, v),</math> we may express the thematic extensions as <math>\varphi = \theta f</math> and <math>\gamma = \theta g.</math>

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[File:Diff Log Dyn Sys -- Figure 21 -- Thematization of Disjunction and Equality.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 21.} ~~ \text{Thematization of Disjunction and Equality}</math>
|}

====Thematization : Truth Tables====

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
That which distorts honest shapes or which creates unearthly beings or places or contingencies is a nuisance and a revolt.
| width="4%" |  
|-
| align="right" colspan="3" | — Walt Whitman, ''Leaves of Grass'', [Whi, 19]
|}

Tables 22 through 25 outline a method for computing the thematic extensions of propositions in terms of their coordinate values.

A preliminary step, as illustrated in Table 22, is to write out the truth table representations of the propositional forms whose thematic extensions one wants to compute, in the present instance, the functions <math>f(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}</math> and <math>g(u, v) = \texttt{((} u, v \texttt{))}.</math>

<br>

{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:50%"
|+ style="height:30px" | <math>\text{Table 22.} ~~ \text{Disjunction}~ f ~\text{and Equality}~ g</math>
|- style="height:35px; background:ghostwhite; width:100%"
| <math>u</math>
| <math>v</math>
| style="border-left:1px solid black" | <math>f</math>
| <math>g</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
0
\\[4pt]
0
\\[4pt]
1
\\[4pt]
1
\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
0
\\[4pt]
1
\\[4pt]
0
\\[4pt]
1
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
0
\\[4pt]
1
\\[4pt]
1
\\[4pt]
1
\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
1
\\[4pt]
0
\\[4pt]
0
\\[4pt]
1
\end{matrix}</math>
|}

<br>

Next, each propositional form is individually represented in the fashion shown in Tables 23-i and 23-ii, using <math>{}^{\backprime\backprime} f {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} g {}^{\prime\prime}</math> as function names and creating new variables <math>x</math> and <math>y</math> to hold the associated functional values. This pair of Tables outlines the first stage in the transition from the <math>2</math>-dimensional universes of <math>f</math> and <math>g</math> to the <math>3</math>-dimensional universes of <math>\theta f</math> and <math>\theta g.</math> The top halves of the Tables replicate the truth table patterns for <math>f</math> and <math>g</math> in the form <math>f : [u, v] \to [x]</math> and <math>g : [u, v] \to [y].</math> The bottom halves of the tables print the negatives of these pictures, as it were, and paste the truth tables for <math>\texttt{(} f \texttt{)}</math> and <math>\texttt{(} g \texttt{)}</math> under the copies for <math>f</math> and <math>g.</math> At this stage, the columns for <math>\theta f</math> and <math>\theta g</math> are appended almost as afterthoughts, amounting to indicator functions for the sets of ordered triples that make up the functions <math>f</math> and <math>g.</math>

<br>

{| align="center" border="0" style="width:90%"
|+ style="height:25px" | <math>\text{Tables 23-i and 23-ii.} ~~ \text{Thematics of Disjunction and Equality (1)}</math>
| width="50%" |
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:25px" | <math>\text{Table 23-i.} ~~ \text{Disjunction}~ f</math>
|- style="height:25px; background:ghostwhite; width:100%"
| <math>u</math>
| <math>v</math>
| style="border-left:1px solid black; border-right:1px solid black" | <math>f</math>
| <math>x</math>
| <math>\varphi</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |
<math>\begin{matrix}\to\\\to\\\to\\\to\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |  
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\0\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}</math>
|}
| width="50%" |
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:25px" | <math>\text{Table 23-ii.} ~~ \text{Equality}~ g</math>
|- style="height:25px; background:ghostwhite; width:100%"
| <math>u</math>
| <math>v</math>
| style="border-left:1px solid black; border-right:1px solid black" | <math>g</math>
| <math>y</math>
| <math>\gamma</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |
<math>\begin{matrix}\to\\\to\\\to\\\to\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |  
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}</math>
|}
|}

<br>

All the data are now in place to give the truth tables for <math>\theta f</math> and <math>\theta g.</math> All that remains to be done is to permute the rows and change the roles of <math>x</math> and <math>y</math> from dependent to independent variables. In Tables 24-i and 24-ii the rows are arranged in such a way as to put the 3-tuples <math>(u, v, x)</math> and <math>(u, v, y)</math> in binary numerical order, suitable for viewing as the arguments of the maps <math>\theta f = \varphi : [u, v, x] \to \mathbb{B}</math> and <math>\theta g = \gamma : [u, v, y] \to \mathbb{B}.</math> Moreover, the structure of the tables is altered slightly, allowing the now vestigial functions <math>\theta f</math> and <math>\theta g</math> to be passed over without further attention and shifting the heavy vertical bars a notch to the right. In effect, this clinches the fact that the thematic variables <math>x := \check{f}</math> and <math>y := \check{g}</math> are now treated as independent variables.

<br>

{| align="center" border="0" style="width:90%"
|+ style="height:25px" | <math>\text{Tables 24-i and 24-ii.} ~~ \text{Thematics of Disjunction and Equality (2)}</math>
| width="50%" |
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:25px" | <math>\text{Table 24-i.} ~~ \text{Disjunction}~ f</math>
|- style="height:25px; background:ghostwhite; width:100%"
| <math>u</math>
| <math>v</math>
| <math>f</math>
| <math>x</math>
| style="border-left:1px solid black" | <math>\varphi</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <font size="+2"> <br></font><math>\begin{matrix}\to\\~\\\to\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
|}
| width="50%" |
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:25px" | <math>\text{Table 24-ii.} ~~ \text{Equality}~ g</math>
|- style="height:25px; background:ghostwhite; width:100%"
| <math>u</math>
| <math>v</math>
| <math>g</math>
| <math>y</math>
| style="border-left:1px solid black" | <math>\gamma</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" |  <br><math>\begin{matrix}\to\\\to\end{matrix}</math><br> 
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}</math>
|}
|}

<br>

An optional reshuffling of the rows brings additional features of the thematic extensions to light. Leaving the columns in place for the sake of comparison, Tables 25-i and 25-ii sort the rows in a different order, in effect treating <math>x</math> and <math>y</math> as the primary variables in their respective 3-tuples. Regarding the thematic extensions in the form <math>\varphi : [x, u, v] \to \mathbb{B}</math> and <math>\gamma : [y, u, v] \to \mathbb{B}</math> makes it easier to see in this tabular setting a property that was graphically obvious in the venn diagrams above. Specifically, when the thematic variable <math>\check{F}</math> is true then <math>\theta F</math> exhibits the pattern of the original <math>F,</math> and when <math>\check{F}</math> is false then <math>\theta F</math> exhibits the pattern of its negation <math>\texttt{(} F \texttt{)}.</math>

<br>

{| align="center" border="0" style="width:90%"
|+ style="height:25px" | <math>\text{Tables 25-i and 25-ii.} ~~ \text{Thematics of Disjunction and Equality (3)}</math>
| width="50%" |
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:25px" | <math>\text{Table 25-i.} ~~ \text{Disjunction}~ f</math>
|- style="height:25px; background:ghostwhite; width:100%"
| <math>u</math>
| <math>v</math>
| <math>f</math>
| <math>x</math>
| style="border-left:1px solid black" | <math>\varphi</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>{\to}</math><br><font size="+2"> <br> <br> <br></font>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}</math>
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\0\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| style="border-top:1px solid black" | <font size="+2"> <br></font><math>\begin{matrix}\to\\\to\\\to\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}</math>
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\1\end{matrix}</math>
|}
| width="50%" |
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:25px" | <math>\text{Table 25-ii.} ~~ \text{Equality}~ g</math>
|- style="height:25px; background:ghostwhite; width:100%"
| <math>u</math>
| <math>v</math>
| <math>g</math>
| <math>y</math>
| style="border-left:1px solid black" | <math>\gamma</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| style="border-top:1px solid black" |  <br><math>\begin{matrix}\to\\\to\end{matrix}</math><br> 
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}</math>
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}</math>
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}</math>
|}
|}

<br>

Finally, Tables 26-i and 26-ii compare the tacit extensions <math>\boldsymbol\varepsilon : [u, v] \to [u, v, x]</math> and <math>\boldsymbol\varepsilon : [u, v] \to [u, v, y]</math> with the thematic extensions of the same types, as applied to the propositions <math>f</math> and <math>g,</math> respectively.

<br>

{| align="center" border="0" style="width:90%"
|+ style="height:25px" | <math>\text{Tables 26-i and 26-ii.} ~~ \text{Tacit Extension and Thematization}</math>
| width="50%" |
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:25px" | <math>\text{Table 26-i.} ~~ \text{Disjunction}~ f</math>
|- style="height:25px; background:ghostwhite; width:100%"
| <math>u</math>
| <math>v</math>
| <math>x</math>
| style="border-left:1px solid black" | <math>\boldsymbol\varepsilon f</math>
| <math>\theta f</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
|}
| width="50%" |
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:25px" | <math>\text{Table 26-ii.} ~~ \text{Equality}~ g</math>
|- style="height:25px; background:ghostwhite; width:100%"
| <math>u</math>
| <math>v</math>
| <math>y</math>
| style="border-left:1px solid black" | <math>\boldsymbol\varepsilon g</math>
| <math>\theta g</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}1\\1\\0\\0\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}</math>
|}
|}

<br>

Table 27 summarizes the thematic extensions of all propositions on two variables. Column 4 lists the equations of form <math>\texttt{((} \check{f_i}, f_i (u, v) \texttt{))}</math> and Column 5 simplifies these equations into the form of algebraic expressions. As always, <math>{}^{\backprime\backprime} + {}^{\prime\prime}</math> refers to exclusive disjunction and each <math>{}^{\backprime\backprime} \check{f} {}^{\prime\prime}</math> appearing in the last two Columns refers to the corresponding variable name <math>{}^{\backprime\backprime} \check{f_i} {}^{\prime\prime}.</math>

<br>

{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 27.} ~~ \text{Thematization of Bivariate Propositions}</math>
|- style="height:30px; background:ghostwhite"
|  
|  
| <math>{f}</math>
| <math>\theta f</math>
| <math>\theta f</math>
|- style="background:ghostwhite"
| align="right" | <math>u\colon</math>
| <math>1~1~0~0</math>
|  
|  
|  
|- style="background:ghostwhite"
| align="right" | <math>v\colon</math>
| <math>1~0~1~0</math>
|  
|  
|  
|-
| <math>f_{0}</math>
| <math>0~0~0~0</math>
| <math>\texttt{(~)}</math>
| align="left" | <math>\texttt{((} \check{f} \texttt{,~(~)~))}</math>
| align="left" | <math>\check{f} + 1</math>
|-
|
<math>\begin{matrix}
f_{1}
\\[4pt]
f_{2}
\\[4pt]
f_{4}
\\[4pt]
f_{8}
\end{matrix}</math>
|
<math>\begin{matrix}
0~0~0~1
\\[4pt]
0~0~1~0
\\[4pt]
0~1~0~0
\\[4pt]
1~0~0~0
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} u \texttt{)(} v \texttt{)}
\\[4pt]
\texttt{(} u \texttt{)~} v \texttt{~}
\\[4pt]
\texttt{~} u \texttt{~(} v \texttt{)}
\\[4pt]
\texttt{~} u \texttt{~~} v \texttt{~}
\end{matrix}</math>
| align="left" |
<math>\begin{array}{l}
\texttt{((} \check{f} \texttt{,~(u)(v)~))}
\\[4pt]
\texttt{((} \check{f} \texttt{,~(u)~v~~))}
\\[4pt]
\texttt{((} \check{f} \texttt{,~~u~(v)~))}
\\[4pt]
\texttt{((} \check{f} \texttt{,~~u~~v~~))}
\end{array}</math>
| align="left" |
<math>\begin{array}{l}
\check{f} + u + v + uv
\\[4pt]
\check{f} + v + uv + 1
\\[4pt]
\check{f} + u + uv + 1
\\[4pt]
\check{f} + uv + 1
\end{array}</math>
|-
|
<math>\begin{matrix}
f_{3}
\\[4pt]
f_{12}
\end{matrix}</math>
|
<math>\begin{matrix}
0~0~1~1
\\[4pt]
1~1~0~0
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} u \texttt{)}
\\[4pt]
\texttt{~} u \texttt{~}
\end{matrix}</math>
| align="left" |
<math>\begin{array}{l}
\texttt{((} \check{f} \texttt{,~(u)~))}
\\[4pt]
\texttt{((} \check{f} \texttt{,~~u~~))}
\end{array}</math>
| align="left" |
<math>\begin{array}{l}
\check{f} + u
\\[4pt]
\check{f} + u + 1
\end{array}</math>
|-
|
<math>\begin{matrix}
f_{6}
\\[4pt]
f_{9}
\end{matrix}</math>
|
<math>\begin{matrix}
0~1~1~0
\\[4pt]
1~0~0~1
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} u \texttt{,} v \texttt{)}
\\[4pt]
\texttt{((} u \texttt{,} v \texttt{))}
\end{matrix}</math>
| align="left" |
<math>\begin{array}{l}
\texttt{((} \check{f} \texttt{,~~(} u \texttt{,} v \texttt{)~~))}
\\[4pt]
\texttt{((} \check{f} \texttt{,~((} u \texttt{,} v \texttt{))~))}
\end{array}</math>
| align="left" |
<math>\begin{array}{l}
\check{f} + u + v + 1
\\[4pt]
\check{f} + u + v
\end{array}</math>
|-
|
<math>\begin{matrix}
f_{5}
\\[4pt]
f_{10}
\end{matrix}</math>
|
<math>\begin{matrix}
0~1~0~1
\\[4pt]
1~0~1~0
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} v \texttt{)}
\\[4pt]
\texttt{~} v \texttt{~}
\end{matrix}</math>
| align="left" |
<math>\begin{array}{l}
\texttt{((} \check{f} \texttt{,~(} v \texttt{)~))}
\\[4pt]
\texttt{((} \check{f} \texttt{,~~} v \texttt{~~))}
\end{array}</math>
| align="left" |
<math>\begin{array}{l}
\check{f} + v
\\[4pt]
\check{f} + v + 1
\end{array}</math>
|-
|
<math>\begin{matrix}
f_{7}
\\[4pt]
f_{11}
\\[4pt]
f_{13}
\\[4pt]
f_{14}
\end{matrix}</math>
|
<math>\begin{matrix}
0~1~1~1
\\[4pt]
1~0~1~1
\\[4pt]
1~1~0~1
\\[4pt]
1~1~1~0
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(~} u \texttt{~~} v \texttt{~)}
\\[4pt]
\texttt{(~} u \texttt{~(} v \texttt{))}
\\[4pt]
\texttt{((} u \texttt{)~} v \texttt{~)}
\\[4pt]
\texttt{((} u \texttt{)(} v \texttt{))}
\end{matrix}</math>
| align="left" |
<math>\begin{array}{l}
\texttt{((} \check{f} \texttt{,~(~} u \texttt{~~} v \texttt{~)~))}
\\[4pt]
\texttt{((} \check{f} \texttt{,~(~} u \texttt{~(} v \texttt{))~))}
\\[4pt]
\texttt{((} \check{f} \texttt{,~((} u \texttt{)~} v \texttt{~)~))}
\\[4pt]
\texttt{((} \check{f} \texttt{,~((} u \texttt{)(} v \texttt{))~))}
\end{array}</math>
| align="left" |
<math>\begin{array}{l}
\check{f} + uv
\\[4pt]
\check{f} + u + uv
\\[4pt]
\check{f} + v + uv
\\[4pt]
\check{f} + u + v + uv + 1
\end{array}</math>
|-
| <math>f_{15}</math>
| <math>1~1~1~1</math>
| <math>\texttt{((~))}</math>
| align="left" | <math>\texttt{((} \check{f} \texttt{,~((~))~))}</math>
| align="left" | <math>\check{f}</math>
|}

<br>

In order to show what all of the thematic extensions from two dimensions to three dimensions look like in terms of coordinates, Tables 28 and 29 present ordinary truth tables for the functions <math>f_i : \mathbb{B}^2 \to \mathbb{B}</math> and for the corresponding thematizations <math>\theta f_i = \varphi_i : \mathbb{B}^3 \to \mathbb{B}.</math>

<br>

{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 28.} ~~ \text{Propositions on Two Variables}</math>
|- style="height:35px; background:ghostwhite"
| style="width:5%; border-bottom:1px solid black" | <math>u</math>
| style="width:5%; border-bottom:1px solid black" | <math>v</math>
| style="width:5%; border-bottom:1px solid black; border-left:1px solid black" | <math>f_{0}</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{1}</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{2}</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{3}</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{4}</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{5}</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{6}</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{7}</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{8}</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{9}</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{10}</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{11}</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{12}</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{13}</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{14}</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{15}</math>
|-
| <math>0</math>
| <math>0</math>
| style="border-left:1px solid black" |  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|-
| <math>0</math>
| <math>1</math>
| style="border-left:1px solid black" |  
|  
| <math>1</math>
| <math>1</math>
|  
|  
| <math>1</math>
| <math>1</math>
|  
|  
| <math>1</math>
| <math>1</math>
|  
|  
| <math>1</math>
| <math>1</math>
|-
| <math>1</math>
| <math>0</math>
| style="border-left:1px solid black" |  
|  
|  
|  
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
|  
|  
|  
|  
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
|-
| <math>1</math>
| <math>1</math>
| style="border-left:1px solid black" |  
|  
|  
|  
|  
|  
|  
|  
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
|}

<br>

{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 29.} ~~ \text{Thematic Extensions of Bivariate Propositions}</math>
|- style="height:35px; background:ghostwhite"
| style="width:5%; border-bottom:1px solid black" | <math>u</math>
| style="width:5%; border-bottom:1px solid black" | <math>v</math>
| style="width:5%; border-bottom:1px solid black" | <math>\check{f}</math>
| style="width:5%; border-bottom:1px solid black; border-left:1px solid black" | <math>\varphi_{0}</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{1}</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{2}</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{3}</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{4}</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{5}</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{6}</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{7}</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{8}</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{9}</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{10}</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{11}</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{12}</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{13}</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{14}</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{15}</math>
|-
| <math>0</math>
| <math>0</math>
| <math>0</math>
| style="border-left:1px solid black" | <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|  
|-
| <math>0</math>
| <math>0</math>
| <math>1</math>
| style="border-left:1px solid black" |  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|-
| <math>0</math>
| <math>1</math>
| <math>0</math>
| style="border-left:1px solid black" | <math>1</math>
| <math>1</math>
|  
|  
| <math>1</math>
| <math>1</math>
|  
|  
| <math>1</math>
| <math>1</math>
|  
|  
| <math>1</math>
| <math>1</math>
|  
|  
|-
| <math>0</math>
| <math>1</math>
| <math>1</math>
| style="border-left:1px solid black" |  
|  
| <math>1</math>
| <math>1</math>
|  
|  
| <math>1</math>
| <math>1</math>
|  
|  
| <math>1</math>
| <math>1</math>
|  
|  
| <math>1</math>
| <math>1</math>
|-
| <math>1</math>
| <math>0</math>
| <math>0</math>
| style="border-left:1px solid black" | <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
|  
|  
|  
|  
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
|  
|  
|  
|  
|-
| <math>1</math>
| <math>0</math>
| <math>1</math>
| style="border-left:1px solid black" |  
|  
|  
|  
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
|  
|  
|  
|  
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
|-
| <math>1</math>
| <math>1</math>
| <math>0</math>
| style="border-left:1px solid black" | <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
|  
|  
|  
|  
|  
|  
|  
|  
|-
| <math>1</math>
| <math>1</math>
| <math>1</math>
| style="border-left:1px solid black" |  
|  
|  
|  
|  
|  
|  
|  
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
|}

<br>

===Propositional Transformations===

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
If only the word ‘artificial’ were associated with the idea of ''art'', or expert skill gained through voluntary apprenticeship (instead of suggesting the factitious and unreal), we might say that ''logical'' refers to artificial thought.
| width="4%" |  
|-
| align="right" colspan="3" | — John Dewey, ''How We Think'', [Dew, 56–57]
|}

Next we develop a set of concepts for dealing with transformations between universes of discourse.  As a general consideration the source and target universes of a transformation are allowed to be different but when we turn to dynamic systems we focus on the special case of transformations mapping universes into themselves.  In effect, we regard each transformation as one of the possible state transitions in a discrete dynamical process and place it among the myriad ways a universe of discourse may change, and by that change turn into itself.

====Alias and Alibi Transformations====

There are customarily two modes of understanding a transformation, at least, when we try to interpret its relevance to something in reality. A transformation always refers to a changing prospect, to say it in a unified but equivocal way, but this can be taken to mean either a subjective change in the interpreting observer's point of view or an objective change in the systematic subject of discussion. In practice these variant uses of the transformation concept are distinguished in the following terms:

# A ''perspectival'' or ''alias'' transformation refers to a shift in perspective or a change in language that takes place in the observer's frame of reference.
# A ''transitional'' or ''alibi'' transformation refers to a change of position or an alteration of state that occurs in the object system as it falls under study.

(For a recent discussion of the alias vs. alibi issue, as it relates to linear transformations in vector spaces and to other issues of an algebraic nature, see [MaB, 256, 582-4].)

Naturally, when we are concerned with the dynamical properties of a system, the transitional aspect of transformation is the factor that comes to the fore, and this involves us in contemplating all of the ways of changing a universe into itself while remaining under the rule of established dynamical laws. In the prospective application to dynamic systems, and to neural networks viewed in this light, our interest lies chiefly with the transformations of a state space into itself that constitute the state transitions of a discrete dynamic process. Nevertheless, many important properties of these transformations, and some constructions that we need to see most clearly, are independent of the transitional interpretation and are likely to be confounded with irrelevant features if presented first and only in that association.

In addition, and in partial contrast, intelligent systems are exactly that species of dynamic agents that have the capacity to have a point of view, and we cannot do justice to their peculiar properties without examining their ability to form and transform their own frames of reference in exposure to the elements of their own experience. In this setting, the perspectival aspect of transformation is the facet that shines most brightly, perhaps too often leaving us fascinated with mere glimmerings of its actual potential. It needs to be emphasized that nothing of the ordinary sort needs be moved in carrying out a transformation under the alias interpretation, that it may only involve a change in the forms of address, an amendment of the terms which are customed to approach and fashioned to describe the very same things in the very same world. But again, working within a discipline of realistic computation, we know how formidably complex and resource-consuming such transformations of perspective can be to implement in practice, much less to endow in the self-governed form of a nascently intelligent dynamical system.

====Transformations of General Type====

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
<i>Es ist passiert</i>, “it just sort of happened”, people said there when other people in other places thought heaven knows what had occurred. It was a peculiar phrase, not known in this sense to the Germans and with no equivalent in other languages, the very breath of it transforming facts and the bludgeonings of fate into something light as eiderdown, as thought itself.
| width="4%" |  
|-
| align="right" colspan="3" | — Robert Musil, <i>The Man Without Qualities</i>, [Mus, 34]
|}

Consider the situation illustrated in Figure 30, where the alphabets <math>\mathcal{U} = \{ u, v \}</math> and <math>\mathcal{X} = \{ x, y, z \}</math> are used to label basic features in two different logical universes, <math>U^\bullet = [u, v]</math> and <math>X^\bullet = [x, y, z].</math>

{| align="center" border="0" cellpadding="10"
|
<pre>
o-------------------------------------------------------o
| U |
| |
| o-----------o o-----------o |
| / \ / \ |
| / o \ |
| / / \ \ |
| / / \ \ |
| o o o o |
| | | | | |
| | u | | v | |
| | | | | |
| o o o o |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| o-----------o o-----------o |
| |
| |
o---------------------------o---------------------------o
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
o-------------------------o o-------------------------o o-------------------------o
| U | | U | | U |
| o---o o---o | | o---o o---o | | o---o o---o |
| / \ / \ | | / \ / \ | | / \ / \ |
| / o \ | | / o \ | | / o \ |
| / / \ \ | | / / \ \ | | / / \ \ |
| o o o o | | o o o o | | o o o o |
| | u | | v | | | | u | | v | | | | u | | v | |
| o o o o | | o o o o | | o o o o |
| \ \ / / | | \ \ / / | | \ \ / / |
| \ o / | | \ o / | | \ o / |
| \ / \ / | | \ / \ / | | \ / \ / |
| o---o o---o | | o---o o---o | | o---o o---o |
| | | | | |
o-------------------------o o-------------------------o o-------------------------o
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ g | \ f / | h /
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ o----------|-----------\-----/-----------|----------o /
\ | X | \ / | | /
\ | | \ / | | /
\ | | o-----o-----o | | /
\| | / \ | |/
\ | / \ | /
|\ | / \ | /|
| \ | / \ | / |
| \ | / \ | / |
| \ | o x o | / |
| \ | | | | / |
| \ | | | | / |
| \ | | | | / |
| \ | | | | / |
| \ | | | | / |
| \| | | |/ |
| o--o--------o o--------o--o |
| / \ \ / / \ |
| / \ \ / / \ |
| / \ o / \ |
| / \ / \ / \ |
| / \ / \ / \ |
| o o--o-----o--o o |
| | | | | |
| | | | | |
| | | | | |
| | y | | z | |
| | | | | |
| | | | | |
| o o o o |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| \ / \ / |
| o-----------o o-----------o |
| |
| |
o---------------------------------------------------o
\ /
\ /
\ /
\ /
\ /
\ p , q /
\ /
\ /
\ /
\ /
\ /
\ /
\ /
o

Figure 30. Generic Frame of a Logical Transformation
</pre>
|}

Enter the picture, as we usually do, in the middle of things, with features like <math>x, y , z</math> which present themselves to be simple enough in their own right and which form a satisfactory, if temporary foundation to provide a basis for discussion.

In that universe and on those terms we find expression for various propositions and questions of principal interest to ourselves, as indicated by the maps <math>p, q : X \to \mathbb{B}.</math>

Then we discover the simple features <math>\{ x, y, z \}</math> are really more complex than we thought at first, and it becomes useful to regard them as functions <math>\{ f, g, h \}</math> of other features <math>\{ u, v \}</math> we locate in a preface to our original discourse, or suppose as topics of a preliminary universe of discourse <math>U^\bullet = [u, v].</math>

It may happen that those late‑blooming but pre‑ambling features are found to lie closer, in a sense it may be our job to determine, to the central nature of the situation of interest, in which case they earn our regard as being more fundamental, but those functions and features are required only to afford a critical stance on the universe of discourse or an alternate perspective on the nature of things in order to be preserved as useful.

A particular transformation <math>F : [u, v] \to [x, y, z]</math> may be expressed by a system of equations, as shown below.  Here, <math>F</math> is defined by its component maps <math>F = (F_1, F_2, F_3) = (f, g, h),</math> where each component map in <math>\{ f, g, h \}</math> is a proposition of type <math>\mathbb{B}^n \to \mathbb{B}^1.</math>

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:50%"
|
<math>\begin{matrix}
x & = & f(u, v)
\\[10pt]
y & = & g(u, v)
\\[10pt]
z & = & h(u, v)
\end{matrix}</math>
|}

<br>

Regarded as a logical statement, this system of equations expresses a relation between a collection of freely chosen propositions <math>\{ f, g, h \}</math> in one universe of discourse and the special collection of simple propositions <math>\{ x, y, z \}</math> on which is founded another universe of discourse.  Growing familiarity with a particular transformation of discourse, and the desire to achieve a ready understanding of its implications, requires that we be able to convert this information about generals and simples into information about all the main subtypes of propositions, including the linear and singular propositions.

===Analytic Expansions : Operators and Functors===

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
Consider what effects that might <i>conceivably</i> have practical bearings you <i>conceive</i> the objects of your <i>conception</i> to have. <br> Then, your <i>conception</i> of those effects is the whole of your <i>conception</i> of the object.
| width="4%" |  
|-
| align="right" colspan="3" | — C.S. Peirce, “The Maxim of Pragmatism”, CP 5.438
|}

Given the barest idea of a logical transformation, as suggested by the sketch in Figure 30, and having conceptualized the universe of discourse, with all of its points and propositions, as a beginning object of discussion, we are ready to enter the next phase of our investigation.

====Operators on Propositions and Transformations====

The next step is naturally inclined toward objects of the next higher order, namely, with operators that take in argument lists of logical transformations and that give back specified types of logical transformations as their results. For our present aims, we do not need to consider the most general class of such operators, nor any one of them for its own sake. Rather, we are interested in the special sorts of operators that arise in the study and analysis of logical transformations. Figuratively speaking, these operators serve as instruments for the live tomography (and hopefully not the vivisection) of the forms of change under view. Beyond that, they open up ways to implement the changes of view that we need to grasp all the variations on a transformational theme, or to appreciate enough of its significant features to “get the drift” of the change occurring, to form a passing acquaintance or a synthetic comprehension of its general character and disposition.

The simplest type of operator is one that takes a single transformation as an argument and returns a single transformation as a result, and most of the operators explicitly considered in our discussion will be of this kind. Figure 31 illustrates the typical situation.

{| align="center" border="0" cellpadding="20"
|
<pre>
o---------------------------------------o
| |
| |
| U% F X% |
| o------------------>o |
| | | |
| | | |
| | | |
| | | |
| !W! | | !W! |
| | | |
| | | |
| | | |
| v v |
| o------------------>o |
| !W!U% !W!F !W!X% |
| |
| |
o---------------------------------------o
Figure 31. Operator Diagram (1)
</pre>
|}

In this Figure <math>{}^{\backprime\backprime} \mathsf{W} {}^{\prime\prime}</math> stands for a generic operator <math>\mathsf{W},</math> in this case one that takes a logical transformation <math>F</math> of type <math>(U^\bullet \to X^\bullet)</math> into a logical transformation <math>\mathsf{W}F</math> of type <math>(\mathsf{W}U^\bullet \to \mathsf{W}X^\bullet).</math> Thus, the operator <math>\mathsf{W}</math> must be viewed as making assignments for both families of objects we have previously considered, that is, for universes of discourse like <math>{U^\bullet}</math> and <math>{X^\bullet}</math> and for logical transformations like <math>F.</math>

'''Note.''' Strictly speaking, an operator like <math>\mathsf{W}</math> works between two whole categories of universes and transformations, which we call the ''source'' and the ''target'' categories of <math>\mathsf{W}.</math> Given this setting, <math>\mathsf{W}</math> specifies for each universe <math>U^\bullet</math> in its source category a definite universe <math>\mathsf{W}U^\bullet</math> in its target category, and to each transformation <math>F</math> in its source category it assigns a unique transformation <math>\mathsf{W}F</math> in its target category. Naturally, this only works if <math>\mathsf{W}</math> takes the source <math>U^\bullet</math> and the target <math>X^\bullet</math> of the map <math>F</math> over to the source <math>\mathsf{W}U^\bullet</math> and the target <math>\mathsf{W}X^\bullet</math> of the map <math>\mathsf{W}F.</math> With luck or care enough, we can avoid ever having to put anything like that in words again, letting diagrams do the work. In the situations of present concern we are usually focused on a single transformation <math>F,</math> and thus we can take it for granted that the assignment of universes under <math>\mathsf{W}</math> is defined appropriately at the source and target ends of <math>F.</math> It is not always the case, though, that we need to use the particular names (like <math>{}^{\backprime\backprime} \mathsf{W}U^\bullet {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} \mathsf{W}X^\bullet {}^{\prime\prime}</math>) that <math>\mathsf{W}</math> assigns by default to its operative image universes. In most contexts we will usually have a prior acquaintance with these universes under other names and it is necessary only that we can tell from the information associated with an operator <math>\mathsf{W}</math> what universes they are.

In Figure 31 the maps <math>F</math> and <math>\mathsf{W}F</math> are displayed horizontally, the way one normally orients functional arrows in a written text, and <math>\mathsf{W}</math> rolls the map <math>F</math> downward into the images that are associated with <math>\mathsf{W}F.</math> In Figure 32 the same information is redrawn so that the maps <math>F</math> and <math>\mathsf{W}F</math> flow down the page, and <math>\mathsf{W}</math> unfurls the map <math>F</math> rightward into domains that are the eminent purview of <math>\mathsf{W}F.</math>

{| align="center" border="0" cellpadding="20"
|
<pre>
o---------------------------------------o
| |
| |
| U% !W! !W!U% |
| o------------------>o |
| | | |
| | | |
| | | |
| | | |
| F | | !W!F |
| | | |
| | | |
| | | |
| v v |
| o------------------>o |
| X% !W! !W!X% |
| |
| |
o---------------------------------------o
Figure 32. Operator Diagram (2)
</pre>
|}

The latter arrangement, as exhibited in Figure 32, is more congruent with the thinking about operators that we shall do in the rest of this discussion, since all logical transformations from here on out will be pictured vertically, after the fashion of Figure 30.

====Differential Analysis of Propositions and Transformations====

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" | The resultant metaphysical problem now is this: ''Does the man go round the squirrel or not?''
| width="4%" |  
|-
| align="right" colspan="3" | — William James, ''Pragmatism'', [Jam, 43]
|}

The approach to the differential analysis of logical propositions and transformations of discourse to be pursued here is carried out in terms of particular operators <math>\mathsf{W}</math> that act on propositions <math>F</math> or on transformations <math>F</math> to yield the corresponding operator maps <math>\mathsf{W}F.</math> The operator results then become the subject of a series of further stages of analysis, which take them apart into their propositional components, rendering them as a set of purely logical constituents. After this is done, all the parts are then re-integrated to reconstruct the original object in the light of a more complete understanding, at least in ways that enable one to appreciate certain aspects of it with fresh insight.

* '''Remark on Strategy.''' At this point we run into a set of conceptual difficulties that force us to make a strategic choice in how we proceed. Part of the problem can be remedied by extending our discussion of tacit extensions to the transformational context. But the troubles that remain are much more obstinate and lead us to try two different types of solution. The approach that we develop first makes use of a variant type of extension operator, the ''trope extension'', to be defined below. This method is more conservative and requires less preparation, but has features which make it seem unsatisfactory in the long run. A more radical approach, but one with a better hope of long term success, makes use of the notion of ''contingency spaces''. These are an even more generous type of extended universe than the kind we currently use, but are defined subject to certain internal constraints. The extra work needed to set up this method forces us to put it off to a later stage. However, as a compromise, and to prepare the ground for the next pass, we call attention to the various conceptual difficulties as they arise along the way and try to give an honest estimate of how well our first approach deals with them.

We now describe in general terms the particular operators that are instrumental to this form of analysis. The main series of operators all have the form:

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"
|
<math>\begin{matrix}
\mathsf{W}
& : &
( U^\bullet \to X^\bullet )
& \to &
( \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet )
\end{matrix}</math>
|}

<br>

If we assume that the source universe <math>U^\bullet</math> and the target universe <math>X^\bullet</math> have finite dimensions <math>n</math> and <math>k,</math> respectively, then each operator <math>\mathsf{W}</math> is encompassed by the same abstract type:

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"
|
<math>\begin{matrix}
\mathsf{W}
& : &
( [\mathbb{B}^n] \to [\mathbb{B}^k] )
& \to &
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k \times \mathbb{D}^k] )
\end{matrix}</math>
|}

<br>

Since the range features of the operator result <math>\mathsf{W}F : [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k \times \mathbb{D}^k]</math> can be sorted by their ordinary versus differential qualities and the component maps can be examined independently, the complete operator <math>\mathsf{W}</math> can be separated accordingly into two components, in the form <math>\mathsf{W} = (\boldsymbol\varepsilon, \mathrm{W}).</math> Given a fixed context of source and target universes, <math>\boldsymbol\varepsilon</math> is always the same type of operator, a multiple component version of the tacit extension operators that were described earlier. In this context <math>\boldsymbol\varepsilon</math> has the form:

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"
|
<math>\begin{array}{lccccc}
\text{Concrete type}
& \boldsymbol\varepsilon
& : &
( U^\bullet \to X^\bullet )
& \to &
( \mathrm{E}U^\bullet \to X^\bullet )
\\[10pt]
\text{Abstract type}
& \boldsymbol\varepsilon
& : &
( [\mathbb{B}^n] \to [\mathbb{B}^k] )
& \to &
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k] )
\end{array}</math>
|}

<br>

On the other hand, the operator <math>\mathrm{W}</math> is specific to each <math>\mathsf{W}.</math> In this context <math>\mathrm{W}</math> always has the form:

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"
|
<math>\begin{array}{lccccc}
\text{Concrete type}
& W
& : &
( U^\bullet \to X^\bullet )
& \to &
( \mathrm{E}U^\bullet \to \mathrm{d}X^\bullet )
\\[10pt]
\text{Abstract type}
& W
& : &
( [\mathbb{B}^n] \to [\mathbb{B}^k] )
& \to &
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{D}^k] )
\end{array}</math>
|}

<br>

In the types just assigned to <math>\boldsymbol\varepsilon</math> and <math>\mathrm{W}</math> and by implication to their results <math>\boldsymbol\varepsilon F</math> and <math>\mathrm{W}F,</math> we have listed the most restrictive ranges defined for them rather than the more expansive target spaces that subsume these ranges. When there is need to recognize both, we may use type indications like the following:

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"
|
<math>\begin{matrix}
\boldsymbol\varepsilon F
& : &
( \mathrm{E}U^\bullet \to X^\bullet \subseteq \mathrm{E}X^\bullet )
& \cong &
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k] \subseteq [\mathbb{B}^k \times \mathbb{D}^k] )
\\[10pt]
WF
& : &
( \mathrm{E}U^\bullet \to \mathrm{d}X^\bullet \subseteq \mathrm{E}X^\bullet )
& \cong &
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{D}^k] \subseteq [\mathbb{B}^k \times \mathbb{D}^k] )
\end{matrix}</math>
|}

<br>

Hopefully, though, a general appreciation of these subsumptions will prevent us from having to make such declarations more often than absolutely necessary.

In giving names to these operators we try to preserve as much of the traditional nomenclature and as many of the classical associations as possible. The chief difficulty in doing this is occasioned by the distinction between the “sans serif” operators <math>\mathsf{W}</math> and their “serified” components <math>\mathrm{W},</math> which forces us to find two distinct but parallel sets of terminology. Here is a plan to that purpose. First, the component operators <math>\mathrm{W}</math> are named by analogy with the corresponding operators in the classical difference calculus. Next, the complete operators <math>\mathsf{W} = (\boldsymbol\varepsilon, \mathrm{W})</math> are assigned titles according to their roles in a geometric or trigonometric allegory, if only to ensure that the tangent functor, that belongs to this family and whose exposition we are still working toward, comes out fit with its customary name. Finally, the operator results <math>\mathsf{W}F</math> and <math>\mathrm{W}F</math> can be fixed in our frame of reference by tethering the operative adjective for <math>\mathsf{W}</math> or <math>\mathrm{W}</math> to the anchoring epithet “map”, in conformity with an already standard practice.

=====The Secant Operator : E=====

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
Mr. Peirce, after pointing out that our beliefs are really rules for action, said that, to develop a thought's meaning, we need only determine what conduct it is fitted to produce: that conduct is for us its sole significance.
| width="4%" |  
|-
| align="right" colspan="3" | — William James, ''Pragmatism'', [Jam, 46]
|}

Figures 33-i and 33-ii depict two stages in the form of analysis that will be applied to transformations throughout the remainder of this study. From now on our interest is staked on an operator denoted <math>{}^{\backprime\backprime} \mathsf{E} {}^{\prime\prime},</math> which receives the principal investment of analytic attention, and on the constituent parts of <math>\mathsf{E},</math> which derive their shares of significance as developed by the analysis. In the sequel, we refer to <math>\mathsf{E}</math> as the ''secant operator'', taking it for granted that a context has been chosen that defines its type. The secant operator has the component description <math>\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}),</math> and its active ingredient <math>\mathrm{E}</math> is known as the ''enlargement operator''. (Here, we name <math>\mathrm{E}</math> after the literal ancestor of the shift operator in the calculus of finite differences, defined so that <math>\mathrm{E}f(x) = f(x+1)</math> for any suitable function <math>f,</math> though of course the logical analogue that we take up here must have a rather different definition.)

{| align="center" border="0" cellpadding="10"
|
<pre>
U% $E$ $E$U% $E$U% $E$U%
o------------------>o============o============o
| | | |
| | | |
| | | |
| | | |
F | | $E$F = | $d$^0.F + | $r$^0.F
| | | |
| | | |
| | | |
v v v v
o------------------>o============o============o
X% $E$ $E$X% $E$X% $E$X%

Figure 33-i. Analytic Diagram (1)
</pre>
|}

{| align="center" border="0" cellpadding="10"
|
<pre>
U% $E$ $E$U% $E$U% $E$U% $E$U%
o------------------>o============o============o============o
| | | | |
| | | | |
| | | | |
| | | | |
F | | $E$F = | $d$^0.F + | $d$^1.F + | $r$^1.F
| | | | |
| | | | |
| | | | |
v v v v v
o------------------>o============o============o============o
X% $E$ $E$X% $E$X% $E$X% $E$X%

Figure 33-ii. Analytic Diagram (2)
</pre>
|}

In its action on universes <math>\mathsf{E}</math> yields the same result as <math>\mathrm{E},</math> a fact that can be expressed in equational form by writing <math>\mathsf{E}U^\bullet = \mathrm{E}U^\bullet</math> for any universe <math>U^\bullet.</math> Notice that the extended universes across the top and bottom of the diagram are indicated to be strictly identical, rather than requiring a corresponding decomposition for them. In a certain sense, the functional parts of <math>\mathsf{E}F</math> are partitioned into separate contexts that have to be re-integrated again, but the best image to use is that of making transparent copies of each universe and then overlapping their functional contents once more at the conclusion of the analysis, as suggested by the graphic conventions that are used at the top of Figure 30.

Acting on a transformation <math>F</math> from universe <math>U^\bullet</math> to universe <math>X^\bullet,</math> the operator <math>\mathsf{E}</math> determines a transformation <math>\mathsf{E}F</math> from <math>\mathsf{E}U^\bullet</math> to <math>\mathsf{E}X^\bullet.</math> The map <math>\mathsf{E}F</math> forms the main body of evidence to be investigated in performing a differential analysis of <math>F.</math> Because we shall frequently be focusing on small pieces of this map for considerable lengths of time, and consequently lose sight of the “big picture”, it is critically important to emphasize that the map <math>\mathsf{E}F</math> is a transformation that determines a relation from one extended universe into another. This means that we should not be satisfied with our understanding of a transformation <math>F</math> until we can lay out the full “parts diagram” of <math>\mathsf{E}F</math> along the lines of the generic frame in Figure 30.

Working within the confines of propositional calculus, it is possible to give an elementary definition of <math>\mathsf{E}F</math> by means of a system of propositional equations, as we now describe.

Given a transformation

{| align="center" cellpadding="6" width="90%"
| <math>F = (F_1, \ldots, F_k) : \mathbb{B}^n \to \mathbb{B}^k</math>
|}

of concrete type

{| align="center" cellpadding="6" width="90%"
| <math>F : [u_1, \ldots, u_n] \to [x_1, \ldots, x_k],</math>
|}

the transformation

{| align="center" cellpadding="6" width="90%"
| <math>\mathsf{E}F = (F_1, \ldots, F_k, \mathrm{E}F_1, \ldots, \mathrm{E}F_k) : \mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B}^k \times \mathbb{D}^k</math>
|}

of concrete type

{| align="center" cellpadding="6" width="90%"
| <math>\mathsf{E}F : [u_1, \dots, u_n, \mathrm{d}u_1, \dots, \mathrm{d}u_n] \to [x_1, \ldots, x_k, \mathrm{d}x_1, \ldots, \mathrm{d}x_k]</math>
|}

is defined by means of the following system of logical equations:

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"
|
<math>\begin{matrix}
x_1
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)
& = & F_1 (u_1, \ldots, u_n)
\\[4pt]
\cdots && \cdots && \cdots
\\[4pt]
x_k
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)
& = & F_k (u_1, \ldots, u_n)
\\[16pt]
\mathrm{d}x_1
& = & \mathrm{E}F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)
& = & F_1 (u_1 + \mathrm{d}u_1, \ldots, u_n + \mathrm{d}u_n)
\\[4pt]
\cdots && \cdots && \cdots
\\[4pt]
\mathrm{d}x_k
& = & \mathrm{E}F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)
& = & F_k (u_1 + \mathrm{d}u_1, \ldots, u_n + \mathrm{d}u_n)
\end{matrix}</math>
|}

<br>

It is important to note that this system of equations can be read as a conjunction of equational propositions, in effect, as a single proposition in the universe of discourse generated by all the named variables. Specifically, this is the universe of discourse over <math>2(n+k)</math> variables denoted by:

{| align="center" cellpadding="6" width="90%"
|
<math>\begin{matrix}
\mathrm{E}[\mathcal{U} \cup \mathcal{X}]
& = &
[u_1, \ldots, u_n, ~ x_1, \ldots, x_k, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n, ~ \mathrm{d}x_1, \ldots, \mathrm{d}x_k].
\end{matrix}</math>
|}

In this light, it should be clear that the system of equations defining <math>\mathsf{E}F</math> embodies, in a higher rank and differentially extended version, an analogy with the process of thematization that we treated earlier for propositions of type <math>F : \mathbb{B}^n \to \mathbb{B}.</math>

The entire collection of constraints that is represented in the above system of equations may be abbreviated by writing <math>\mathsf{E}F = (\boldsymbol\varepsilon F, \mathrm{E}F),</math> for any map <math>F.</math> This is tantamount to regarding <math>\mathsf{E}</math> as a complex operator, <math>\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}),</math> with a form of application that distributes each component of the operator to work on each component of the operand, as follows:

{| align="center" cellpadding="6" width="90%"
|
<math>\begin{matrix}
\mathsf{E}F
& = &
(\boldsymbol\varepsilon, \mathrm{E})F
& = &
(\boldsymbol\varepsilon F, \mathrm{E}F)
& = &
(\boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k, ~ \mathrm{E}F_1, \ldots, \mathrm{E}F_k).
\end{matrix}</math>
|}

Quite a lot of “thematic infrastructure” or interpretive information is being swept under the rug in the use of such abbreviations. When confusion arises about the meaning of such constructions, one always has recourse to the defining system of equations, in its totality a purely propositional expression. This means that the parenthesized argument lists, that were used in this context to build an image of multi-component transformations, should not be expected to determine a well-defined product in themselves but only to serve as reminders of the prior thematic decisions (choices of variable names, etc.) that have to be made in order to determine one. Accordingly, the argument list notation can be regarded as a kind of ''thematic frame'', an interpretive storage device that preserves the proper associations of concrete logical features between the extended universes at the source and target of <math>\mathsf{E}F.</math>

The generic notations <math>\mathsf{d}^0\!F, \mathsf{d}^1\!F, \ldots, \mathsf{d}^m\!F</math> in Figure 33 refer to the increasing orders of differentials that are extracted in the course of analyzing <math>F.</math> When the analysis is halted at a partial stage of development, notations like <math>\mathsf{r}^0\!F, \mathsf{r}^1\!F, \ldots, \mathsf{r}^m\!F</math> may be used to summarize the contributions to <math>\mathsf{E}F</math> that remain to be analyzed. The Figure illustrates a convention that makes <math>\mathsf{r}^m\!F,</math> in effect, the sum of all differentials of order strictly greater than <math>m.</math>

We next discuss the operators that figure into this form of analysis, describing their effects on transformations. In simplified or specialized contexts these operators tend to take on a variety of different names and notations, some of whose number we introduce along the way.

=====The Radius Operator : e=====

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
And the tangible fact at the root of all our thought-distinctions, however subtle, is that there is no one of them so fine as to consist in anything but a possible difference of practice.
| width="4%" |  
|-
| align="right" colspan="3" | — William James, ''Pragmatism'', [Jam, 46]
|}

The operator identified as <math>\mathrm{d}^0</math> in the analytic diagram (Figure 33) has the sole purpose of creating a proxy for <math>F</math> in the appropriately extended context. Construed in terms of its broadest components, <math>\mathrm{d}^0</math> is equivalent to the doubly tacit extension operator <math>(\boldsymbol\varepsilon, \boldsymbol\varepsilon),</math> in recognition of which let us redub it as <math>{}^{\backprime\backprime} \mathsf{e} {}^{\prime\prime}.</math> Pursuing a geometric analogy, we may refer to <math>\mathsf{e} =(\boldsymbol\varepsilon, \boldsymbol\varepsilon) = \mathrm{d}^0</math> as the ''radius operator''. The operation intended by all of these forms is defined by the following equation:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{lll}
\mathsf{e}F
& = &
(\boldsymbol\varepsilon, \boldsymbol\varepsilon)F
\\[4pt]
& = &
(\boldsymbol\varepsilon F, ~ \boldsymbol\varepsilon F)
\\[4pt]
& = &
(\boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k, ~ \boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k).
\end{array}</math>
|}

which is tantamount to the system of equations below.

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"
|
<math>\begin{matrix}
x_1
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)
& = & F_1 (u_1, \ldots, u_n)
\\[4pt]
\cdots && \cdots && \cdots
\\[4pt]
x_k
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)
& = & F_k (u_1, \ldots, u_n)
\\[16pt]
\mathrm{d}x_1
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)
& = & F_1 (u_1, \ldots, u_n)
\\[4pt]
\cdots && \cdots && \cdots
\\[4pt]
\mathrm{d}x_k
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)
& = & F_k (u_1, \ldots, u_n)
\end{matrix}</math>
|}

<br>

=====The Phantom of the Operators : η=====

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
I was wondering what the reason could be, when I myself raised my head and everything within me seemed drawn towards the Unseen, ''which was playing the most perfect music''!
| width="4%" |  
|-
| align="right" colspan="3" | — Gaston Leroux, ''The Phantom of the Opera'', [Ler, 81]
|}

We now describe an operator whose persistent but elusive action behind the scenes, whose slightly twisted and ambivalent character, and whose fugitive disposition, caught somewhere in flight between the arrantly negative and the positive but errant intent, has cost us some painstaking trouble to detect. In the end we shall place it among the other extensions and projections, as a shade among shadows, of muted tones and motley hue, that adumbrates its own thematic frame and paradoxically lights the way toward a whole new spectrum of values.

Given a transformation <math>F : [u_1, \ldots, u_n] \to [x_1, \dots, x_k],</math> we often have call to consider a family of related transformations, all having the form:

{| align="center" cellpadding="8" width="90%"
| <math>F^\dagger : [u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n] \to [\mathrm{d}x_1, \dots, \mathrm{d}x_k].</math>
|}

The operator <math>\eta</math> is introduced to deal with the simplest one of these maps:

{| align="center" cellpadding="8" width="90%"
| <math>\eta F : [u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n] \to [\mathrm{d}x_1, \ldots \mathrm{d}x_k],</math>
|}

which is defined by the following equations:

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"
|
<math>\begin{matrix}
\mathrm{d}x_1
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n)
& = & F_1 (u_1, \ldots, u_n)
\\[4pt]
\cdots && \cdots && \cdots
\\[4pt]
\mathrm{d}x_k
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n)
& = & F_k (u_1, \ldots, u_n)
\end{matrix}</math>
|}

<br>

In effect, the operator <math>\eta</math> is nothing but the stand-alone version of a procedure that is otherwise invoked subordinate to the work of the radius operator <math>\mathsf{e}.</math> Operating independently, <math>\eta</math> achieves precisely the same results that the second <math>\boldsymbol\varepsilon</math> in <math>(\boldsymbol\varepsilon, \boldsymbol\varepsilon)</math> accomplishes by working within the context of its ordered pair thematic frame. From this point on, because the use of <math>\boldsymbol\varepsilon</math> and <math>\eta</math> in this setting combines the aims of both the tacit and the thematic extensions, and because <math>\eta</math> reflects in regard to <math>\boldsymbol\varepsilon</math> little more than the application of a differential twist, a mere turn of phrase, we refer to <math>\eta</math> as the ''trope extension'' operator.

=====The Chord Operator : D=====

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
What difference would it practically make to any one if this notion rather than that notion were true?  If no practical difference whatever can be traced, then the alternatives mean practically the same thing, and all dispute is idle.
| width="4%" |  
|-
| align="right" colspan="3" | — William James, ''Pragmatism'', [Jam, 45]
|}

We come to an operator always immanent in this form of analysis, and remaining implicitly present in the entire proceeding.  It may appear once as a record:  a relic or revenant reprising the reminders of an earlier stage of development.  Or it may appear always as a resource:  a reserve or redoubt caching in advance an echo of what remains to be played out, cleared up, and requited in full at a future stage.  And all of this remains true whether or not we recall the key at any time, and whether or not the subtending theme is recited explicitly at any stage of play.

This is the operator that is referred to as <math>\mathsf{r}^0</math> in the initial stage of analysis (Figure 33-i) and that is expanded as <math>\mathsf{d}^1 + \mathsf{r}^1</math> in the subsequent step (Figure 33-ii). In congruence, but not quite harmony with our allusions of analogy that are not quite geometry, we call this the ''chord operator'' and denote it <math>\mathsf{D}.</math> In the more casual terms that are here introduced, <math>\mathsf{D}</math> is defined as the remainder of <math>\mathsf{E}</math> and <math>\mathsf{e}</math> and it assigns a due measure to each undertone of accord or discord that is struck between the note of enterprise <math>\mathsf{E}</math> and the bar of exigency <math>\mathsf{e}.</math>

The tension between these counterposed notions, in balance transient but regular in stridence, may be refracted along familiar lines, though never by any such fraction resolved. In this style we write <math>\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}),</math> calling <math>\mathrm{D}</math> the ''difference operator'' and noting that it plays a role in this realm of mutable and diverse discourse that is analogous to the part taken by the discrete difference operator in the ordinary difference calculus. Finally, we should note that the chord <math>\mathsf{D}</math> is not one that need be lost at any stage of development. At the <math>m^\text{th}</math> stage of play it can always be reconstituted in the following form:

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"
|
<math>\begin{array}{lll}
\mathsf{D}
& = & \mathsf{E} - \mathsf{e}
\\[6pt]
& = & \mathsf{r}^0
\\[6pt]
& = & \mathsf{d}^1 + \mathsf{r}^1
\\[6pt]
& = & \mathsf{d}^1 + \ldots + \mathsf{d}^m + \mathsf{r}^m
\\[6pt]
& = & \displaystyle \sum_{i=1}^m \mathsf{d}^i + \mathsf{r}^m
\end{array}</math>
|}

<br>

=====The Tangent Operator : T=====

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
They take part in scenes of whose significance they have no inkling. They are merely tangent to curves of history the beginnings and ends and forms of which pass wholly beyond their ken. So we are tangent to the wider life of things.
| width="4%" |  
|-
| align="right" colspan="3" | — William James, ''Pragmatism'', [Jam, 300]
|}

The operator tagged as <math>\mathsf{d}^1</math> in the analytic diagram (Figure 33) is called the ''tangent operator'' and is usually denoted in this text as <math>\mathsf{d}</math> or <math>\mathsf{T}.</math> Because it has the properties required to qualify as a functor, namely, preserving the identity element of the composition operation and the articulated form of every composition of transformations, it also earns the title of a ''tangent functor''. According to the custom adopted here, we dissect it as <math>\mathsf{T} = \mathsf{d} = (\boldsymbol\varepsilon, \mathrm{d}),</math> where <math>\mathrm{d}</math> is the operator that yields the first order differential <math>\mathrm{d}F</math> when applied to a transformation <math>F,</math> and whose name is legion.

Figure 34 illustrates a stage of analysis where we ignore everything but the tangent functor <math>\mathsf{T}</math> and attend to it chiefly as it bears on the first order differential <math>\mathrm{d}F</math> in the analytic expansion of <math>F.</math> In this situation we often refer to the extended universes <math>\mathrm{E}U^\bullet</math> and <math>\mathrm{E}X^\bullet</math> under the equivalent designations <math>\mathsf{T}U^\bullet</math> and <math>\mathsf{T}X^\bullet,</math> respectively. The purpose of the tangent functor <math>\mathsf{T}</math> is to extract the tangent map <math>\mathsf{T}F</math> at each point of <math>U^\bullet,</math> and the tangent map <math>\mathsf{T}F = (\boldsymbol\varepsilon, \mathrm{d})F</math> tells us not only what the transformation <math>F</math> is doing at each point of the universe <math>U^\bullet</math> but also what <math>F</math> is doing to states in the neighborhood of that point, approximately, linearly, and relatively speaking.

{| align="center" border="0" cellpadding="10"
|
<pre>
U% $T$ $T$U% $T$U%
o------------------>o============o
| | |
| | |
| | |
| | |
F | | $T$F = | <!e!, d> F
| | |
| | |
| | |
v v v
o------------------>o============o
X% $T$ $T$X% $T$X%

Figure 34. Tangent Functor Diagram
</pre>
|}

<ul><li><b>NB.</b>  There is one aspect of the preceding construction which remains especially problematic.  Why did we define the operators <math>\mathrm{W}</math> in <math>\{ \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}</math> so that the ranges of their resulting maps all fall within the realms of differential quality, even fabricating a variant of the tacit extension operator to have that character?  Clearly, not all of the operator maps <math>\mathrm{W}F</math> have equally good reasons for placing their values in differential stocks.  The reason for it appears to be that, without doing this, we cannot justify the comparison and combination of their functional values in the various analytic steps.  By default, only those values in the same functional component can be brought into algebraic modes of interaction.  Up till now the only mechanism provided for their broader association has been a purely logical one, their common placement in a target universe of discourse, but the task of converting that logical circumstance into algebraic forms of application has not yet been taken up.</li></ul>

<br>

----
<div align="center">
• [[Differential Logic and Dynamic Systems • Overview|Overview]]
• [[Differential Logic and Dynamic Systems • Part 1|Part 1]]
• [[Differential Logic and Dynamic Systems • Part 2|Part 2]]
• [[Differential Logic and Dynamic Systems • Part 3|Part 3]]
• [[Differential Logic and Dynamic Systems • Part 4|Part 4]]
• [[Differential Logic and Dynamic Systems • Part 5|Part 5]]
• [[Differential Logic and Dynamic Systems • Appendices|Appendices]]
• [[Differential Logic and Dynamic Systems • References|References]]
• [[Differential Logic and Dynamic Systems • Document History|Document History]]
•
</div>
----

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[[Category:Boolean algebra]]
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[[Category:Combinatorics]]
[[Category:Computation theory]]
[[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]]
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Differential Logic and Dynamic Systems • Part 3

2026-02-09T11:54:25Z

Jon Awbrey: + Differential Logic and Dynamic Systems • Part 3

<b>Author: [[User:Jon Awbrey|Jon Awbrey]]</b>

----
<div align="center">
• [[Differential Logic and Dynamic Systems • Overview|Overview]]
• [[Differential Logic and Dynamic Systems • Part 1|Part 1]]
• [[Differential Logic and Dynamic Systems • Part 2|Part 2]]
• [[Differential Logic and Dynamic Systems • Part 3|Part 3]]
• [[Differential Logic and Dynamic Systems • Part 4|Part 4]]
• [[Differential Logic and Dynamic Systems • Part 5|Part 5]]
• [[Differential Logic and Dynamic Systems • Appendices|Appendices]]
• [[Differential Logic and Dynamic Systems • References|References]]
• [[Differential Logic and Dynamic Systems • Document History|Document History]]
•
</div>
----

==Transformations of Discourse==

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
It is understandable that an engineer should be completely absorbed in his speciality, instead of pouring himself out into the freedom and vastness of the world of thought, even though his machines are being sent off to the ends of the earth;  for he no more needs to be capable of applying to his own personal soul what is daring and new in the soul of his subject than a machine is in fact capable of applying to itself the differential calculus on which it is based.  The same thing cannot, however, be said about mathematics;  for here we have the new method of thought, pure intellect, the very well‑spring of the times, the <i>fons et origo</i> of an unfathomable transformation.
| width="4%" |  
|-
| align="right" colspan="3" | — Robert Musil, ''The Man Without Qualities'', [Mus, 39]
|}

Here we take up the general study of <i>logical transformations</i>, or maps relating one universe of discourse to another.  In many ways, and especially as applied to the subject of intelligent dynamic systems, the argument will develop the antithesis of the statement just quoted.  Along the way, if incidental to my ends, I hope the present essay can pose a fittingly irenic epitaph to the frankly ironic epigraph inscribed at its head.

The goal is to answer a single question:  <i>What is a propositional tangent functor?</i>  In other words, the aim is to develop a clear conception of what manner of thing would pass in the logical realm for a genuine analogue of the tangent functor, an object conceived to generalize as far as possible in the abstract terms of category theory the ordinary notions of functional differentiation and the all too familiar operations of taking derivatives.

As a first step we examine the types of transformations we already know as <i>extensions</i> and <i>projections</i> and we use their special cases to illustrate several styles of logical and visual representation which figure in the sequel.

===Foreshadowing Transformations : Extensions and Projections of Discourse===

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
And, despite the care which she took to look behind her at every moment, she failed to see a shadow which followed her like her own shadow, which stopped when she stopped, which started again when she did, and which made no more noise than a well‑conducted shadow should.
| width="4%" |  
|-
| align="right" colspan="3" | — Gaston Leroux, ''The Phantom of the Opera'', [Ler, 126]
|}

Many times in our discussion we have occasion to place one universe of discourse in the context of a larger universe of discourse.  An embedding of the type <math>[\mathcal{X}] \to [\mathcal{Y}]</math> is implied any time we make use of one basis <math>\mathcal{X}</math> which happens to be included in another basis <math>\mathcal{Y}.</math>  When discussing differential relations we usually have in mind the extended alphabet <math>\mathfrak{Y}</math> has a special construction or a specific lexical relation with respect to the initial alphabet <math>\mathfrak{X},</math> one which is marked by characteristic types of accents, indices, or inflected forms.

====Extension from 1 to 2 Dimensions====

Figure 18-a lays out the ''angular form'' of venn diagram for universes of 1 and 2 dimensions, indicating the embedding map of type <math>\mathbb{B}^1 \to \mathbb{B}^2</math> and detailing the coordinates that are associated with individual cells.  Because all points, cells, or logical interpretations are represented as connected geometric areas, we can say these pictures provide us with an ''areal view'' of each universe of discourse.

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[File:Diff Log Dyn Sys -- Figure 18-a -- Extension from 1 to 2 Dimensions.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 18-a.} ~~ \text{Extension from 1 to 2 Dimensions : Areal}</math>
|}

Figure 18-b shows the differential extension from <math>X^\bullet = [x]</math> to <math>\mathrm{E}X^\bullet = [x, \mathrm{d}x]</math> in a ''bundle of boxes'' form of venn diagram. As awkward as it may seem at first, this type of picture is often the most natural and the most easily available representation when we want to conceptualize the localized information or momentary knowledge of an intelligent dynamic system. It gives a ready picture of a ''proposition at a point'', in the present instance, of a proposition about changing states which is itself associated with a particular dynamic state of a system. It is easy to see how this application might be extended to conceive of more general types of instantaneous knowledge that are possessed by a system.

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[File:Diff Log Dyn Sys -- Figure 18-b -- Extension from 1 to 2 Dimensions.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 18-b.} ~~ \text{Extension from 1 to 2 Dimensions : Bundle}</math>
|}

Figure 18-c shows the same extension in a ''compact'' style of venn diagram, where the differential features at each position are represented by arrows extending from that position that cross or do not cross, as the case may be, the corresponding feature boundaries.

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[File:Diff Log Dyn Sys -- Figure 18-c -- Extension from 1 to 2 Dimensions.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 18-c.} ~~ \text{Extension from 1 to 2 Dimensions : Compact}</math>
|}

Figure 18-d compresses the picture of the differential extension even further, yielding a directed graph or ''digraph'' form of representation. (Notice that my definition of a digraph allows for loops or ''slings'' at individual points, in addition to arcs or ''arrows'' between the points.)

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[File:Diff Log Dyn Sys -- Figure 18-d -- Extension from 1 to 2 Dimensions.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 18-d.} ~~ \text{Extension from 1 to 2 Dimensions : Digraph}</math>
|}

====Extension from 2 to 4 Dimensions====

Figure 19-a lays out the ''areal view'' or the ''angular form'' of venn diagram for universes of 2 and 4 dimensions, indicating the embedding map of type <math>\mathbb{B}^2 \to \mathbb{B}^4.</math> In many ways these pictures are the best kind there is, giving full canvass to an ideal vista. Their style allows the clearest, the fairest, and the plainest view that we can form of a universe of discourse, affording equal representation to all dispositions and maintaining a balance with respect to ordinary and differential features. If only we could extend this view! Unluckily, an obvious difficulty beclouds this prospect, and that is how precipitately we run into the limits of our plane and visual intuitions. Even within the scope of the spare few dimensions that we have scanned up to this point subtle discrepancies have crept in already. The circumstances that bind us and the frameworks that block us, the flat distortion of the planar projection and the inevitable ineffability that precludes us from wrapping its rhomb figure into rings around a torus, all of these factors disguise the underlying but true connectivity of the universe of discourse.

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[File:Diff Log Dyn Sys -- Figure 19-a -- Extension from 2 to 4 Dimensions.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 19-a.} ~~ \text{Extension from 2 to 4 Dimensions : Areal}</math>
|}

Figure 19-b shows the differential extension from <math>U^\bullet = [u, v]</math> to <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v]</math> in the ''bundle of boxes'' form of venn diagram.

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[File:Diff Log Dyn Sys -- Figure 19-b -- Extension from 2 to 4 Dimensions.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 19-b.} ~~ \text{Extension from 2 to 4 Dimensions : Bundle}</math>
|}

As dimensions increase, this factorization of the extended universe along the lines that are marked out by the bundle picture begins to look more and more like a practical necessity. But whenever we use a propositional model to address a real situation in the context of nature we need to remain aware that this articulation into factors, affecting our description, may be wholly artificial in nature and cleave to nothing, no joint in nature, nor any juncture in time to be in or out of joint.

Figure 19-c illustrates the extension from 2 to 4 dimensions in the ''compact'' style of venn diagram. Here, just the changes with respect to the center cell are shown.

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[File:Diff Log Dyn Sys -- Figure 19-c -- Extension from 2 to 4 Dimensions.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 19-c.} ~~ \text{Extension from 2 to 4 Dimensions : Compact}</math>
|}

Figure 19-d gives the ''digraph'' form of representation for the differential extension <math>U^\bullet \to \mathrm{E}U^\bullet,</math> where the 4 nodes marked with a circle <math>{}^{\bigcirc}</math> are the cells <math>uv,\, u \texttt{(} v \texttt{)},\, \texttt{(} u \texttt{)} v,\, \texttt{(} u \texttt{)(} v \texttt{)},</math> respectively, and where a 2-headed arc counts as 2 arcs of the differential digraph.

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[File:Diff Log Dyn Sys -- Figure 19-d -- Extension from 2 to 4 Dimensions.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 19-d.} ~~ \text{Extension from 2 to 4 Dimensions : Digraph}</math>
|}

===Thematization of Functions : And a Declaration of Independence for Variables===

{| width="100%"
| align="left" |
''And as imagination bodies forth''<br>
''The forms of things unknown, the poet's pen''<br>
''Turns them to shapes, and gives to airy nothing''<br>
''A local habitation and a name.''
| align="right" valign="bottom" | A Midsummer Night's Dream, 5.1.18
|}

In the representation of propositions as functions it is possible to notice different degrees of explicitness in the way their functional character is symbolized. To indicate what I mean by this, the next series of Figures illustrates a set of graphic conventions that will be put to frequent use in the remainder of this discussion, both to mark the relevant distinctions and to help us convert between related expressions at different levels of explicitness in their functionality.

====Thematization : Venn Diagrams====

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
The known universe has one complete lover and that is the greatest poet. He consumes an eternal passion and is indifferent which chance happens and which possible contingency of fortune or misfortune and persuades daily and hourly his delicious pay.
| width="4%" |  
|-
| align="right" colspan="3" | — Walt Whitman, ''Leaves of Grass'', [Whi, 11–12]
|}

Figure 20-i traces the first couple of steps in this order of ''thematic'' progression, that will gradually run the gamut through a complete series of degrees of functional explicitness in the expression of logical propositions. The first venn diagram represents a situation where the function is indicated by a shaded figure and a logical expression. At this stage one may be thinking of the proposition only as expressed by a formula in a particular language and its content only as a subset of the universe of discourse, as when considering the proposition <math>u\!\cdot\!v</math> in the universe <math>[u, v].</math> The second venn diagram depicts a situation in which two significant steps have been taken. First, one has taken the trouble to give the proposition <math>u\!\cdot\!v</math> a distinctive functional name <math>{}^{\backprime\backprime} J {}^{\prime\prime}.</math> Second, one has come to think explicitly about the target domain that contains the functional values of <math>J,</math> as when writing <math>J : \langle u, v \rangle \to \mathbb{B}.</math>

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[File:Diff Log Dyn Sys -- Figure 20-i -- Thematization of Conjunction (Stage 1).gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 20-i.} ~~ \text{Thematization of Conjunction (Stage 1)}</math>
|}

In Figure 20-ii the proposition <math>J</math> is viewed explicitly as a transformation from one universe of discourse to another.

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[File:Diff Log Dyn Sys -- Figure 20-ii -- Thematization of Conjunction (Stage 2).gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 20-ii.} ~~ \text{Thematization of Conjunction (Stage 2)}</math>
|}

{| align="center" border="0" cellpadding="10"
|
<pre>
o-------------------------------o o-------------------------------o
| | | |
| o-----o o-----o | | o-----o o-----o |
| / \ / \ | | / \ / \ |
| / o \ | | / o \ |
| / /`\ \ | | / /`\ \ |
| o o```o o | | o o```o o |
| | u |```| v | | | | u |```| v | |
| o o```o o | | o o```o o |
| \ \`/ / | | \ \`/ / |
| \ o / | | \ o / |
| \ / \ / | | \ / \ / |
| o-----o o-----o | | o-----o o-----o |
| | | |
o-------------------------------o o-------------------------------o
\ / \ /
\ / \ /
\ / \ J /
\ / \ /
\ / \ /
o----------\---------/----------o o----------\---------/----------o
| \ / | | \ / |
| \ / | | \ / |
| o-----@-----o | | o-----@-----o |
| /`````````````\ | | /`````````````\ |
| /```````````````\ | | /```````````````\ |
| /`````````````````\ | | /`````````````````\ |
| o```````````````````o | | o```````````````````o |
| |```````````````````| | | |```````````````````| |
| |```````` J ````````| | | |```````` x ````````| |
| |```````````````````| | | |```````````````````| |
| o```````````````````o | | o```````````````````o |
| \`````````````````/ | | \`````````````````/ |
| \```````````````/ | | \```````````````/ |
| \`````````````/ | | \`````````````/ |
| o-----------o | | o-----------o |
| | | |
| | | |
o-------------------------------o o-------------------------------o
J = u v x = J<u, v>

Figure 20-ii. Thematization of Conjunction (Stage 2)
</pre>
|}

In the first venn diagram the name that is assigned to a composite proposition, function, or region in the source universe is delegated to a simple feature in the target universe. This can result in a single character or term exceeding the responsibilities it can carry off well. Allowing the name of a function <math>J : \langle u, v \rangle \to \mathbb{B}</math> to serve as the name of its dependent variable <math>J : \mathbb{B}</math> does not mean that one has to confuse a function with any of its values, but it does put one at risk for a number of obvious problems, and we should not be surprised, on numerous and limiting occasions, when quibbling arises from the attempts of a too original syntax to serve these two masters.

The second venn diagram circumvents these difficulties by introducing a new variable name for each basic feature of the target universe, as when writing <math>J : \langle u, v \rangle \to \langle x \rangle,</math> and thereby assigns a concrete type <math>\langle x \rangle</math> to the abstract codomain <math>\mathbb{B}.</math> To make this induction of variables more formal one can append subscripts, as in <math>x_J,</math> to indicate the origin or derivation of the new characters. Or we may use a lexical modifier to convert function names into variable names, for example, associating the function name <math>J</math> with the variable name <math>\check{J}.</math> Thus we may think of <math>x = x_J = \check{J}</math> as the ''cache variable'' corresponding to the function <math>J</math> or the symbol <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> considered as a contingent variable.

In Figure 20-iii we arrive at a stage where the functional equations <math>J = u\!\cdot\!v</math> and <math>x = u\!\cdot\!v</math> are regarded as propositions in their own right, reigning in and ruling over the 3-feature universes of discourse <math>[u, v, J]</math> and <math>[u, v, x],</math> respectively. Subject to the cautions already noted, the function name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> can be reinterpreted as the name of a feature <math>\check{J}</math> and the equation <math>J = u\!\cdot\!v</math> can be read as the logical equivalence <math>\texttt{((} J, u ~ v \texttt{))}.</math> To give it a generic name let us call this newly expressed, collateral proposition the ''thematization'' or the ''thematic extension'' of the original proposition <math>J.</math>

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[File:Diff Log Dyn Sys -- Figure 20-iii -- Thematization of Conjunction (Stage 3).gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 20-iii.} ~~ \text{Thematization of Conjunction (Stage 3)}</math>
|}

The first venn diagram represents the thematization of the conjunction <math>J</math> with shading in the appropriate regions of the universe <math>[u, v, J].</math> Also, it illustrates a quick way of constructing a thematic extension. First, draw a line, in practice or the imagination, that bisects every cell of the original universe, placing half of each cell under the aegis of the thematized proposition and the other half under its antithesis. Next, on the scene where the theme applies leave the shade wherever it lies, and off the stage, where it plays otherwise, stagger the pattern in a harlequin guise.

In the final venn diagram of this sequence the thematic progression comes full circle and completes one round of its development. The ambiguities that were occasioned by the changing role of the name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> are resolved by introducing a new variable name <math>{}^{\backprime\backprime} x {}^{\prime\prime}</math> to take the place of <math>\check{J},</math> and the region that represents this fresh featured <math>x</math> is circumscribed in a more conventional symmetry of form and placement. Just as we once gave the name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> to the proposition <math>u\!\cdot\!v,</math> we now give the name <math>{}^{\backprime\backprime} \iota {}^{\prime\prime}</math> to its thematization <math>\texttt{((} x, u ~ v \texttt{))}.</math> Already, again, at this culminating stage of reflection, we begin to think of the newly named proposition as a distinctive individual, a particular function <math>\iota : \langle u, v, x \rangle \to \mathbb{B}.</math>

From now on, the terms ''thematic extension'' and ''thematization'' will be used to describe both the process and degree of explication that progresses through this series of pictures, both the operation of increasingly explicit symbolization and the dimension of variation that is swept out by it. To speak of this change in general, that takes us in our current example from <math>J</math> to <math>\iota,</math> we introduce a class of operators symbolized by the Greek letter <math>\theta,</math> writing <math>\iota = \theta J</math> in the present instance. The operator <math>\theta,</math> in the present situation bearing the type <math>\theta : [u, v] \to [u, v, x],</math> provides us with a convenient way of recapitulating and summarizing the complete cycle of thematic developments.

Figure 21 shows how the thematic extension operator <math>\theta</math> acts on two further examples, the disjunction <math>\texttt{((} u \texttt{)(} v \texttt{))}</math> and the equality <math>\texttt{((} u, v \texttt{))}.</math> Referring to the disjunction as <math>f(u, v)</math> and the equality as <math>f(u, v),</math> we may express the thematic extensions as <math>\varphi = \theta f</math> and <math>\gamma = \theta g.</math>

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[File:Diff Log Dyn Sys -- Figure 21 -- Thematization of Disjunction and Equality.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 21.} ~~ \text{Thematization of Disjunction and Equality}</math>
|}

====Thematization : Truth Tables====

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
That which distorts honest shapes or which creates unearthly beings or places or contingencies is a nuisance and a revolt.
| width="4%" |  
|-
| align="right" colspan="3" | — Walt Whitman, ''Leaves of Grass'', [Whi, 19]
|}

Tables 22 through 25 outline a method for computing the thematic extensions of propositions in terms of their coordinate values.

A preliminary step, as illustrated in Table 22, is to write out the truth table representations of the propositional forms whose thematic extensions one wants to compute, in the present instance, the functions <math>f(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}</math> and <math>g(u, v) = \texttt{((} u, v \texttt{))}.</math>

<br>

{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:50%"
|+ style="height:30px" | <math>\text{Table 22.} ~~ \text{Disjunction}~ f ~\text{and Equality}~ g</math>
|- style="height:35px; background:ghostwhite; width:100%"
| <math>u</math>
| <math>v</math>
| style="border-left:1px solid black" | <math>f</math>
| <math>g</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
0
\\[4pt]
0
\\[4pt]
1
\\[4pt]
1
\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
0
\\[4pt]
1
\\[4pt]
0
\\[4pt]
1
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
0
\\[4pt]
1
\\[4pt]
1
\\[4pt]
1
\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
1
\\[4pt]
0
\\[4pt]
0
\\[4pt]
1
\end{matrix}</math>
|}

<br>

Next, each propositional form is individually represented in the fashion shown in Tables 23-i and 23-ii, using <math>{}^{\backprime\backprime} f {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} g {}^{\prime\prime}</math> as function names and creating new variables <math>x</math> and <math>y</math> to hold the associated functional values. This pair of Tables outlines the first stage in the transition from the <math>2</math>-dimensional universes of <math>f</math> and <math>g</math> to the <math>3</math>-dimensional universes of <math>\theta f</math> and <math>\theta g.</math> The top halves of the Tables replicate the truth table patterns for <math>f</math> and <math>g</math> in the form <math>f : [u, v] \to [x]</math> and <math>g : [u, v] \to [y].</math> The bottom halves of the tables print the negatives of these pictures, as it were, and paste the truth tables for <math>\texttt{(} f \texttt{)}</math> and <math>\texttt{(} g \texttt{)}</math> under the copies for <math>f</math> and <math>g.</math> At this stage, the columns for <math>\theta f</math> and <math>\theta g</math> are appended almost as afterthoughts, amounting to indicator functions for the sets of ordered triples that make up the functions <math>f</math> and <math>g.</math>

<br>

{| align="center" border="0" style="width:90%"
|+ style="height:25px" | <math>\text{Tables 23-i and 23-ii.} ~~ \text{Thematics of Disjunction and Equality (1)}</math>
| width="50%" |
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:25px" | <math>\text{Table 23-i.} ~~ \text{Disjunction}~ f</math>
|- style="height:25px; background:ghostwhite; width:100%"
| <math>u</math>
| <math>v</math>
| style="border-left:1px solid black; border-right:1px solid black" | <math>f</math>
| <math>x</math>
| <math>\varphi</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |
<math>\begin{matrix}\to\\\to\\\to\\\to\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |  
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\0\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}</math>
|}
| width="50%" |
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:25px" | <math>\text{Table 23-ii.} ~~ \text{Equality}~ g</math>
|- style="height:25px; background:ghostwhite; width:100%"
| <math>u</math>
| <math>v</math>
| style="border-left:1px solid black; border-right:1px solid black" | <math>g</math>
| <math>y</math>
| <math>\gamma</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |
<math>\begin{matrix}\to\\\to\\\to\\\to\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |  
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}</math>
|}
|}

<br>

All the data are now in place to give the truth tables for <math>\theta f</math> and <math>\theta g.</math> All that remains to be done is to permute the rows and change the roles of <math>x</math> and <math>y</math> from dependent to independent variables. In Tables 24-i and 24-ii the rows are arranged in such a way as to put the 3-tuples <math>(u, v, x)</math> and <math>(u, v, y)</math> in binary numerical order, suitable for viewing as the arguments of the maps <math>\theta f = \varphi : [u, v, x] \to \mathbb{B}</math> and <math>\theta g = \gamma : [u, v, y] \to \mathbb{B}.</math> Moreover, the structure of the tables is altered slightly, allowing the now vestigial functions <math>\theta f</math> and <math>\theta g</math> to be passed over without further attention and shifting the heavy vertical bars a notch to the right. In effect, this clinches the fact that the thematic variables <math>x := \check{f}</math> and <math>y := \check{g}</math> are now treated as independent variables.

<br>

{| align="center" border="0" style="width:90%"
|+ style="height:25px" | <math>\text{Tables 24-i and 24-ii.} ~~ \text{Thematics of Disjunction and Equality (2)}</math>
| width="50%" |
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:25px" | <math>\text{Table 24-i.} ~~ \text{Disjunction}~ f</math>
|- style="height:25px; background:ghostwhite; width:100%"
| <math>u</math>
| <math>v</math>
| <math>f</math>
| <math>x</math>
| style="border-left:1px solid black" | <math>\varphi</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <font size="+2"> <br></font><math>\begin{matrix}\to\\~\\\to\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
|}
| width="50%" |
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:25px" | <math>\text{Table 24-ii.} ~~ \text{Equality}~ g</math>
|- style="height:25px; background:ghostwhite; width:100%"
| <math>u</math>
| <math>v</math>
| <math>g</math>
| <math>y</math>
| style="border-left:1px solid black" | <math>\gamma</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" |  <br><math>\begin{matrix}\to\\\to\end{matrix}</math><br> 
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}</math>
|}
|}

<br>

An optional reshuffling of the rows brings additional features of the thematic extensions to light. Leaving the columns in place for the sake of comparison, Tables 25-i and 25-ii sort the rows in a different order, in effect treating <math>x</math> and <math>y</math> as the primary variables in their respective 3-tuples. Regarding the thematic extensions in the form <math>\varphi : [x, u, v] \to \mathbb{B}</math> and <math>\gamma : [y, u, v] \to \mathbb{B}</math> makes it easier to see in this tabular setting a property that was graphically obvious in the venn diagrams above. Specifically, when the thematic variable <math>\check{F}</math> is true then <math>\theta F</math> exhibits the pattern of the original <math>F,</math> and when <math>\check{F}</math> is false then <math>\theta F</math> exhibits the pattern of its negation <math>\texttt{(} F \texttt{)}.</math>

<br>

{| align="center" border="0" style="width:90%"
|+ style="height:25px" | <math>\text{Tables 25-i and 25-ii.} ~~ \text{Thematics of Disjunction and Equality (3)}</math>
| width="50%" |
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:25px" | <math>\text{Table 25-i.} ~~ \text{Disjunction}~ f</math>
|- style="height:25px; background:ghostwhite; width:100%"
| <math>u</math>
| <math>v</math>
| <math>f</math>
| <math>x</math>
| style="border-left:1px solid black" | <math>\varphi</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>{\to}</math><br><font size="+2"> <br> <br> <br></font>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}</math>
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\0\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| style="border-top:1px solid black" | <font size="+2"> <br></font><math>\begin{matrix}\to\\\to\\\to\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}</math>
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\1\end{matrix}</math>
|}
| width="50%" |
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:25px" | <math>\text{Table 25-ii.} ~~ \text{Equality}~ g</math>
|- style="height:25px; background:ghostwhite; width:100%"
| <math>u</math>
| <math>v</math>
| <math>g</math>
| <math>y</math>
| style="border-left:1px solid black" | <math>\gamma</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| style="border-top:1px solid black" |  <br><math>\begin{matrix}\to\\\to\end{matrix}</math><br> 
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}</math>
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}</math>
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}</math>
|}
|}

<br>

Finally, Tables 26-i and 26-ii compare the tacit extensions <math>\boldsymbol\varepsilon : [u, v] \to [u, v, x]</math> and <math>\boldsymbol\varepsilon : [u, v] \to [u, v, y]</math> with the thematic extensions of the same types, as applied to the propositions <math>f</math> and <math>g,</math> respectively.

<br>

{| align="center" border="0" style="width:90%"
|+ style="height:25px" | <math>\text{Tables 26-i and 26-ii.} ~~ \text{Tacit Extension and Thematization}</math>
| width="50%" |
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:25px" | <math>\text{Table 26-i.} ~~ \text{Disjunction}~ f</math>
|- style="height:25px; background:ghostwhite; width:100%"
| <math>u</math>
| <math>v</math>
| <math>x</math>
| style="border-left:1px solid black" | <math>\boldsymbol\varepsilon f</math>
| <math>\theta f</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
|}
| width="50%" |
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:25px" | <math>\text{Table 26-ii.} ~~ \text{Equality}~ g</math>
|- style="height:25px; background:ghostwhite; width:100%"
| <math>u</math>
| <math>v</math>
| <math>y</math>
| style="border-left:1px solid black" | <math>\boldsymbol\varepsilon g</math>
| <math>\theta g</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}1\\1\\0\\0\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}</math>
|}
|}

<br>

Table 27 summarizes the thematic extensions of all propositions on two variables. Column 4 lists the equations of form <math>\texttt{((} \check{f_i}, f_i (u, v) \texttt{))}</math> and Column 5 simplifies these equations into the form of algebraic expressions. As always, <math>{}^{\backprime\backprime} + {}^{\prime\prime}</math> refers to exclusive disjunction and each <math>{}^{\backprime\backprime} \check{f} {}^{\prime\prime}</math> appearing in the last two Columns refers to the corresponding variable name <math>{}^{\backprime\backprime} \check{f_i} {}^{\prime\prime}.</math>

<br>

{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 27.} ~~ \text{Thematization of Bivariate Propositions}</math>
|- style="height:30px; background:ghostwhite"
|  
|  
| <math>{f}</math>
| <math>\theta f</math>
| <math>\theta f</math>
|- style="background:ghostwhite"
| align="right" | <math>u\colon</math>
| <math>1~1~0~0</math>
|  
|  
|  
|- style="background:ghostwhite"
| align="right" | <math>v\colon</math>
| <math>1~0~1~0</math>
|  
|  
|  
|-
| <math>f_{0}</math>
| <math>0~0~0~0</math>
| <math>\texttt{(~)}</math>
| align="left" | <math>\texttt{((} \check{f} \texttt{,~(~)~))}</math>
| align="left" | <math>\check{f} + 1</math>
|-
|
<math>\begin{matrix}
f_{1}
\\[4pt]
f_{2}
\\[4pt]
f_{4}
\\[4pt]
f_{8}
\end{matrix}</math>
|
<math>\begin{matrix}
0~0~0~1
\\[4pt]
0~0~1~0
\\[4pt]
0~1~0~0
\\[4pt]
1~0~0~0
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} u \texttt{)(} v \texttt{)}
\\[4pt]
\texttt{(} u \texttt{)~} v \texttt{~}
\\[4pt]
\texttt{~} u \texttt{~(} v \texttt{)}
\\[4pt]
\texttt{~} u \texttt{~~} v \texttt{~}
\end{matrix}</math>
| align="left" |
<math>\begin{array}{l}
\texttt{((} \check{f} \texttt{,~(u)(v)~))}
\\[4pt]
\texttt{((} \check{f} \texttt{,~(u)~v~~))}
\\[4pt]
\texttt{((} \check{f} \texttt{,~~u~(v)~))}
\\[4pt]
\texttt{((} \check{f} \texttt{,~~u~~v~~))}
\end{array}</math>
| align="left" |
<math>\begin{array}{l}
\check{f} + u + v + uv
\\[4pt]
\check{f} + v + uv + 1
\\[4pt]
\check{f} + u + uv + 1
\\[4pt]
\check{f} + uv + 1
\end{array}</math>
|-
|
<math>\begin{matrix}
f_{3}
\\[4pt]
f_{12}
\end{matrix}</math>
|
<math>\begin{matrix}
0~0~1~1
\\[4pt]
1~1~0~0
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} u \texttt{)}
\\[4pt]
\texttt{~} u \texttt{~}
\end{matrix}</math>
| align="left" |
<math>\begin{array}{l}
\texttt{((} \check{f} \texttt{,~(u)~))}
\\[4pt]
\texttt{((} \check{f} \texttt{,~~u~~))}
\end{array}</math>
| align="left" |
<math>\begin{array}{l}
\check{f} + u
\\[4pt]
\check{f} + u + 1
\end{array}</math>
|-
|
<math>\begin{matrix}
f_{6}
\\[4pt]
f_{9}
\end{matrix}</math>
|
<math>\begin{matrix}
0~1~1~0
\\[4pt]
1~0~0~1
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} u \texttt{,} v \texttt{)}
\\[4pt]
\texttt{((} u \texttt{,} v \texttt{))}
\end{matrix}</math>
| align="left" |
<math>\begin{array}{l}
\texttt{((} \check{f} \texttt{,~~(} u \texttt{,} v \texttt{)~~))}
\\[4pt]
\texttt{((} \check{f} \texttt{,~((} u \texttt{,} v \texttt{))~))}
\end{array}</math>
| align="left" |
<math>\begin{array}{l}
\check{f} + u + v + 1
\\[4pt]
\check{f} + u + v
\end{array}</math>
|-
|
<math>\begin{matrix}
f_{5}
\\[4pt]
f_{10}
\end{matrix}</math>
|
<math>\begin{matrix}
0~1~0~1
\\[4pt]
1~0~1~0
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} v \texttt{)}
\\[4pt]
\texttt{~} v \texttt{~}
\end{matrix}</math>
| align="left" |
<math>\begin{array}{l}
\texttt{((} \check{f} \texttt{,~(} v \texttt{)~))}
\\[4pt]
\texttt{((} \check{f} \texttt{,~~} v \texttt{~~))}
\end{array}</math>
| align="left" |
<math>\begin{array}{l}
\check{f} + v
\\[4pt]
\check{f} + v + 1
\end{array}</math>
|-
|
<math>\begin{matrix}
f_{7}
\\[4pt]
f_{11}
\\[4pt]
f_{13}
\\[4pt]
f_{14}
\end{matrix}</math>
|
<math>\begin{matrix}
0~1~1~1
\\[4pt]
1~0~1~1
\\[4pt]
1~1~0~1
\\[4pt]
1~1~1~0
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(~} u \texttt{~~} v \texttt{~)}
\\[4pt]
\texttt{(~} u \texttt{~(} v \texttt{))}
\\[4pt]
\texttt{((} u \texttt{)~} v \texttt{~)}
\\[4pt]
\texttt{((} u \texttt{)(} v \texttt{))}
\end{matrix}</math>
| align="left" |
<math>\begin{array}{l}
\texttt{((} \check{f} \texttt{,~(~} u \texttt{~~} v \texttt{~)~))}
\\[4pt]
\texttt{((} \check{f} \texttt{,~(~} u \texttt{~(} v \texttt{))~))}
\\[4pt]
\texttt{((} \check{f} \texttt{,~((} u \texttt{)~} v \texttt{~)~))}
\\[4pt]
\texttt{((} \check{f} \texttt{,~((} u \texttt{)(} v \texttt{))~))}
\end{array}</math>
| align="left" |
<math>\begin{array}{l}
\check{f} + uv
\\[4pt]
\check{f} + u + uv
\\[4pt]
\check{f} + v + uv
\\[4pt]
\check{f} + u + v + uv + 1
\end{array}</math>
|-
| <math>f_{15}</math>
| <math>1~1~1~1</math>
| <math>\texttt{((~))}</math>
| align="left" | <math>\texttt{((} \check{f} \texttt{,~((~))~))}</math>
| align="left" | <math>\check{f}</math>
|}

<br>

In order to show what all of the thematic extensions from two dimensions to three dimensions look like in terms of coordinates, Tables 28 and 29 present ordinary truth tables for the functions <math>f_i : \mathbb{B}^2 \to \mathbb{B}</math> and for the corresponding thematizations <math>\theta f_i = \varphi_i : \mathbb{B}^3 \to \mathbb{B}.</math>

<br>

{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 28.} ~~ \text{Propositions on Two Variables}</math>
|- style="height:35px; background:ghostwhite"
| style="width:5%; border-bottom:1px solid black" | <math>u</math>
| style="width:5%; border-bottom:1px solid black" | <math>v</math>
| style="width:5%; border-bottom:1px solid black; border-left:1px solid black" | <math>f_{0}</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{1}</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{2}</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{3}</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{4}</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{5}</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{6}</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{7}</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{8}</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{9}</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{10}</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{11}</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{12}</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{13}</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{14}</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{15}</math>
|-
| <math>0</math>
| <math>0</math>
| style="border-left:1px solid black" |  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|-
| <math>0</math>
| <math>1</math>
| style="border-left:1px solid black" |  
|  
| <math>1</math>
| <math>1</math>
|  
|  
| <math>1</math>
| <math>1</math>
|  
|  
| <math>1</math>
| <math>1</math>
|  
|  
| <math>1</math>
| <math>1</math>
|-
| <math>1</math>
| <math>0</math>
| style="border-left:1px solid black" |  
|  
|  
|  
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
|  
|  
|  
|  
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
|-
| <math>1</math>
| <math>1</math>
| style="border-left:1px solid black" |  
|  
|  
|  
|  
|  
|  
|  
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
|}

<br>

{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 29.} ~~ \text{Thematic Extensions of Bivariate Propositions}</math>
|- style="height:35px; background:ghostwhite"
| style="width:5%; border-bottom:1px solid black" | <math>u</math>
| style="width:5%; border-bottom:1px solid black" | <math>v</math>
| style="width:5%; border-bottom:1px solid black" | <math>\check{f}</math>
| style="width:5%; border-bottom:1px solid black; border-left:1px solid black" | <math>\varphi_{0}</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{1}</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{2}</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{3}</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{4}</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{5}</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{6}</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{7}</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{8}</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{9}</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{10}</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{11}</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{12}</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{13}</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{14}</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{15}</math>
|-
| <math>0</math>
| <math>0</math>
| <math>0</math>
| style="border-left:1px solid black" | <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|  
|-
| <math>0</math>
| <math>0</math>
| <math>1</math>
| style="border-left:1px solid black" |  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|  
| <math>1</math>
|-
| <math>0</math>
| <math>1</math>
| <math>0</math>
| style="border-left:1px solid black" | <math>1</math>
| <math>1</math>
|  
|  
| <math>1</math>
| <math>1</math>
|  
|  
| <math>1</math>
| <math>1</math>
|  
|  
| <math>1</math>
| <math>1</math>
|  
|  
|-
| <math>0</math>
| <math>1</math>
| <math>1</math>
| style="border-left:1px solid black" |  
|  
| <math>1</math>
| <math>1</math>
|  
|  
| <math>1</math>
| <math>1</math>
|  
|  
| <math>1</math>
| <math>1</math>
|  
|  
| <math>1</math>
| <math>1</math>
|-
| <math>1</math>
| <math>0</math>
| <math>0</math>
| style="border-left:1px solid black" | <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
|  
|  
|  
|  
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
|  
|  
|  
|  
|-
| <math>1</math>
| <math>0</math>
| <math>1</math>
| style="border-left:1px solid black" |  
|  
|  
|  
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
|  
|  
|  
|  
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
|-
| <math>1</math>
| <math>1</math>
| <math>0</math>
| style="border-left:1px solid black" | <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
|  
|  
|  
|  
|  
|  
|  
|  
|-
| <math>1</math>
| <math>1</math>
| <math>1</math>
| style="border-left:1px solid black" |  
|  
|  
|  
|  
|  
|  
|  
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
| <math>1</math>
|}

<br>

===Propositional Transformations===

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
If only the word ‘artificial’ were associated with the idea of ''art'', or expert skill gained through voluntary apprenticeship (instead of suggesting the factitious and unreal), we might say that ''logical'' refers to artificial thought.
| width="4%" |  
|-
| align="right" colspan="3" | — John Dewey, ''How We Think'', [Dew, 56–57]
|}

Next we develop a set of concepts for dealing with transformations between universes of discourse.  As a general consideration the source and target universes of a transformation are allowed to be different but when we turn to dynamic systems we focus on the special case of transformations mapping universes into themselves.  In effect, we regard each transformation as one of the possible state transitions in a discrete dynamical process and place it among the myriad ways a universe of discourse may change, and by that change turn into itself.

====Alias and Alibi Transformations====

There are customarily two modes of understanding a transformation, at least, when we try to interpret its relevance to something in reality. A transformation always refers to a changing prospect, to say it in a unified but equivocal way, but this can be taken to mean either a subjective change in the interpreting observer's point of view or an objective change in the systematic subject of discussion. In practice these variant uses of the transformation concept are distinguished in the following terms:

# A ''perspectival'' or ''alias'' transformation refers to a shift in perspective or a change in language that takes place in the observer's frame of reference.
# A ''transitional'' or ''alibi'' transformation refers to a change of position or an alteration of state that occurs in the object system as it falls under study.

(For a recent discussion of the alias vs. alibi issue, as it relates to linear transformations in vector spaces and to other issues of an algebraic nature, see [MaB, 256, 582-4].)

Naturally, when we are concerned with the dynamical properties of a system, the transitional aspect of transformation is the factor that comes to the fore, and this involves us in contemplating all of the ways of changing a universe into itself while remaining under the rule of established dynamical laws. In the prospective application to dynamic systems, and to neural networks viewed in this light, our interest lies chiefly with the transformations of a state space into itself that constitute the state transitions of a discrete dynamic process. Nevertheless, many important properties of these transformations, and some constructions that we need to see most clearly, are independent of the transitional interpretation and are likely to be confounded with irrelevant features if presented first and only in that association.

In addition, and in partial contrast, intelligent systems are exactly that species of dynamic agents that have the capacity to have a point of view, and we cannot do justice to their peculiar properties without examining their ability to form and transform their own frames of reference in exposure to the elements of their own experience. In this setting, the perspectival aspect of transformation is the facet that shines most brightly, perhaps too often leaving us fascinated with mere glimmerings of its actual potential. It needs to be emphasized that nothing of the ordinary sort needs be moved in carrying out a transformation under the alias interpretation, that it may only involve a change in the forms of address, an amendment of the terms which are customed to approach and fashioned to describe the very same things in the very same world. But again, working within a discipline of realistic computation, we know how formidably complex and resource-consuming such transformations of perspective can be to implement in practice, much less to endow in the self-governed form of a nascently intelligent dynamical system.

====Transformations of General Type====

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
''Es ist passiert'', “it just sort of happened”, people said there when other people in other places thought heaven knows what had occurred. It was a peculiar phrase, not known in this sense to the Germans and with no equivalent in other languages, the very breath of it transforming facts and the bludgeonings of fate into something light as eiderdown, as thought itself.
| width="4%" |  
|-
| align="right" colspan="3" | — Robert Musil, ''The Man Without Qualities'', [Mus, 34]
|}

Consider the situation illustrated in Figure 30, where the alphabets <math>\mathcal{U} = \{ u, v \}</math> and <math>\mathcal{X} = \{ x, y, z \}</math> are used to label basic features in two different logical universes, <math>U^\bullet = [u, v]</math> and <math>X^\bullet = [x, y, z].</math>

{| align="center" border="0" cellpadding="10"
|
<pre>
o-------------------------------------------------------o
| U |
| |
| o-----------o o-----------o |
| / \ / \ |
| / o \ |
| / / \ \ |
| / / \ \ |
| o o o o |
| | | | | |
| | u | | v | |
| | | | | |
| o o o o |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| o-----------o o-----------o |
| |
| |
o---------------------------o---------------------------o
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
o-------------------------o o-------------------------o o-------------------------o
| U | | U | | U |
| o---o o---o | | o---o o---o | | o---o o---o |
| / \ / \ | | / \ / \ | | / \ / \ |
| / o \ | | / o \ | | / o \ |
| / / \ \ | | / / \ \ | | / / \ \ |
| o o o o | | o o o o | | o o o o |
| | u | | v | | | | u | | v | | | | u | | v | |
| o o o o | | o o o o | | o o o o |
| \ \ / / | | \ \ / / | | \ \ / / |
| \ o / | | \ o / | | \ o / |
| \ / \ / | | \ / \ / | | \ / \ / |
| o---o o---o | | o---o o---o | | o---o o---o |
| | | | | |
o-------------------------o o-------------------------o o-------------------------o
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ g | \ f / | h /
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ o----------|-----------\-----/-----------|----------o /
\ | X | \ / | | /
\ | | \ / | | /
\ | | o-----o-----o | | /
\| | / \ | |/
\ | / \ | /
|\ | / \ | /|
| \ | / \ | / |
| \ | / \ | / |
| \ | o x o | / |
| \ | | | | / |
| \ | | | | / |
| \ | | | | / |
| \ | | | | / |
| \ | | | | / |
| \| | | |/ |
| o--o--------o o--------o--o |
| / \ \ / / \ |
| / \ \ / / \ |
| / \ o / \ |
| / \ / \ / \ |
| / \ / \ / \ |
| o o--o-----o--o o |
| | | | | |
| | | | | |
| | | | | |
| | y | | z | |
| | | | | |
| | | | | |
| o o o o |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| \ / \ / |
| o-----------o o-----------o |
| |
| |
o---------------------------------------------------o
\ /
\ /
\ /
\ /
\ /
\ p , q /
\ /
\ /
\ /
\ /
\ /
\ /
\ /
o

Figure 30. Generic Frame of a Logical Transformation
</pre>
|}

Enter the picture, as we usually do, in the middle of things, with features like <math>x, y , z</math> that present themselves to be simple enough in their own right and that form a satisfactory, if temporary foundation to provide a basis for discussion. In this universe and on these terms we find expression for various propositions and questions of principal interest to ourselves, as indicated by the maps <math>p, q : X \to \mathbb{B}.</math> Then we discover that the simple features <math>\{ x, y, z \}</math> are really more complex than we thought at first, and it becomes useful to regard them as functions <math>\{ f, g, h \}</math> of other features <math>\{ u, v \}</math> that we place in a preface to our original discourse, or suppose as topics of a preliminary universe of discourse <math>U^\bullet = [u, v].</math> It may happen that these late-blooming but pre-ambling features are found to lie closer, in a sense that may be our job to determine, to the central nature of the situation of interest, in which case they earn our regard as being more fundamental, but these functions and features are only required to supply a critical stance on the universe of discourse or an alternate perspective on the nature of things in order to be preserved as useful.

A particular transformation <math>F : [u, v] \to [x, y, z]</math> may be expressed by a system of equations, as shown below. Here, <math>F</math> is defined by its component maps <math>F = (F_1, F_2, F_3) = (f, g, h),</math> where each component map in <math>\{ f, g, h \}</math> is a proposition of type <math>\mathbb{B}^n \to \mathbb{B}^1.</math>

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:50%"
|
<math>\begin{matrix}
x & = & f(u, v)
\\[10pt]
y & = & g(u, v)
\\[10pt]
z & = & h(u, v)
\end{matrix}</math>
|}

<br>

Regarded as a logical statement, this system of equations expresses a relation between a collection of freely chosen propositions <math>\{ f, g, h \}</math> in one universe of discourse and the special collection of simple propositions <math>\{ x, y, z \}</math> on which is founded another universe of discourse. Growing familiarity with a particular transformation of discourse, and the desire to achieve a ready understanding of its implications, requires that we be able to convert this information about generals and simples into information about all the main subtypes of propositions, including the linear and singular propositions.

===Analytic Expansions : Operators and Functors===

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
Consider what effects that might <i>conceivably</i> have practical bearings you <i>conceive</i> the objects of your <i>conception</i> to have. <br> Then, your <i>conception</i> of those effects is the whole of your <i>conception</i> of the object.
| width="4%" |  
|-
| align="right" colspan="3" | — C.S. Peirce, “The Maxim of Pragmatism”, CP 5.438
|}

Given the barest idea of a logical transformation, as suggested by the sketch in Figure 30, and having conceptualized the universe of discourse, with all of its points and propositions, as a beginning object of discussion, we are ready to enter the next phase of our investigation.

====Operators on Propositions and Transformations====

The next step is naturally inclined toward objects of the next higher order, namely, with operators that take in argument lists of logical transformations and that give back specified types of logical transformations as their results. For our present aims, we do not need to consider the most general class of such operators, nor any one of them for its own sake. Rather, we are interested in the special sorts of operators that arise in the study and analysis of logical transformations. Figuratively speaking, these operators serve as instruments for the live tomography (and hopefully not the vivisection) of the forms of change under view. Beyond that, they open up ways to implement the changes of view that we need to grasp all the variations on a transformational theme, or to appreciate enough of its significant features to “get the drift” of the change occurring, to form a passing acquaintance or a synthetic comprehension of its general character and disposition.

The simplest type of operator is one that takes a single transformation as an argument and returns a single transformation as a result, and most of the operators explicitly considered in our discussion will be of this kind. Figure 31 illustrates the typical situation.

{| align="center" border="0" cellpadding="20"
|
<pre>
o---------------------------------------o
| |
| |
| U% F X% |
| o------------------>o |
| | | |
| | | |
| | | |
| | | |
| !W! | | !W! |
| | | |
| | | |
| | | |
| v v |
| o------------------>o |
| !W!U% !W!F !W!X% |
| |
| |
o---------------------------------------o
Figure 31. Operator Diagram (1)
</pre>
|}

In this Figure <math>{}^{\backprime\backprime} \mathsf{W} {}^{\prime\prime}</math> stands for a generic operator <math>\mathsf{W},</math> in this case one that takes a logical transformation <math>F</math> of type <math>(U^\bullet \to X^\bullet)</math> into a logical transformation <math>\mathsf{W}F</math> of type <math>(\mathsf{W}U^\bullet \to \mathsf{W}X^\bullet).</math> Thus, the operator <math>\mathsf{W}</math> must be viewed as making assignments for both families of objects we have previously considered, that is, for universes of discourse like <math>{U^\bullet}</math> and <math>{X^\bullet}</math> and for logical transformations like <math>F.</math>

'''Note.''' Strictly speaking, an operator like <math>\mathsf{W}</math> works between two whole categories of universes and transformations, which we call the ''source'' and the ''target'' categories of <math>\mathsf{W}.</math> Given this setting, <math>\mathsf{W}</math> specifies for each universe <math>U^\bullet</math> in its source category a definite universe <math>\mathsf{W}U^\bullet</math> in its target category, and to each transformation <math>F</math> in its source category it assigns a unique transformation <math>\mathsf{W}F</math> in its target category. Naturally, this only works if <math>\mathsf{W}</math> takes the source <math>U^\bullet</math> and the target <math>X^\bullet</math> of the map <math>F</math> over to the source <math>\mathsf{W}U^\bullet</math> and the target <math>\mathsf{W}X^\bullet</math> of the map <math>\mathsf{W}F.</math> With luck or care enough, we can avoid ever having to put anything like that in words again, letting diagrams do the work. In the situations of present concern we are usually focused on a single transformation <math>F,</math> and thus we can take it for granted that the assignment of universes under <math>\mathsf{W}</math> is defined appropriately at the source and target ends of <math>F.</math> It is not always the case, though, that we need to use the particular names (like <math>{}^{\backprime\backprime} \mathsf{W}U^\bullet {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} \mathsf{W}X^\bullet {}^{\prime\prime}</math>) that <math>\mathsf{W}</math> assigns by default to its operative image universes. In most contexts we will usually have a prior acquaintance with these universes under other names and it is necessary only that we can tell from the information associated with an operator <math>\mathsf{W}</math> what universes they are.

In Figure 31 the maps <math>F</math> and <math>\mathsf{W}F</math> are displayed horizontally, the way one normally orients functional arrows in a written text, and <math>\mathsf{W}</math> rolls the map <math>F</math> downward into the images that are associated with <math>\mathsf{W}F.</math> In Figure 32 the same information is redrawn so that the maps <math>F</math> and <math>\mathsf{W}F</math> flow down the page, and <math>\mathsf{W}</math> unfurls the map <math>F</math> rightward into domains that are the eminent purview of <math>\mathsf{W}F.</math>

{| align="center" border="0" cellpadding="20"
|
<pre>
o---------------------------------------o
| |
| |
| U% !W! !W!U% |
| o------------------>o |
| | | |
| | | |
| | | |
| | | |
| F | | !W!F |
| | | |
| | | |
| | | |
| v v |
| o------------------>o |
| X% !W! !W!X% |
| |
| |
o---------------------------------------o
Figure 32. Operator Diagram (2)
</pre>
|}

The latter arrangement, as exhibited in Figure 32, is more congruent with the thinking about operators that we shall do in the rest of this discussion, since all logical transformations from here on out will be pictured vertically, after the fashion of Figure 30.

====Differential Analysis of Propositions and Transformations====

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" | The resultant metaphysical problem now is this: ''Does the man go round the squirrel or not?''
| width="4%" |  
|-
| align="right" colspan="3" | — William James, ''Pragmatism'', [Jam, 43]
|}

The approach to the differential analysis of logical propositions and transformations of discourse to be pursued here is carried out in terms of particular operators <math>\mathsf{W}</math> that act on propositions <math>F</math> or on transformations <math>F</math> to yield the corresponding operator maps <math>\mathsf{W}F.</math> The operator results then become the subject of a series of further stages of analysis, which take them apart into their propositional components, rendering them as a set of purely logical constituents. After this is done, all the parts are then re-integrated to reconstruct the original object in the light of a more complete understanding, at least in ways that enable one to appreciate certain aspects of it with fresh insight.

* '''Remark on Strategy.''' At this point we run into a set of conceptual difficulties that force us to make a strategic choice in how we proceed. Part of the problem can be remedied by extending our discussion of tacit extensions to the transformational context. But the troubles that remain are much more obstinate and lead us to try two different types of solution. The approach that we develop first makes use of a variant type of extension operator, the ''trope extension'', to be defined below. This method is more conservative and requires less preparation, but has features which make it seem unsatisfactory in the long run. A more radical approach, but one with a better hope of long term success, makes use of the notion of ''contingency spaces''. These are an even more generous type of extended universe than the kind we currently use, but are defined subject to certain internal constraints. The extra work needed to set up this method forces us to put it off to a later stage. However, as a compromise, and to prepare the ground for the next pass, we call attention to the various conceptual difficulties as they arise along the way and try to give an honest estimate of how well our first approach deals with them.

We now describe in general terms the particular operators that are instrumental to this form of analysis. The main series of operators all have the form:

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"
|
<math>\begin{matrix}
\mathsf{W}
& : &
( U^\bullet \to X^\bullet )
& \to &
( \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet )
\end{matrix}</math>
|}

<br>

If we assume that the source universe <math>U^\bullet</math> and the target universe <math>X^\bullet</math> have finite dimensions <math>n</math> and <math>k,</math> respectively, then each operator <math>\mathsf{W}</math> is encompassed by the same abstract type:

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"
|
<math>\begin{matrix}
\mathsf{W}
& : &
( [\mathbb{B}^n] \to [\mathbb{B}^k] )
& \to &
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k \times \mathbb{D}^k] )
\end{matrix}</math>
|}

<br>

Since the range features of the operator result <math>\mathsf{W}F : [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k \times \mathbb{D}^k]</math> can be sorted by their ordinary versus differential qualities and the component maps can be examined independently, the complete operator <math>\mathsf{W}</math> can be separated accordingly into two components, in the form <math>\mathsf{W} = (\boldsymbol\varepsilon, \mathrm{W}).</math> Given a fixed context of source and target universes, <math>\boldsymbol\varepsilon</math> is always the same type of operator, a multiple component version of the tacit extension operators that were described earlier. In this context <math>\boldsymbol\varepsilon</math> has the form:

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"
|
<math>\begin{array}{lccccc}
\text{Concrete type}
& \boldsymbol\varepsilon
& : &
( U^\bullet \to X^\bullet )
& \to &
( \mathrm{E}U^\bullet \to X^\bullet )
\\[10pt]
\text{Abstract type}
& \boldsymbol\varepsilon
& : &
( [\mathbb{B}^n] \to [\mathbb{B}^k] )
& \to &
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k] )
\end{array}</math>
|}

<br>

On the other hand, the operator <math>\mathrm{W}</math> is specific to each <math>\mathsf{W}.</math> In this context <math>\mathrm{W}</math> always has the form:

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"
|
<math>\begin{array}{lccccc}
\text{Concrete type}
& W
& : &
( U^\bullet \to X^\bullet )
& \to &
( \mathrm{E}U^\bullet \to \mathrm{d}X^\bullet )
\\[10pt]
\text{Abstract type}
& W
& : &
( [\mathbb{B}^n] \to [\mathbb{B}^k] )
& \to &
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{D}^k] )
\end{array}</math>
|}

<br>

In the types just assigned to <math>\boldsymbol\varepsilon</math> and <math>\mathrm{W}</math> and by implication to their results <math>\boldsymbol\varepsilon F</math> and <math>\mathrm{W}F,</math> we have listed the most restrictive ranges defined for them rather than the more expansive target spaces that subsume these ranges. When there is need to recognize both, we may use type indications like the following:

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"
|
<math>\begin{matrix}
\boldsymbol\varepsilon F
& : &
( \mathrm{E}U^\bullet \to X^\bullet \subseteq \mathrm{E}X^\bullet )
& \cong &
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k] \subseteq [\mathbb{B}^k \times \mathbb{D}^k] )
\\[10pt]
WF
& : &
( \mathrm{E}U^\bullet \to \mathrm{d}X^\bullet \subseteq \mathrm{E}X^\bullet )
& \cong &
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{D}^k] \subseteq [\mathbb{B}^k \times \mathbb{D}^k] )
\end{matrix}</math>
|}

<br>

Hopefully, though, a general appreciation of these subsumptions will prevent us from having to make such declarations more often than absolutely necessary.

In giving names to these operators we try to preserve as much of the traditional nomenclature and as many of the classical associations as possible. The chief difficulty in doing this is occasioned by the distinction between the “sans serif” operators <math>\mathsf{W}</math> and their “serified” components <math>\mathrm{W},</math> which forces us to find two distinct but parallel sets of terminology. Here is a plan to that purpose. First, the component operators <math>\mathrm{W}</math> are named by analogy with the corresponding operators in the classical difference calculus. Next, the complete operators <math>\mathsf{W} = (\boldsymbol\varepsilon, \mathrm{W})</math> are assigned titles according to their roles in a geometric or trigonometric allegory, if only to ensure that the tangent functor, that belongs to this family and whose exposition we are still working toward, comes out fit with its customary name. Finally, the operator results <math>\mathsf{W}F</math> and <math>\mathrm{W}F</math> can be fixed in our frame of reference by tethering the operative adjective for <math>\mathsf{W}</math> or <math>\mathrm{W}</math> to the anchoring epithet “map”, in conformity with an already standard practice.

=====The Secant Operator : E=====

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
Mr. Peirce, after pointing out that our beliefs are really rules for action, said that, to develop a thought's meaning, we need only determine what conduct it is fitted to produce: that conduct is for us its sole significance.
| width="4%" |  
|-
| align="right" colspan="3" | — William James, ''Pragmatism'', [Jam, 46]
|}

Figures 33-i and 33-ii depict two stages in the form of analysis that will be applied to transformations throughout the remainder of this study. From now on our interest is staked on an operator denoted <math>{}^{\backprime\backprime} \mathsf{E} {}^{\prime\prime},</math> which receives the principal investment of analytic attention, and on the constituent parts of <math>\mathsf{E},</math> which derive their shares of significance as developed by the analysis. In the sequel, we refer to <math>\mathsf{E}</math> as the ''secant operator'', taking it for granted that a context has been chosen that defines its type. The secant operator has the component description <math>\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}),</math> and its active ingredient <math>\mathrm{E}</math> is known as the ''enlargement operator''. (Here, we name <math>\mathrm{E}</math> after the literal ancestor of the shift operator in the calculus of finite differences, defined so that <math>\mathrm{E}f(x) = f(x+1)</math> for any suitable function <math>f,</math> though of course the logical analogue that we take up here must have a rather different definition.)

{| align="center" border="0" cellpadding="10"
|
<pre>
U% $E$ $E$U% $E$U% $E$U%
o------------------>o============o============o
| | | |
| | | |
| | | |
| | | |
F | | $E$F = | $d$^0.F + | $r$^0.F
| | | |
| | | |
| | | |
v v v v
o------------------>o============o============o
X% $E$ $E$X% $E$X% $E$X%

Figure 33-i. Analytic Diagram (1)
</pre>
|}

{| align="center" border="0" cellpadding="10"
|
<pre>
U% $E$ $E$U% $E$U% $E$U% $E$U%
o------------------>o============o============o============o
| | | | |
| | | | |
| | | | |
| | | | |
F | | $E$F = | $d$^0.F + | $d$^1.F + | $r$^1.F
| | | | |
| | | | |
| | | | |
v v v v v
o------------------>o============o============o============o
X% $E$ $E$X% $E$X% $E$X% $E$X%

Figure 33-ii. Analytic Diagram (2)
</pre>
|}

In its action on universes <math>\mathsf{E}</math> yields the same result as <math>\mathrm{E},</math> a fact that can be expressed in equational form by writing <math>\mathsf{E}U^\bullet = \mathrm{E}U^\bullet</math> for any universe <math>U^\bullet.</math> Notice that the extended universes across the top and bottom of the diagram are indicated to be strictly identical, rather than requiring a corresponding decomposition for them. In a certain sense, the functional parts of <math>\mathsf{E}F</math> are partitioned into separate contexts that have to be re-integrated again, but the best image to use is that of making transparent copies of each universe and then overlapping their functional contents once more at the conclusion of the analysis, as suggested by the graphic conventions that are used at the top of Figure 30.

Acting on a transformation <math>F</math> from universe <math>U^\bullet</math> to universe <math>X^\bullet,</math> the operator <math>\mathsf{E}</math> determines a transformation <math>\mathsf{E}F</math> from <math>\mathsf{E}U^\bullet</math> to <math>\mathsf{E}X^\bullet.</math> The map <math>\mathsf{E}F</math> forms the main body of evidence to be investigated in performing a differential analysis of <math>F.</math> Because we shall frequently be focusing on small pieces of this map for considerable lengths of time, and consequently lose sight of the “big picture”, it is critically important to emphasize that the map <math>\mathsf{E}F</math> is a transformation that determines a relation from one extended universe into another. This means that we should not be satisfied with our understanding of a transformation <math>F</math> until we can lay out the full “parts diagram” of <math>\mathsf{E}F</math> along the lines of the generic frame in Figure 30.

Working within the confines of propositional calculus, it is possible to give an elementary definition of <math>\mathsf{E}F</math> by means of a system of propositional equations, as we now describe.

Given a transformation

{| align="center" cellpadding="6" width="90%"
| <math>F = (F_1, \ldots, F_k) : \mathbb{B}^n \to \mathbb{B}^k</math>
|}

of concrete type

{| align="center" cellpadding="6" width="90%"
| <math>F : [u_1, \ldots, u_n] \to [x_1, \ldots, x_k],</math>
|}

the transformation

{| align="center" cellpadding="6" width="90%"
| <math>\mathsf{E}F = (F_1, \ldots, F_k, \mathrm{E}F_1, \ldots, \mathrm{E}F_k) : \mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B}^k \times \mathbb{D}^k</math>
|}

of concrete type

{| align="center" cellpadding="6" width="90%"
| <math>\mathsf{E}F : [u_1, \dots, u_n, \mathrm{d}u_1, \dots, \mathrm{d}u_n] \to [x_1, \ldots, x_k, \mathrm{d}x_1, \ldots, \mathrm{d}x_k]</math>
|}

is defined by means of the following system of logical equations:

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"
|
<math>\begin{matrix}
x_1
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)
& = & F_1 (u_1, \ldots, u_n)
\\[4pt]
\cdots && \cdots && \cdots
\\[4pt]
x_k
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)
& = & F_k (u_1, \ldots, u_n)
\\[16pt]
\mathrm{d}x_1
& = & \mathrm{E}F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)
& = & F_1 (u_1 + \mathrm{d}u_1, \ldots, u_n + \mathrm{d}u_n)
\\[4pt]
\cdots && \cdots && \cdots
\\[4pt]
\mathrm{d}x_k
& = & \mathrm{E}F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)
& = & F_k (u_1 + \mathrm{d}u_1, \ldots, u_n + \mathrm{d}u_n)
\end{matrix}</math>
|}

<br>

It is important to note that this system of equations can be read as a conjunction of equational propositions, in effect, as a single proposition in the universe of discourse generated by all the named variables. Specifically, this is the universe of discourse over <math>2(n+k)</math> variables denoted by:

{| align="center" cellpadding="6" width="90%"
|
<math>\begin{matrix}
\mathrm{E}[\mathcal{U} \cup \mathcal{X}]
& = &
[u_1, \ldots, u_n, ~ x_1, \ldots, x_k, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n, ~ \mathrm{d}x_1, \ldots, \mathrm{d}x_k].
\end{matrix}</math>
|}

In this light, it should be clear that the system of equations defining <math>\mathsf{E}F</math> embodies, in a higher rank and differentially extended version, an analogy with the process of thematization that we treated earlier for propositions of type <math>F : \mathbb{B}^n \to \mathbb{B}.</math>

The entire collection of constraints that is represented in the above system of equations may be abbreviated by writing <math>\mathsf{E}F = (\boldsymbol\varepsilon F, \mathrm{E}F),</math> for any map <math>F.</math> This is tantamount to regarding <math>\mathsf{E}</math> as a complex operator, <math>\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}),</math> with a form of application that distributes each component of the operator to work on each component of the operand, as follows:

{| align="center" cellpadding="6" width="90%"
|
<math>\begin{matrix}
\mathsf{E}F
& = &
(\boldsymbol\varepsilon, \mathrm{E})F
& = &
(\boldsymbol\varepsilon F, \mathrm{E}F)
& = &
(\boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k, ~ \mathrm{E}F_1, \ldots, \mathrm{E}F_k).
\end{matrix}</math>
|}

Quite a lot of “thematic infrastructure” or interpretive information is being swept under the rug in the use of such abbreviations. When confusion arises about the meaning of such constructions, one always has recourse to the defining system of equations, in its totality a purely propositional expression. This means that the parenthesized argument lists, that were used in this context to build an image of multi-component transformations, should not be expected to determine a well-defined product in themselves but only to serve as reminders of the prior thematic decisions (choices of variable names, etc.) that have to be made in order to determine one. Accordingly, the argument list notation can be regarded as a kind of ''thematic frame'', an interpretive storage device that preserves the proper associations of concrete logical features between the extended universes at the source and target of <math>\mathsf{E}F.</math>

The generic notations <math>\mathsf{d}^0\!F, \mathsf{d}^1\!F, \ldots, \mathsf{d}^m\!F</math> in Figure 33 refer to the increasing orders of differentials that are extracted in the course of analyzing <math>F.</math> When the analysis is halted at a partial stage of development, notations like <math>\mathsf{r}^0\!F, \mathsf{r}^1\!F, \ldots, \mathsf{r}^m\!F</math> may be used to summarize the contributions to <math>\mathsf{E}F</math> that remain to be analyzed. The Figure illustrates a convention that makes <math>\mathsf{r}^m\!F,</math> in effect, the sum of all differentials of order strictly greater than <math>m.</math>

We next discuss the operators that figure into this form of analysis, describing their effects on transformations. In simplified or specialized contexts these operators tend to take on a variety of different names and notations, some of whose number we introduce along the way.

=====The Radius Operator : e=====

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
And the tangible fact at the root of all our thought-distinctions, however subtle, is that there is no one of them so fine as to consist in anything but a possible difference of practice.
| width="4%" |  
|-
| align="right" colspan="3" | — William James, ''Pragmatism'', [Jam, 46]
|}

The operator identified as <math>\mathrm{d}^0</math> in the analytic diagram (Figure 33) has the sole purpose of creating a proxy for <math>F</math> in the appropriately extended context. Construed in terms of its broadest components, <math>\mathrm{d}^0</math> is equivalent to the doubly tacit extension operator <math>(\boldsymbol\varepsilon, \boldsymbol\varepsilon),</math> in recognition of which let us redub it as <math>{}^{\backprime\backprime} \mathsf{e} {}^{\prime\prime}.</math> Pursuing a geometric analogy, we may refer to <math>\mathsf{e} =(\boldsymbol\varepsilon, \boldsymbol\varepsilon) = \mathrm{d}^0</math> as the ''radius operator''. The operation intended by all of these forms is defined by the following equation:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{lll}
\mathsf{e}F
& = &
(\boldsymbol\varepsilon, \boldsymbol\varepsilon)F
\\[4pt]
& = &
(\boldsymbol\varepsilon F, ~ \boldsymbol\varepsilon F)
\\[4pt]
& = &
(\boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k, ~ \boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k).
\end{array}</math>
|}

which is tantamount to the system of equations below.

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"
|
<math>\begin{matrix}
x_1
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)
& = & F_1 (u_1, \ldots, u_n)
\\[4pt]
\cdots && \cdots && \cdots
\\[4pt]
x_k
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)
& = & F_k (u_1, \ldots, u_n)
\\[16pt]
\mathrm{d}x_1
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)
& = & F_1 (u_1, \ldots, u_n)
\\[4pt]
\cdots && \cdots && \cdots
\\[4pt]
\mathrm{d}x_k
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)
& = & F_k (u_1, \ldots, u_n)
\end{matrix}</math>
|}

<br>

=====The Phantom of the Operators : η=====

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
I was wondering what the reason could be, when I myself raised my head and everything within me seemed drawn towards the Unseen, ''which was playing the most perfect music''!
| width="4%" |  
|-
| align="right" colspan="3" | — Gaston Leroux, ''The Phantom of the Opera'', [Ler, 81]
|}

We now describe an operator whose persistent but elusive action behind the scenes, whose slightly twisted and ambivalent character, and whose fugitive disposition, caught somewhere in flight between the arrantly negative and the positive but errant intent, has cost us some painstaking trouble to detect. In the end we shall place it among the other extensions and projections, as a shade among shadows, of muted tones and motley hue, that adumbrates its own thematic frame and paradoxically lights the way toward a whole new spectrum of values.

Given a transformation <math>F : [u_1, \ldots, u_n] \to [x_1, \dots, x_k],</math> we often have call to consider a family of related transformations, all having the form:

{| align="center" cellpadding="8" width="90%"
| <math>F^\dagger : [u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n] \to [\mathrm{d}x_1, \dots, \mathrm{d}x_k].</math>
|}

The operator <math>\eta</math> is introduced to deal with the simplest one of these maps:

{| align="center" cellpadding="8" width="90%"
| <math>\eta F : [u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n] \to [\mathrm{d}x_1, \ldots \mathrm{d}x_k],</math>
|}

which is defined by the following equations:

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"
|
<math>\begin{matrix}
\mathrm{d}x_1
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n)
& = & F_1 (u_1, \ldots, u_n)
\\[4pt]
\cdots && \cdots && \cdots
\\[4pt]
\mathrm{d}x_k
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n)
& = & F_k (u_1, \ldots, u_n)
\end{matrix}</math>
|}

<br>

In effect, the operator <math>\eta</math> is nothing but the stand-alone version of a procedure that is otherwise invoked subordinate to the work of the radius operator <math>\mathsf{e}.</math> Operating independently, <math>\eta</math> achieves precisely the same results that the second <math>\boldsymbol\varepsilon</math> in <math>(\boldsymbol\varepsilon, \boldsymbol\varepsilon)</math> accomplishes by working within the context of its ordered pair thematic frame. From this point on, because the use of <math>\boldsymbol\varepsilon</math> and <math>\eta</math> in this setting combines the aims of both the tacit and the thematic extensions, and because <math>\eta</math> reflects in regard to <math>\boldsymbol\varepsilon</math> little more than the application of a differential twist, a mere turn of phrase, we refer to <math>\eta</math> as the ''trope extension'' operator.

=====The Chord Operator : D=====

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
What difference would it practically make to any one if this notion rather than that notion were true?  If no practical difference whatever can be traced, then the alternatives mean practically the same thing, and all dispute is idle.
| width="4%" |  
|-
| align="right" colspan="3" | — William James, ''Pragmatism'', [Jam, 45]
|}

We come to an operator always immanent in this form of analysis, and remaining implicitly present in the entire proceeding.  It may appear once as a record:  a relic or revenant reprising the reminders of an earlier stage of development.  Or it may appear always as a resource:  a reserve or redoubt caching in advance an echo of what remains to be played out, cleared up, and requited in full at a future stage.  And all of this remains true whether or not we recall the key at any time, and whether or not the subtending theme is recited explicitly at any stage of play.

This is the operator that is referred to as <math>\mathsf{r}^0</math> in the initial stage of analysis (Figure 33-i) and that is expanded as <math>\mathsf{d}^1 + \mathsf{r}^1</math> in the subsequent step (Figure 33-ii). In congruence, but not quite harmony with our allusions of analogy that are not quite geometry, we call this the ''chord operator'' and denote it <math>\mathsf{D}.</math> In the more casual terms that are here introduced, <math>\mathsf{D}</math> is defined as the remainder of <math>\mathsf{E}</math> and <math>\mathsf{e}</math> and it assigns a due measure to each undertone of accord or discord that is struck between the note of enterprise <math>\mathsf{E}</math> and the bar of exigency <math>\mathsf{e}.</math>

The tension between these counterposed notions, in balance transient but regular in stridence, may be refracted along familiar lines, though never by any such fraction resolved. In this style we write <math>\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}),</math> calling <math>\mathrm{D}</math> the ''difference operator'' and noting that it plays a role in this realm of mutable and diverse discourse that is analogous to the part taken by the discrete difference operator in the ordinary difference calculus. Finally, we should note that the chord <math>\mathsf{D}</math> is not one that need be lost at any stage of development. At the <math>m^\text{th}</math> stage of play it can always be reconstituted in the following form:

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"
|
<math>\begin{array}{lll}
\mathsf{D}
& = & \mathsf{E} - \mathsf{e}
\\[6pt]
& = & \mathsf{r}^0
\\[6pt]
& = & \mathsf{d}^1 + \mathsf{r}^1
\\[6pt]
& = & \mathsf{d}^1 + \ldots + \mathsf{d}^m + \mathsf{r}^m
\\[6pt]
& = & \displaystyle \sum_{i=1}^m \mathsf{d}^i + \mathsf{r}^m
\end{array}</math>
|}

<br>

=====The Tangent Operator : T=====

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
They take part in scenes of whose significance they have no inkling. They are merely tangent to curves of history the beginnings and ends and forms of which pass wholly beyond their ken. So we are tangent to the wider life of things.
| width="4%" |  
|-
| align="right" colspan="3" | — William James, ''Pragmatism'', [Jam, 300]
|}

The operator tagged as <math>\mathsf{d}^1</math> in the analytic diagram (Figure 33) is called the ''tangent operator'' and is usually denoted in this text as <math>\mathsf{d}</math> or <math>\mathsf{T}.</math> Because it has the properties required to qualify as a functor, namely, preserving the identity element of the composition operation and the articulated form of every composition of transformations, it also earns the title of a ''tangent functor''. According to the custom adopted here, we dissect it as <math>\mathsf{T} = \mathsf{d} = (\boldsymbol\varepsilon, \mathrm{d}),</math> where <math>\mathrm{d}</math> is the operator that yields the first order differential <math>\mathrm{d}F</math> when applied to a transformation <math>F,</math> and whose name is legion.

Figure 34 illustrates a stage of analysis where we ignore everything but the tangent functor <math>\mathsf{T}</math> and attend to it chiefly as it bears on the first order differential <math>\mathrm{d}F</math> in the analytic expansion of <math>F.</math> In this situation we often refer to the extended universes <math>\mathrm{E}U^\bullet</math> and <math>\mathrm{E}X^\bullet</math> under the equivalent designations <math>\mathsf{T}U^\bullet</math> and <math>\mathsf{T}X^\bullet,</math> respectively. The purpose of the tangent functor <math>\mathsf{T}</math> is to extract the tangent map <math>\mathsf{T}F</math> at each point of <math>U^\bullet,</math> and the tangent map <math>\mathsf{T}F = (\boldsymbol\varepsilon, \mathrm{d})F</math> tells us not only what the transformation <math>F</math> is doing at each point of the universe <math>U^\bullet</math> but also what <math>F</math> is doing to states in the neighborhood of that point, approximately, linearly, and relatively speaking.

{| align="center" border="0" cellpadding="10"
|
<pre>
U% $T$ $T$U% $T$U%
o------------------>o============o
| | |
| | |
| | |
| | |
F | | $T$F = | <!e!, d> F
| | |
| | |
| | |
v v v
o------------------>o============o
X% $T$ $T$X% $T$X%

Figure 34. Tangent Functor Diagram
</pre>
|}

<ul><li><b>NB.</b>  There is one aspect of the preceding construction which remains especially problematic.  Why did we define the operators <math>\mathrm{W}</math> in <math>\{ \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}</math> so that the ranges of their resulting maps all fall within the realms of differential quality, even fabricating a variant of the tacit extension operator to have that character?  Clearly, not all of the operator maps <math>\mathrm{W}F</math> have equally good reasons for placing their values in differential stocks.  The reason for it appears to be that, without doing this, we cannot justify the comparison and combination of their functional values in the various analytic steps.  By default, only those values in the same functional component can be brought into algebraic modes of interaction.  Up till now the only mechanism provided for their broader association has been a purely logical one, their common placement in a target universe of discourse, but the task of converting that logical circumstance into algebraic forms of application has not yet been taken up.</li></ul>

----
<div align="center">
• [[Differential Logic and Dynamic Systems • Overview|Overview]]
• [[Differential Logic and Dynamic Systems • Part 1|Part 1]]
• [[Differential Logic and Dynamic Systems • Part 2|Part 2]]
• [[Differential Logic and Dynamic Systems • Part 3|Part 3]]
• [[Differential Logic and Dynamic Systems • Part 4|Part 4]]
• [[Differential Logic and Dynamic Systems • Part 5|Part 5]]
• [[Differential Logic and Dynamic Systems • Appendices|Appendices]]
• [[Differential Logic and Dynamic Systems • References|References]]
• [[Differential Logic and Dynamic Systems • Document History|Document History]]
•
</div>
----

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Differential Logic and Dynamic Systems • Part 2

2026-02-02T16:30:29Z

Jon Awbrey: del {{anchor| ,,,

Differential Logic and Dynamic Systems • Part 2

2026-02-02T16:26:10Z

Jon Awbrey: + Differential Logic and Dynamic Systems • Part 2

Differential Logic and Dynamic Systems • Overview

2026-02-02T16:24:11Z

Jon Awbrey: File:Diff Log Dyn Sys • Tangent Functor Ferris Wheel 2.0.png

__NOTOC__<b>Author: [[User:Jon Awbrey|Jon Awbrey]]</b>

----
<div align="center">
• [[Differential Logic and Dynamic Systems • Overview|Overview]]
• [[Differential Logic and Dynamic Systems • Part 1|Part 1]]
• [[Differential Logic and Dynamic Systems • Part 2|Part 2]]
• [[Differential Logic and Dynamic Systems • Part 3|Part 3]]
• [[Differential Logic and Dynamic Systems • Part 4|Part 4]]
• [[Differential Logic and Dynamic Systems • Part 5|Part 5]]
• [[Differential Logic and Dynamic Systems • Appendices|Appendices]]
• [[Differential Logic and Dynamic Systems • References|References]]
• [[Differential Logic and Dynamic Systems • Document History|Document History]]
•
</div>
----

{| align="center" cellpadding="10"
| [[File:Diff Log Dyn Sys • Tangent Functor Ferris Wheel 2.0.png|448px]]
|}

{| style="height:36px; width:100%"
| align="left" | ''Stand and unfold yourself.''
| align="right" | Hamlet: Francisco—1.1.2
|}

In modeling intelligent systems, whether we are trying to understand a natural system or engineer an artificial system, there has long been a tension or trade‑off between dynamic paradigms and symbolic paradigms.  Dynamic models take their cue from physics, using quantitative measures and differential equations to model the evolution of a system's state through time.  Symbolic models use logical methods to describe systems and their agents in qualitative terms, deriving logical consequences of a system's description or an agent's state of information.  Logic-based systems have tended to be static in character, largely because we have lacked a proper logical analogue of differential calculus.  The work laid out in this report is intended to address that lack.

This article develops a differential extension of propositional calculus and applies it to the analysis of dynamic systems whose states are described in qualitative logical terms.  The work pursued here is coordinated with a parallel application focusing on neural network systems but the dependencies are arranged to make the present article the main and the more self-contained work, to serve as a conceptual frame and a technical background for the network project.

<h2>[[Differential Logic and Dynamic Systems • Part 1#Review and Transition|Review and Transition]]</h2>

<h2>[[Differential Logic and Dynamic Systems • Part 1#A Functional Conception of Propositional Calculus|A Functional Conception of Propositional Calculus]]</h2>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 1#Qualitative Logic and Quantitative Analogy|Qualitative Logic and Quantitative Analogy]]</h3>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 1#Philosophy of Notation : Formal Terms and Flexible Types|Philosophy of Notation : Formal Terms and Flexible Types]]</h3>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 1#Special Classes of Propositions|Special Classes of Propositions]]</h3>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 1#Basis Relativity and Type Ambiguity|Basis Relativity and Type Ambiguity]]</h3>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 1#The Analogy Between Real and Boolean Types|The Analogy Between Real and Boolean Types]]</h3>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 1#Theory of Control and Control of Theory|Theory of Control and Control of Theory]]</h3>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 1#Propositions as Types and Higher Order Types|Propositions as Types and Higher Order Types]]</h3>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 1#Reality at the Threshold of Logic|Reality at the Threshold of Logic]]</h3>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 1#Tables of Propositional Forms|Tables of Propositional Forms]]</h3>

<h2>[[Differential Logic and Dynamic Systems • Part 1#A Differential Extension of Propositional Calculus|A Differential Extension of Propositional Calculus]]</h2>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 1#Differential Propositions : Qualitative Analogues of Differential Equations|Differential Propositions : Qualitative Analogues of Differential Equations]]</h3>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 1#An Interlude on the Path|An Interlude on the Path]]</h3>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 1#The Extended Universe of Discourse|The Extended Universe of Discourse]]</h3>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 1#Intentional Propositions|Intentional Propositions]]</h3>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 1#Life on Easy Street|Life on Easy Street]]</h3>

<h2>[[Differential Logic and Dynamic Systems • Part 2#Back to the Beginning : Exemplary Universes|Back to the Beginning : Exemplary Universes]]</h2>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 2#A One-Dimensional Universe|A One-Dimensional Universe]]</h3>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 2#Example 1. A Square Rigging|Example 1. A Square Rigging]]</h3>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 2#Back to the Feature|Back to the Feature]]</h3>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 2#Tacit Extensions|Tacit Extensions]]</h3>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 2#Example 2. Drives and Their Vicissitudes|Example 2. Drives and Their Vicissitudes]]</h3>

<h2>[[Differential Logic and Dynamic Systems • Part 3#Transformations of Discourse|Transformations of Discourse]]</h2>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 3#Foreshadowing Transformations : Extensions and Projections of Discourse|Foreshadowing Transformations : Extensions and Projections of Discourse]]</h3>

<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Part 3#Extension from 1 to 2 Dimensions|Extension from 1 to 2 Dimensions]]</h4>

<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Part 3#Extension from 2 to 4 Dimensions|Extension from 2 to 4 Dimensions]]</h4>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 3#Thematization of Functions : And a Declaration of Independence for Variables|Thematization of Functions : And a Declaration of Independence for Variables]]</h3>

<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Part 3#Thematization : Venn Diagrams|Thematization : Venn Diagrams]]</h4>

<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Part 3#Thematization : Truth Tables|Thematization : Truth Tables]]</h4>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 3#Propositional Transformations|Propositional Transformations]]</h3>

<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Part 3#Alias and Alibi Transformations|Alias and Alibi Transformations]]</h4>

<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Part 3#Transformations of General Type|Transformations of General Type]]</h4>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 3#Analytic Expansions : Operators and Functors|Analytic Expansions : Operators and Functors]]</h3>

<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Part 3#Operators on Propositions and Transformations|Operators on Propositions and Transformations]]</h4>

<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Part 3#Differential Analysis of Propositions and Transformations|Differential Analysis of Propositions and Transformations]]</h4>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Part 3#The Secant Operator : E|The Secant Operator : E]]</h5>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Part 3#The Radius Operator : e|The Radius Operator : e]]</h5>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Part 3#The Phantom of the Operators : η|The Phantom of the Operators : η]]</h5>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Part 3#The Chord Operator : D|The Chord Operator : D]]</h5>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Part 3#The Tangent Operator : T|The Tangent Operator : T]]</h5>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 4#Transformations of Type B² → B¹|Transformations of Type B² → B¹]]</h3>

<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Part 4#Analytic Expansion of Conjunction|Analytic Expansion of Conjunction]]</h4>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Part 4#Tacit Extension of Conjunction|Tacit Extension of Conjunction]]</h5>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Part 4#Enlargement Map of Conjunction|Enlargement Map of Conjunction]]</h5>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Part 4#Digression : Reflection on Use and Mention|Digression : Reflection on Use and Mention]]</h5>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Part 4#Difference Map of Conjunction|Difference Map of Conjunction]]</h5>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Part 4#Differential of Conjunction|Differential of Conjunction]]</h5>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Part 4#Remainder of Conjunction|Remainder of Conjunction]]</h5>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Part 4#Summary of Conjunction|Summary of Conjunction]]</h5>

<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Part 4#Analytic Series : Coordinate Method|Analytic Series : Coordinate Method]]</h4>

<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Part 4#Analytic Series : Recap|Analytic Series : Recap]]</h4>

<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Part 4#Terminological Interlude|Terminological Interlude]]</h4>

<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Part 4#End of Perfunctory Chatter : Time to Roll the Clip!|End of Perfunctory Chatter : Time to Roll the Clip!]]</h4>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Part 4#Operator Maps : Areal Views|Operator Maps : Areal Views]]</h5>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Part 4#Operator Maps : Box Views|Operator Maps : Box Views]]</h5>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Part 4#Operator Diagrams for the Conjunction J = uv|Operator Diagrams for the Conjunction ''J'' = ''uv'']]</h5>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 5#Taking Aim at Higher Dimensional Targets|Taking Aim at Higher Dimensional Targets]]</h3>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 5#Transformations of Type B² → B²|Transformations of Type B² → B²]]</h3>

<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Part 5#Logical Transformations|Logical Transformations]]</h4>

<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Part 5#Local Transformations|Local Transformations]]</h4>

<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Part 5#Difference Operators and Tangent Functors|Difference Operators and Tangent Functors]]</h4>

<h2>[[Differential Logic and Dynamic Systems • Part 5#Epilogue, Enchoiry, Exodus|Epilogue, Enchoiry, Exodus]]</h2>

<h2>[[Differential Logic and Dynamic Systems • Appendices#Appendices|Appendices]]</h2>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Appendices#Appendix 1. Propositional Forms and Differential Expansions|Appendix 1. Propositional Forms and Differential Expansions]]</h3>

<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Appendices#Table A1. Propositional Forms on Two Variables|Table A1. Propositional Forms on Two Variables]]</h4>

<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Appendices#Table A2. Propositional Forms on Two Variables|Table A2. Propositional Forms on Two Variables]]</h4>

<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Appendices#Table A3. Ef Expanded Over Differential Features|Table A3. E''f'' Expanded Over Differential Features]]</h4>

<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Appendices#Table A4. Df Expanded Over Differential Features|Table A4. D''f'' Expanded Over Differential Features]]</h4>

<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Appendices#Table A5. Ef Expanded Over Ordinary Features|Table A5. E''f'' Expanded Over Ordinary Features]]</h4>

<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Appendices#Table A6. Df Expanded Over Ordinary Features|Table A6. D''f'' Expanded Over Ordinary Features]]</h4>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Appendices#Appendix 2. Differential Forms|Appendix 2. Differential Forms]]</h3>

<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Appendices#Table A7. Differential Forms Expanded on a Logical Basis|Table A7. Differential Forms Expanded on a Logical Basis]]</h4>

<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Appendices#Table A8. Differential Forms Expanded on an Algebraic Basis|Table A8. Differential Forms Expanded on an Algebraic Basis]]</h4>

<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Appendices#Table A9. Tangent Proposition as Pointwise Linear Approximation|Table A9. Tangent Proposition as Pointwise Linear Approximation]]</h4>

<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Appendices#Table A10. Taylor Series Expansion Df = df + d²f|Table A10. Taylor Series Expansion D''f'' = d''f'' + d²''f'']]</h4>

<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Appendices#Table A11. Partial Differentials and Relative Differentials|Table A11. Partial Differentials and Relative Differentials]]</h4>

<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Appendices#Table A12. Detail of Calculation for the Difference Map|Table A12. Detail of Calculation for the Difference Map]]</h4>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Appendices#Appendix 3. Computational Details|Appendix 3. Computational Details]]</h3>

<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Appendices#Operator Maps for the Logical Conjunction f8(u, v)|Operator Maps for the Logical Conjunction ''f''<sub>8</sub>(''u'', ''v'')]]</h4>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of εf8|Computation of ε''f''<sub>8</sub>]]</h5>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of Ef8|Computation of E''f''<sub>8</sub>]]</h5>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of Df8|Computation of D''f''<sub>8</sub>]]</h5>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of df8 2|Computation of d''f''<sub>8</sub>]]</h5>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of rf8|Computation of r''f''<sub>8</sub>]]</h5>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation Summary for Conjunction|Computation Summary for Conjunction]]</h5>

<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Appendices#Operator Maps for the Logical Equality f9(u, v)|Operator Maps for the Logical Equality ''f''<sub>9</sub>(''u'', ''v'')]]</h4>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of εf9|Computation of ε''f''<sub>9</sub>]]</h5>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of Ef9|Computation of E''f''<sub>9</sub>]]</h5>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of Df9|Computation of D''f''<sub>9</sub>]]</h5>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of df9 2|Computation of d''f''<sub>9</sub>]]</h5>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of rf9|Computation of r''f''<sub>9</sub>]]</h5>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation Summary for Equality|Computation Summary for Equality]]</h5>

<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Appendices#Operator Maps for the Logical Implication f11(u, v)|Operator Maps for the Logical Implication ''f''<sub>11</sub>(''u'', ''v'')]]</h4>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of εf11|Computation of ε''f''<sub>11</sub>]]</h5>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of Ef11|Computation of E''f''<sub>11</sub>]]</h5>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of Df11|Computation of D''f''<sub>11</sub>]]</h5>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of df11 2|Computation of d''f''<sub>11</sub>]]</h5>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of rf11|Computation of r''f''<sub>11</sub>]]</h5>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation Summary for Implication|Computation Summary for Implication]]</h5>

<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Appendices#Operator Maps for the Logical Disjunction f14(u, v)|Operator Maps for the Logical Disjunction ''f''<sub>14</sub>(''u'', ''v'')]]</h4>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of εf14|Computation of ε''f''<sub>14</sub>]]</h5>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of Ef14|Computation of E''f''<sub>14</sub>]]</h5>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of Df14|Computation of D''f''<sub>14</sub>]]</h5>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of df14 2|Computation of d''f''<sub>14</sub>]]</h5>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of rf14|Computation of r''f''<sub>14</sub>]]</h5>

<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation Summary for Disjunction|Computation Summary for Disjunction]]</h5>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Appendices#Appendix 4. Source Materials|Appendix 4. Source Materials]]</h3>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Appendices#Appendix 5. Various Definitions of the Tangent Vector|Appendix 5. Various Definitions of the Tangent Vector]]</h3>

<h2>[[Differential Logic and Dynamic Systems • References#References|References]]</h2>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • References#Works Cited|Works Cited]]</h3>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • References#Works Consulted|Works Consulted]]</h3>

<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • References#Incidental Works|Incidental Works]]</h3>

<h2>[[Differential Logic and Dynamic Systems • Document History#Document History|Document History]]</h2>

----
<div align="center">
• [[Differential Logic and Dynamic Systems • Overview|Overview]]
• [[Differential Logic and Dynamic Systems • Part 1|Part 1]]
• [[Differential Logic and Dynamic Systems • Part 2|Part 2]]
• [[Differential Logic and Dynamic Systems • Part 3|Part 3]]
• [[Differential Logic and Dynamic Systems • Part 4|Part 4]]
• [[Differential Logic and Dynamic Systems • Part 5|Part 5]]
• [[Differential Logic and Dynamic Systems • Appendices|Appendices]]
• [[Differential Logic and Dynamic Systems • References|References]]
• [[Differential Logic and Dynamic Systems • Document History|Document History]]
•
</div>
----

[[Category:Adaptive systems]]
[[Category:Artificial intelligence]]
[[Category:Boolean algebra]]
[[Category:Boolean functions]]
[[Category:Category theory]]
[[Category:Combinatorics]]
[[Category:Computation theory]]
[[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:Logic]]
[[Category:Logical graphs]]
[[Category:Neural networks]]
[[Category:Peirce, Charles Sanders]]
[[Category:Semiotics]]
[[Category:Systems theory]]
[[Category:Visualization]]

Differential Logic and Dynamic Systems • Part 1

2026-02-02T16:22:45Z

Jon Awbrey: File:Diff Log Dyn Sys • Tangent Functor Ferris Wheel 2.0.png

<b>Author: [[User:Jon Awbrey|Jon Awbrey]]</b>

----
<div align="center">
• [[Differential Logic and Dynamic Systems • Overview|Overview]]
• [[Differential Logic and Dynamic Systems • Part 1|Part 1]]
• [[Differential Logic and Dynamic Systems • Part 2|Part 2]]
• [[Differential Logic and Dynamic Systems • Part 3|Part 3]]
• [[Differential Logic and Dynamic Systems • Part 4|Part 4]]
• [[Differential Logic and Dynamic Systems • Part 5|Part 5]]
• [[Differential Logic and Dynamic Systems • Appendices|Appendices]]
• [[Differential Logic and Dynamic Systems • References|References]]
• [[Differential Logic and Dynamic Systems • Document History|Document History]]
•
</div>
----

{| align="center" cellpadding="10"
| [[File:Diff Log Dyn Sys • Tangent Functor Ferris Wheel 2.0.png|448px]]
|}

{| style="height:36px; width:100%"
| align="left" | ''Stand and unfold yourself.''
| align="right" | Hamlet: Francisco—1.1.2
|}

In modeling intelligent systems, whether we are trying to understand a natural system or engineer an artificial system, there has long been a tension or trade‑off between dynamic paradigms and symbolic paradigms.  Dynamic models take their cue from physics, using quantitative measures and differential equations to model the evolution of a system's state through time.  Symbolic models use logical methods to describe systems and their agents in qualitative terms, deriving logical consequences of a system's description or an agent's state of information.  Logic-based systems have tended to be static in character, largely because we have lacked a proper logical analogue of differential calculus.  The work laid out in this report is intended to address that lack.

This article develops a differential extension of propositional calculus and applies it to the analysis of dynamic systems whose states are described in qualitative logical terms.  The work pursued here is coordinated with a parallel application focusing on neural network systems but the dependencies are arranged to make the present article the main and the more self-contained work, to serve as a conceptual frame and a technical background for the network project.

==Review and Transition==

This note continues a previous discussion on the problem of dealing with change and diversity in logic-based intelligent systems.  It is useful to begin by summarizing essential material from previous reports.

Table 1 outlines a notation for propositional calculus based on two types of logical connectives, both of variable <math>k</math>-ary scope.

* A bracketed list of propositional expressions in the form <math>\texttt{(} e_1 \texttt{,} e_2 \texttt{,} \ldots \texttt{,} e_{k-1} \texttt{,} e_k \texttt{)}</math> indicates that exactly one of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> is false.

* A concatenation of propositional expressions in the form <math>e_1 ~ e_2 ~ \ldots ~ e_{k-1} ~ e_k</math> indicates that all of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> are true, in other words, that their [[logical conjunction]] is true.

All other propositional connectives can be obtained in a very efficient style of representation through combinations of these two forms.  Strictly speaking, the concatenation form is dispensable in light of the bracketed form but it is convenient to maintain it as an abbreviation of more complicated bracket expressions.

{| align="center" style="font-size:larger; text-align:center; width:100%"
| height="20px" | <math>\text{Table 1. Syntax and Semantics of a Calculus for Propositional Logic}</math>
|-
| [[File:Syntax and Semantics of a Calculus for Propositional Logic 4.0.png|600px]]
|}

Table 1 is the first of several “Rosetta Stones” we'll use in this discussion to translate between different languages for the same subject matters.  In this case the Table displays equivalent expressions for simple examples of propositional forms in four notations for propositional calculus.

* Column 1 shows the logical graphs used to represent a number of simple propositional forms.
* Column 2 shows the traverse strings corresponding to the logical graphs in Column 1.
* Column 3 interprets the graph and string by means of conventional verbal formulas.
* Column 4 translates the interpretation into a number of symbolic notations.

This treatment of propositional logic is derived from the work of C.S. Peirce [P1, P2], who gave this approach an extensive development in his graphical systems of predicate, relational, and modal logic [Rob]. More recently, these ideas were revived and supplemented in an alternative interpretation by George Spencer-Brown [SpB]. Both of these authors used other forms of enclosure where I use parentheses, but the structural topologies of expression and the functional varieties of interpretation are fundamentally the same.

While working with expressions solely in propositional calculus, it is easiest to use plain parentheses for logical connectives. In contexts where parentheses are needed for other purposes “teletype” parentheses <math>\texttt{(} \ldots \texttt{)}</math> or barred parentheses <math>(\!| \ldots |\!)</math> may be used for logical operators.

The briefest expression for logical truth is the empty word, usually denoted by <math>{}^{\backprime\backprime} \boldsymbol\varepsilon {}^{\prime\prime}</math> or <math>{}^{\backprime\backprime} \boldsymbol\lambda {}^{\prime\prime}</math> in formal languages, where it forms the identity element for concatenation. To make it visible in this text, it may be denoted by the equivalent expression <math>{}^{\backprime\backprime} \texttt{((} ~ \texttt{))} {}^{\prime\prime},</math> or, especially if operating in an algebraic context, by a simple <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}.</math> Also when working in an algebraic mode, the plus sign <math>{}^{\backprime\backprime} + {}^{\prime\prime}</math> may be used for [[exclusive disjunction]]. For example, we have the following paraphrases of algebraic expressions by bracket expressions:

{| align="center" cellpadding="6" style="text-align:center"
|
<math>\begin{matrix}
x + y ~=~ \texttt{(} x, y \texttt{)}
\\[6pt]
x + y + z ~=~ \texttt{((} x, y \texttt{)}, z \texttt{)} ~=~ \texttt{(} x, \texttt{(} y, z \texttt{))}
\end{matrix}</math>
|}

It is important to note that the last expressions are not equivalent to the triple bracket <math>\texttt{(} x, y, z \texttt{)}.</math>

<b>Note.</b> The usage that one often sees, of a plus sign "<math>+</math>" to represent inclusive disjunction, and the reference to this operation as ''boolean addition'', is a misnomer on at least two counts. Boole used the plus sign to represent exclusive disjunction (at any rate, an operation of aggregation restricted in its logical interpretation to cases where the represented sets are disjoint (Boole, 32)), as any mathematician with a sensitivity to the ring and field properties of algebra would do:

<blockquote>
The expression <math>x + y</math> seems indeed uninterpretable, unless it be assumed that the things represented by <math>x</math> and the things represented by <math>y</math> are entirely separate; that they embrace no individuals in common. (Boole, 66).
</blockquote>

It was only later that Peirce and Jevons treated inclusive disjunction as a fundamental operation, but these authors, with a respect for the algebraic properties that were already associated with the plus sign, used a variety of other symbols for inclusive disjunction (Sty, 177, 189). It seems to have been Schröder who later reassigned the plus sign to inclusive disjunction (Sty, 208). Additional information, discussion, and references can be found in (Boole) and (Sty, 177–263). Aside from these historical points, which never really count against a current practice that has gained a life of its own, this usage does have a further disadvantage of cutting or confounding the lines of communication between algebra and logic. For this reason, it will be avoided here.

==A Functional Conception of Propositional Calculus==

{| width="100%" cellpadding="0" cellspacing="0"
| width="40%" |  
| width="60%" |
Out of the dimness opposite equals advance . . . .<br>
     Always substance and increase,<br>
Always a knit of identity . . . . always distinction . . . .<br>
     always a breed of life.
|-
|  
| align="right" | — Walt Whitman, ''Leaves of Grass'', [Whi, 28]
|}

In the general case, we start with a set of logical features <math>\{a_1, \ldots, a_n\}</math> that represent properties of objects or propositions about the world. In concrete examples the features <math>\{a_i\}</math> commonly appear as capital letters from an ''alphabet'' like <math>\{A, B, C, \ldots\}</math> or as meaningful words from a linguistic ''vocabulary'' of codes. This language can be drawn from any sources, whether natural, technical, or artificial in character and interpretation. In the application to dynamic systems we tend to use the letters <math>\{x_1, \ldots, x_n\}</math> as our coordinate propositions, and to interpret them as denoting properties of a system's ''state'', that is, as propositions about its location in configuration space. Because I have to consider non-deterministic systems from the outset, I often use the word ''state'' in a loose sense, to denote the position or configuration component of a contemplated state vector, whether or not it ever gets a deterministic completion.

The set of logical features <math>\{a_1, \ldots, a_n\}</math> provides a basis for generating an <math>n</math>-dimensional ''[[universe of discourse]]'' that I denote as <math>[a_1, \ldots, a_n].</math> It is useful to consider each universe of discourse as a unified categorical object that incorporates both the set of points <math>\langle a_1, \ldots, a_n \rangle</math> and the set of propositions <math>f : \langle a_1, \ldots, a_n \rangle \to \mathbb{B}</math> that are implicit with the ordinary picture of a venn diagram on <math>n</math> features. Thus, we may regard the universe of discourse <math>[a_1, \ldots, a_n]</math> as an ordered pair having the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}),</math> and we may abbreviate this last type designation as <math>\mathbb{B}^n\ +\!\to \mathbb{B},</math> or even more succinctly as <math>[\mathbb{B}^n].</math> (Used this way, the angle brackets <math>\langle\ldots\rangle</math> are referred to as ''generator brackets''.)

Table 2 exhibits the scheme of notation I use to formalize the domain of propositional calculus, corresponding to the logical content of truth tables and venn diagrams. Although it overworks the square brackets a bit, I also use either one of the equivalent notations <math>[n]</math> or <math>\mathbf{n}</math> to denote the data type of a finite set on <math>n</math> elements.

<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 2.} ~~ \text{Propositional Calculus : Basic Notation}</math>
|- style="height:40px; background:ghostwhite"
| <math>\text{Symbol}</math>
| <math>\text{Notation}</math>
| <math>\text{Description}</math>
| <math>\text{Type}</math>
|-
| <math>\mathfrak{A}</math>
| <math>\{ {}^{\backprime\backprime} a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} a_n {}^{\prime\prime} \}</math>
| <math>\text{Alphabet}</math>
| <math>[n] = \mathbf{n}</math>
|-
| <math>\mathcal{A}</math>
| <math>\{ a_1, \ldots, a_n \}</math>
| <math>\text{Basis}</math>
| <math>[n] = \mathbf{n}</math>
|-
| <math>A_i</math>
| <math>\{ \texttt{(} a_i \texttt{)}, a_i \}</math>
| <math>\text{Dimension}~ i</math>
| <math>\mathbb{B}</math>
|-
| <math>A</math>
|
<math>\begin{matrix}
\langle \mathcal{A} \rangle
\\[2pt]
\langle a_1, \ldots, a_n \rangle
\\[2pt]
\{ (a_1, \ldots, a_n) \}
\\[2pt]
A_1 \times \ldots \times A_n
\\[2pt]
\textstyle \prod_{i=1}^n A_i
\end{matrix}</math>
|
<math>\begin{matrix}
\text{Set of cells},
\\[2pt]
\text{coordinate tuples},
\\[2pt]
\text{points, or vectors}
\\[2pt]
\text{in the universe}
\\[2pt]
\text{of discourse}
\end{matrix}</math>
| <math>\mathbb{B}^n</math>
|-
| <math>A^*</math>
| <math>(\mathrm{hom} : A \to \mathbb{B})</math>
| <math>\text{Linear functions}</math>
| <math>(\mathbb{B}^n)^* \cong \mathbb{B}^n</math>
|-
| <math>A^\uparrow</math>
| <math>(A \to \mathbb{B})</math>
| <math>\text{Boolean functions}</math>
| <math>\mathbb{B}^n \to \mathbb{B}</math>
|-
| <math>A^\bullet</math>
|
<math>\begin{matrix}
[\mathcal{A}]
\\[2pt]
(A, A^\uparrow)
\\[2pt]
(A ~+\!\to \mathbb{B})
\\[2pt]
(A, (A \to \mathbb{B}))
\\[2pt]
[a_1, \ldots, a_n]
\end{matrix}</math>
|
<math>\begin{matrix}
\text{Universe of discourse}
\\[2pt]
\text{based on the features}
\\[2pt]
\{ a_1, \ldots, a_n \}
\end{matrix}</math>
|
<math>\begin{matrix}
(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))
\\[2pt]
(\mathbb{B}^n ~+\!\to \mathbb{B})
\\[2pt]
[\mathbb{B}^n]
\end{matrix}</math>
|}

<br>

===Qualitative Logic and Quantitative Analogy===

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
''Logical'', however, is used in a third sense, which is at once more vital and more practical; to denote, namely, the systematic care, negative and positive, taken to safeguard reflection so that it may yield the best results under the given conditions.
| width="4%" |  
|-
| align="right" colspan="3" | — John Dewey, ''How We Think'', [Dew, 56]
|}

These concepts and notations may now be explained in greater detail. In order to begin as simply as possible, let us distinguish two levels of analysis and set out initially on the easier path. On the first level of analysis we take spaces like <math>\mathbb{B},</math> <math>\mathbb{B}^n,</math> and <math>(\mathbb{B}^n \to \mathbb{B})</math> at face value and treat them as the primary objects of interest. On the second level of analysis we use these spaces as coordinate charts for talking about points and functions in more fundamental spaces.

A pair of spaces, of types <math>\mathbb{B}^n</math> and <math>(\mathbb{B}^n \to \mathbb{B}),</math> give typical expression to everything we commonly associate with the ordinary picture of a venn diagram. The dimension, <math>n,</math> counts the number of “circles” or simple closed curves that are inscribed in the universe of discourse, corresponding to its relevant logical features or basic propositions. Elements of type <math>\mathbb{B}^n</math> correspond to what are often called propositional ''interpretations'' in logic, that is, the different assignments of truth values to sentence letters. Relative to a given universe of discourse, these interpretations are visualized as its ''cells'', in other words, the smallest enclosed areas or undivided regions of the venn diagram. The functions <math>f : \mathbb{B}^n \to \mathbb{B}</math> correspond to the different ways of shading the venn diagram to indicate arbitrary propositions, regions, or sets. Regions included under a shading indicate the ''models'', and regions excluded represent the ''non-models'' of a proposition. To recognize and formalize the natural cohesion of these two layers of concepts into a single universe of discourse, we introduce the type notations <math>[\mathbb{B}^n] = \mathbb{B}^n\ +\!\to \mathbb{B}</math> to stand for the pair of types <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})).</math> The resulting “stereotype” serves to frame the universe of discourse as a unified categorical object, and makes it subject to prescribed sets of evaluations and transformations (categorical morphisms or ''arrows'') that affect the universe of discourse as an integrated whole.

Most of the time we can serve the algebraic, geometric, and logical interests of our study without worrying about their occasional conflicts and incidental divergences. The conventions and definitions already set down will continue to cover most of the algebraic and functional aspects of our discussion, but to handle the logical and qualitative aspects we will need to add a few more. In general, abstract sets may be denoted by gothic, greek, or script capital variants of <math>A, B, C,</math> and so on, with their elements being denoted by a corresponding set of subscripted letters in plain lower case, for example, <math>\mathcal{A} = \{a_i\}.</math> Most of the time, a set such as <math>\mathcal{A} = \{a_i\}</math> will be employed as the ''alphabet'' of a [[formal language]]. These alphabet letters serve to name the logical features (properties or propositions) that generate a particular universe of discourse. When we want to discuss the particular features of a universe of discourse, beyond the abstract designation of a type like <math>(\mathbb{B}^n\ +\!\to \mathbb{B}),</math> then we may use the following notations. If <math>\mathcal{A} = \{a_1, \ldots, a_n\}</math> is an alphabet of logical features, then <math>A = \langle \mathcal{A} \rangle = \langle a_1, \ldots, a_n \rangle</math> is the set of interpretations, <math>A^\uparrow = (A \to \mathbb{B})</math> is the set of propositions, and <math>A^\bullet = [\mathcal{A}] = [a_1, \ldots, a_n]</math> is the combination of these interpretations and propositions into the universe of discourse that is based on the features <math>\{a_1, \ldots, a_n\}.</math>

As always, especially in concrete examples, these rules may be dropped whenever necessary, reverting to a free assortment of feature labels. However, when we need to talk about the logical aspects of a space that is already named as a vector space, it will be necessary to make special provisions. At any rate, these elaborations can be deferred until actually needed.

==={{anchor|Philosophy of Notation}}Philosophy of Notation : Formal Terms and Flexible Types===

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
Where number is irrelevant, regimented mathematical technique has hitherto tended to be lacking. Thus it is that the progress of natural science has depended so largely upon the discernment of measurable quantity of one sort or another.
| width="4%" |  
|-
| align="right" colspan="3" | — W.V. Quine, ''Mathematical Logic'', [Qui, 7]
|}

For much of our discussion propositions and boolean functions are treated as the same formal objects, or as different interpretations of the same formal calculus. This rule of interpretation has exceptions, though. There is a distinctively logical interest in the use of propositional calculus that is not exhausted by its functional interpretation. It is part of our task in this study to deal with these uniquely logical characteristics as they present themselves both in our subject matter and in our formal calculus. Just to provide a hint of what's at stake: In logic, as opposed to the more imaginative realms of mathematics, we consider it a good thing to always know what we are talking about. Where mathematics encourages tolerance for uninterpreted symbols as intermediate terms, logic exerts a keener effort to interpret directly each oblique carrier of meaning, no matter how slight, and to unfold the complicities of every indirection in the flow of information. Translated into functional terms, this means that we want to maintain a continual, immediate, and persistent sense of both the converse relation <math>f^{-1} \subseteq \mathbb{B} \times \mathbb{B}^n,</math> or what is the same thing, <math>f^{-1} : \mathbb{B} \to \mathcal{P}(\mathbb{B}^n),</math> and the ''fibers'' or inverse images <math>f^{-1}(0)</math> and <math>f^{-1}(1),</math> associated with each boolean function <math>f : \mathbb{B}^n \to \mathbb{B}</math> that we use. In practical terms, the desired implementation of a propositional interpreter should incorporate our intuitive recognition that the induced partition of the functional domain into level sets <math>f^{-1}(b),</math> for <math>b \in \mathbb{B},</math> is part and parcel of understanding the denotative uses of each propositional function <math>f.</math>

==={{anchor|Special Classes}}Special Classes of Propositions===

It is important to remember that the coordinate propositions <math>\{a_i\},</math> besides being projection maps <math>a_i : \mathbb{B}^n \to \mathbb{B},</math> are propositions on an equal footing with all others, even though employed as a basis in a particular moment. This set of <math>n</math> propositions may sometimes be referred to as the ''basic propositions'', the ''coordinate propositions'', or the ''simple propositions'' that found a universe of discourse. Either one of the equivalent notations, <math>\{a_i : \mathbb{B}^n \to \mathbb{B}\}</math> or <math>(\mathbb{B}^n \xrightarrow{i} \mathbb{B}),</math> may be used to indicate the adoption of the propositions <math>a_i</math> as a basis for describing a universe of discourse.

Among the <math>2^{2^n}</math> propositions in <math>[a_1, \ldots, a_n]</math> are several families of <math>2^n</math> propositions each that take on special forms with respect to the basis <math>\{ a_1, \ldots, a_n \}.</math> Three of these families are especially prominent in the present context, the ''linear'', the ''positive'', and the ''singular'' propositions. Each family is naturally parameterized by the coordinate <math>n</math>-tuples in <math>\mathbb{B}^n</math> and falls into <math>n + 1</math> ranks, with a binomial coefficient <math>\tbinom{n}{k}</math> giving the number of propositions that have rank or weight <math>k.</math>

<ul>
{{anchor|Linear Propositions}}<li>
<p>The ''linear propositions'', <math>\{ \ell : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{\ell} \mathbb{B}),</math> may be written as sums:</p>

{| align="center" cellpadding="8" width="90%"
|
<math>\sum_{i=1}^n e_i ~=~ e_1 + \ldots + e_n
~\text{where}~
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 0 \end{matrix}\right\}
~\text{for}~ i = 1 ~\text{to}~ n.</math>
|}</li>

{{anchor|Positive Propositions}}<li>
<p>The ''positive propositions'', <math>\{ p : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{p} \mathbb{B}),</math> may be written as products:</p>

{| align="center" cellpadding="8" width="90%"
|
<math>\prod_{i=1}^n e_i ~=~ e_1 \cdot \ldots \cdot e_n
~\text{where}~
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 1 \end{matrix}\right\}
~\text{for}~ i = 1 ~\text{to}~ n.</math>
|}</li>

{{anchor|Singular Propositions}}<li>
<p>The ''singular propositions'', <math>\{ \mathbf{x} : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{s} \mathbb{B}),</math> may be written as products:</p>

{| align="center" cellpadding="8" width="90%"
|
<math>\prod_{i=1}^n e_i ~=~ e_1 \cdot \ldots \cdot e_n
~\text{where}~
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = \texttt{(} a_i \texttt{)} \end{matrix}\right\}
~\text{for}~ i = 1 ~\text{to}~ n.</math>
|}</li>
</ul>

In each case the rank <math>k</math> ranges from <math>0</math> to <math>n</math> and counts the number of positive appearances of the coordinate propositions <math>a_1, \ldots, a_n</math> in the resulting expression. For example, for <math>{n = 3},</math> the linear proposition of rank <math>0</math> is <math>0,</math> the positive proposition of rank <math>0</math> is <math>1,</math> and the singular proposition of rank <math>0</math> is <math>\texttt{(} a_1 \texttt{)(} a_2 \texttt{)(} a_3\texttt{)}.</math>

The basic propositions <math>a_i : \mathbb{B}^n \to \mathbb{B}</math> are both linear and positive. So these two kinds of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions.

Linear propositions and positive propositions are generated by taking boolean sums and products, respectively, over selected subsets of basic propositions, so both families of propositions are parameterized by the powerset <math>\mathcal{P}(\mathcal{I}),</math> that is, the set of all subsets <math>J</math> of the basic index set <math>\mathcal{I} = \{1, \ldots, n\}.</math>

Let us define <math>\mathcal{A}_J</math> as the subset of <math>\mathcal{A}</math> that is given by <math>\{a_i : i \in J\}.</math> Then we may comprehend the action of the linear and the positive propositions in the following terms:

* The linear proposition <math>\ell_J : \mathbb{B}^n \to \mathbb{B}</math> evaluates each cell <math>\mathbf{x}</math> of <math>\mathbb{B}^n</math> by looking at the coefficients of <math>\mathbf{x}</math> with respect to the features that <math>\ell_J</math> "likes", namely those in <math>\mathcal{A}_J,</math> and then adds them up in <math>\mathbb{B}.</math> Thus, <math>\ell_J(\mathbf{x})</math> computes the parity of the number of features that <math>\mathbf{x}</math> has in <math>\mathcal{A}_J,</math> yielding one for odd and zero for even. Expressed in this idiom, <math>\ell_J(\mathbf{x}) = 1</math> says that <math>\mathbf{x}</math> seems ''odd'' (or ''oddly true'') to <math>\mathcal{A}_J,</math> whereas <math>\ell_J(\mathbf{x}) = 0</math> says that <math>\mathbf{x}</math> seems ''even'' (or ''evenly true'') to <math>\mathcal{A}_J,</math> so long as we recall that ''zero times'' is evenly often, too.

* The positive proposition <math>p_J : \mathbb{B}^n \to \mathbb{B}</math> evaluates each cell <math>\mathbf{x}</math> of <math>\mathbb{B}^n</math> by looking at the coefficients of <math>\mathbf{x}</math> with regard to the features that <math>p_J</math> "likes", namely those in <math>\mathcal{A}_J,</math> and then takes their product in <math>\mathbb{B}.</math> Thus, <math>p_J(\mathbf{x})</math> assesses the unanimity of the multitude of features that <math>\mathbf{x}</math> has in <math>\mathcal{A}_J,</math> yielding one for all and aught for else. In these consensual or contractual terms, <math>p_J(\mathbf{x}) = 1</math> means that <math>\mathbf{x}</math> is ''AOK'' or congruent with all of the conditions of <math>\mathcal{A}_J,</math> while <math>p_J(\mathbf{x}) = 0</math> means that <math>\mathbf{x}</math> defaults or dissents from some condition of <math>\mathcal{A}_J.</math>

===Basis Relativity and Type Ambiguity===

Finally, two things are important to keep in mind with regard to the simplicity, linearity, positivity, and singularity of propositions.

First, all of these properties are relative to a particular basis. For example, a singular proposition with respect to a basis <math>\mathcal{A}</math> will not remain singular if <math>\mathcal{A}</math> is extended by a number of new and independent features. Even if we stick to the original set of pairwise options <math>\{a_i\} \cup \{ \texttt{(} a_i \texttt{)} \}</math> to select a new basis, the sets of linear and positive propositions are determined by the choice of simple propositions, and this determination is tantamount to the conventional choice of a cell as origin.

Second, the singular propositions <math>\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B},</math> picking out as they do a single cell or a coordinate tuple <math>\mathbf{x}</math> of <math>\mathbb{B}^n,</math> become the carriers or the vehicles of a certain type-ambiguity that vacillates between the dual forms <math>\mathbb{B}^n</math> and <math>(\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B})</math> and infects the whole hierarchy of types built on them. In other words, the terms that signify the interpretations <math>\mathbf{x} : \mathbb{B}^n</math> and the singular propositions <math>\mathbf{x} : \mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B}</math> are fully equivalent in information, and this means that every token of the type <math>\mathbb{B}^n</math> can be reinterpreted as an appearance of the subtype <math>\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B}.</math> And vice versa, the two types can be exchanged with each other everywhere that they turn up. In practical terms, this allows the use of singular propositions as a way of denoting points, forming an alternative to coordinate tuples.

For example, relative to the universe of discourse <math>[a_1, a_2, a_3]</math> the singular proposition <math>a_1 a_2 a_3 : \mathbb{B}^3 \xrightarrow{s} \mathbb{B}</math> could be explicitly retyped as <math>a_1 a_2 a_3 : \mathbb{B}^3</math> to indicate the point <math>(1, 1, 1)</math> but in most cases the proper interpretation could be gathered from context. Both notations remain dependent on a particular basis, but the code that is generated under the singular option has the advantage in its self-commenting features, in other words, it constantly reminds us of its basis in the process of denoting points. When the time comes to put a multiplicity of different bases into play, and to search for objects and properties that remain invariant under the transformations between them, this infinitesimal potential advantage may well evolve into an overwhelming practical necessity.

===The Analogy Between Real and Boolean Types===

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
Measurement consists in correlating our subject matter with the series of real numbers; and such correlations are desirable because, once they are set up, all the well-worked theory of numerical mathematics lies ready at hand as a tool for our further reasoning.
| width="4%" |  
|-
| align="right" colspan="3" | — W.V. Quine, ''Mathematical Logic'', [Qui, 7]
|}

There are two further reasons why it useful to spend time on a careful treatment of types, and they both have to do with our being able to take full computational advantage of certain dimensions of flexibility in the types that apply to terms. First, the domains of differential geometry and logic programming are connected by analogies between real and boolean types of the same pattern. Second, the types involved in these patterns have important isomorphisms connecting them that apply on both the real and the boolean sides of the picture.

Amazingly enough, these isomorphisms are themselves schematized by the axioms and theorems of propositional logic. This fact is known as the ''propositions as types'' analogy or the Curry–Howard isomorphism [How]. In another formulation it says that terms are to types as proofs are to propositions. See [LaS, 42–46] and [SeH] for a good discussion and further references. To anticipate the bearing of these issues on our immediate topic, Table 3 sketches a partial overview of the Real to Boolean analogy that may serve to illustrate the paradigm in question.

<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 3.} ~~ \text{Analogy Between Real and Boolean Types}</math>
|- style="height:40px; background:ghostwhite"
| <math>\text{Real Domain} ~ \mathbb{R}</math>
| <math>\longleftrightarrow</math>
| <math>\text{Boolean Domain} ~ \mathbb{B}</math>
|-
| <math>\mathbb{R}^n</math>
| <math>\text{Basic Space}</math>
| <math>\mathbb{B}^n</math>
|-
| <math>\mathbb{R}^n \to \mathbb{R}</math>
| <math>\text{Function Space}</math>
| <math>\mathbb{B}^n \to \mathbb{B}</math>
|-
| <math>(\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R}</math>
| <math>\text{Tangent Vector}</math>
| <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math>
|-
| <math>\mathbb{R}^n \to ((\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R})</math>
| <math>\text{Vector Field}</math>
| <math>\mathbb{B}^n \to ((\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B})</math>
|-
| <math>(\mathbb{R}^n \times (\mathbb{R}^n \to \mathbb{R})) \to \mathbb{R}</math>
| '''<font size="4">"</font>'''
| <math>(\mathbb{B}^n \times (\mathbb{B}^n \to \mathbb{B})) \to \mathbb{B}</math>
|-
| <math>((\mathbb{R}^n \to \mathbb{R}) \times \mathbb{R}^n) \to \mathbb{R}</math>
| '''<font size="4">"</font>'''
| <math>((\mathbb{B}^n \to \mathbb{B}) \times \mathbb{B}^n) \to \mathbb{B}</math>
|-
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^n \to \mathbb{R})</math>
| <math>\text{Derivation}</math>
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^n \to \mathbb{B})</math>
|-
| <math>\mathbb{R}^n \to \mathbb{R}^m</math>
| <math>\begin{matrix}\text{Basic}\\[2pt]\text{Transformation}\end{matrix}</math>
| <math>\mathbb{B}^n \to \mathbb{B}^m</math>
|-
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^m \to \mathbb{R})</math>
| <math>\begin{matrix}\text{Function}\\[2pt]\text{Transformation}\end{matrix}</math>
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^m \to \mathbb{B})</math>
|}

<br>

The Table exhibits a sample of likely parallels between the real and boolean domains. The central column gives a selection of terminology that is borrowed from differential geometry and extended in its meaning to the logical side of the Table. These are the varieties of spaces that come up when we turn to analyzing the dynamics of processes that pursue their courses through the states of an arbitrary space <math>X.</math> Moreover, when it becomes necessary to approach situations of overwhelming dynamic complexity in a succession of qualitative reaches, then the methods of logic that are afforded by the boolean domains, with their declarative means of synthesis and deductive modes of analysis, supply a natural battery of tools for the task.

It is usually expedient to take these spaces two at a time, in dual pairs of the form <math>X</math> and <math>(X \to \mathbb{K}).</math> In general, one creates pairs of type schemas by replacing any space <math>X</math> with its dual <math>(X \to \mathbb{K}),</math> for example, pairing the type <math>X \to Y</math> with the type <math>(X \to \mathbb{K}) \to (Y \to \mathbb{K}),</math> and <math>X \times Y</math> with <math>(X \to \mathbb{K}) \times (Y \to \mathbb{K}).</math> The word ''dual'' is used here in its broader sense to mean all of the functionals, not just the linear ones. Given any function <math>f : X \to \mathbb{K},</math> the ''converse'' or inverse relation corresponding to <math>f</math> is denoted <math>f^{-1},</math> and the subsets of <math>X</math> that are defined by <math>f^{-1}(k),</math> taken over <math>k</math> in <math>\mathbb{K},</math> are called the ''fibers'' or the ''level sets'' of the function <math>f.</math>

===Theory of Control and Control of Theory===

{| width="100%" cellpadding="0" cellspacing="0"
| width="40%" |  
| width="60%" |
You will hardly know who I am or what I mean,<br>
But I shall be good health to you nevertheless,<br>
And filter and fibre your blood.
|-
|  
| align="right" | — Walt Whitman, ''Leaves of Grass'', [Whi, 88]
|}

In the boolean context a function <math>f : X \to \mathbb{B}</math> is tantamount to a ''proposition'' about elements of <math>X,</math> and the elements of <math>X</math> constitute the ''interpretations'' of that proposition. The fiber <math>f^{-1}(1)</math> comprises the set of ''models'' of <math>f,</math> or examples of elements in <math>X</math> satisfying the proposition <math>f.</math> The fiber <math>f^{-1}(0)</math> collects the complementary set of ''anti-models'', or the exceptions to the proposition <math>f</math> that exist in <math>X.</math> Of course, the space of functions <math>(X \to \mathbb{B})</math> is isomorphic to the set of all subsets of <math>X,</math> called the ''power set'' of <math>X,</math> and often denoted <math>\mathcal{P}(X)</math> or <math>2^X.</math>

The operation of replacing <math>X</math> by <math>(X \to \mathbb{B})</math> in a type schema corresponds to a certain shift of attitude towards the space <math>X,</math> in which one passes from a focus on the ostensibly individual elements of <math>X</math> to a concern with the states of information and uncertainty one possesses about objects and situations in <math>X.</math> The conceptual obstacles in the path of this transition can be smoothed over by using singular functions <math>(\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B})</math> as stepping stones. First of all, it's an easy step from an element <math>\mathbf{x}</math> of type <math>\mathbb{B}^n</math> to the equivalent information of a singular proposition <math>\mathbf{x} : X \xrightarrow{s} \mathbb{B}, </math> and then only a small jump of generalization remains to reach the type of an arbitrary proposition <math>f : X \to \mathbb{B},</math> perhaps understood to indicate a relaxed constraint on the singularity of points or a neighborhood circumscribing the original <math>\mathbf{x}.</math> This is frequently a useful transformation, communicating between the ''objective'' and the ''intentional'' perspectives, in spite perhaps of the open objection that this distinction is transient in the mean time and ultimately superficial.

It is hoped that this measure of flexibility, allowing us to stretch a point into a proposition, can be useful in the examination of inquiry driven systems, where the differences between empirical, intentional, and theoretical propositions constitute the discrepancies and the distributions that drive experimental activity. I can give this model of inquiry a cybernetic cast by realizing that theory change and theory evolution, as well as the choice and the evaluation of experiments, are actions that are taken by a system or its agent in response to the differences that are detected between observational contents and theoretical coverage.

All of the above notwithstanding, there are several points that distinguish these two tasks, namely, the ''theory of control'' and the ''control of theory'', features that are often obscured by too much precipitation in the quickness with which we understand their similarities. In the control of uncertainty through inquiry, some of the actuators that we need to be concerned with are axiom changers and theory modifiers, operators with the power to compile and to revise the theories that generate expectations and predictions, effectors that form and edit our grammars for the languages of observational data, and agencies that rework the proposed model to fit the actual sequences of events and the realized relationships of values that are observed in the environment. Moreover, when steps must be taken to carry out an experimental action, there must be something about the particular shape of our uncertainty that guides us in choosing what directions to explore, and this impression is more than likely influenced by previous accumulations of experience. Thus it must be anticipated that much of what goes into scientific progress, or any sustainable effort toward a goal of knowledge, is necessarily predicated on long term observation and modal expectations, not only on the more local or short term prediction and correction.

===Propositions as Types and Higher Order Types===

The types collected in Table 3 (repeated below) serve to illustrate the themes of ''higher order propositional expressions'' and the ''propositions as types'' (PAT) analogy.

<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 3.} ~~ \text{Analogy Between Real and Boolean Types}</math>
|- style="height:40px; background:ghostwhite"
| <math>\text{Real Domain} ~ \mathbb{R}</math>
| <math>\longleftrightarrow</math>
| <math>\text{Boolean Domain} ~ \mathbb{B}</math>
|-
| <math>\mathbb{R}^n</math>
| <math>\text{Basic Space}</math>
| <math>\mathbb{B}^n</math>
|-
| <math>\mathbb{R}^n \to \mathbb{R}</math>
| <math>\text{Function Space}</math>
| <math>\mathbb{B}^n \to \mathbb{B}</math>
|-
| <math>(\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R}</math>
| <math>\text{Tangent Vector}</math>
| <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math>
|-
| <math>\mathbb{R}^n \to ((\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R})</math>
| <math>\text{Vector Field}</math>
| <math>\mathbb{B}^n \to ((\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B})</math>
|-
| <math>(\mathbb{R}^n \times (\mathbb{R}^n \to \mathbb{R})) \to \mathbb{R}</math>
| '''<font size="4">"</font>'''
| <math>(\mathbb{B}^n \times (\mathbb{B}^n \to \mathbb{B})) \to \mathbb{B}</math>
|-
| <math>((\mathbb{R}^n \to \mathbb{R}) \times \mathbb{R}^n) \to \mathbb{R}</math>
| '''<font size="4">"</font>'''
| <math>((\mathbb{B}^n \to \mathbb{B}) \times \mathbb{B}^n) \to \mathbb{B}</math>
|-
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^n \to \mathbb{R})</math>
| <math>\text{Derivation}</math>
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^n \to \mathbb{B})</math>
|-
| <math>\mathbb{R}^n \to \mathbb{R}^m</math>
| <math>\begin{matrix}\text{Basic}\\[2pt]\text{Transformation}\end{matrix}</math>
| <math>\mathbb{B}^n \to \mathbb{B}^m</math>
|-
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^m \to \mathbb{R})</math>
| <math>\begin{matrix}\text{Function}\\[2pt]\text{Transformation}\end{matrix}</math>
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^m \to \mathbb{B})</math>
|}

<br>

First, observe that the type of a ''tangent vector at a point'', also known as a ''directional derivative'' at that point, has the form <math>(\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K},</math> where <math>\mathbb{K}</math> is the chosen ground field, in the present case either <math>\mathbb{R}</math> or <math>\mathbb{B}.</math> At a point in a space of type <math>\mathbb{K}^n,</math> a directional derivative operator <math>\vartheta</math> takes a function on that space, an <math>f</math> of type <math>(\mathbb{K}^n \to \mathbb{K}),</math> and maps it to a ground field value of type <math>\mathbb{K}.</math> This value is known as the ''derivative'' of <math>f</math> in the direction <math>\vartheta</math> [Che46, 76–77]. In the boolean case <math>\vartheta : (\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math> has the form of a proposition about propositions, in other words, a proposition of the next higher type.

Next, by way of illustrating the propositions as types idea, consider a proposition of the form <math>X \Rightarrow (Y \Rightarrow Z).</math> One knows from propositional calculus that this is logically equivalent to a proposition of the form <math>(X \land Y) \Rightarrow Z.</math> But this equivalence should remind us of the functional isomorphism that exists between a construction of the type <math>X \to (Y \to Z)</math> and a construction of the type <math>(X \times Y) \to Z.</math> The propositions as types analogy permits us to take a functional type like this and, under the right conditions, replace the functional arrows “<math>\to</math>” and products “<math>\times</math>” with the respective logical arrows “<math>\Rightarrow</math>” and products “<math>\land</math>”. Accordingly, viewing the result as a proposition, we can employ axioms and theorems of propositional calculus to suggest appropriate isomorphisms among the categorical and functional constructions.

Finally, examine the middle four rows of Table 3. These display a series of isomorphic types that stretch from the categories that are labeled ''Vector Field'' to those that are labeled ''Derivation''. A ''vector field'', also known as an ''infinitesimal transformation'', associates a tangent vector at a point with each point of a space. In symbols, a vector field is a function of the form <math>\textstyle \xi : X \to \bigcup_{x \in X} \xi_x</math> that assigns to each point <math>x</math> of the space <math>X</math> a tangent vector to <math>X</math> at that point, namely, the tangent vector <math>\xi_x</math> [Che46, 82–83]. If <math>X</math> is of the type <math>\mathbb{K}^n,</math> then <math>\xi</math> is of the type <math>\mathbb{K}^n \to ((\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K}).</math> This has the pattern <math>X \to (Y \to Z),</math> with <math>X = \mathbb{K}^n,</math> <math>Y = (\mathbb{K}^n \to \mathbb{K}),</math> and <math>Z = \mathbb{K}.</math>

Applying the propositions as types analogy, one can follow this pattern through a series of metamorphoses from the type of a vector field to the type of a derivation, as traced out in Table 4. Observe how the function <math>f : X \to \mathbb{K},</math> associated with the place of <math>Y</math> in the pattern, moves through its paces from the second to the first position. In this way, the vector field <math>\xi,</math> initially viewed as attaching each tangent vector <math>\xi_x</math> to the site <math>x</math> where it acts in <math>X,</math> now comes to be seen as acting on each scalar potential <math>f : X \to \mathbb{K}</math> like a generalized species of differentiation, producing another function <math>\xi f : X \to \mathbb{K}</math> of the same type.

<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 4.} ~~ \text{An Equivalence Based on the Propositions as Types Analogy}</math>
|- style="height:40px; background:ghostwhite"
| <math>\text{Pattern}</math>
| <math>\text{Construct}</math>
| <math>\text{Instance}</math>
|-
| <math>X \to (Y \to Z)</math>
| <math>\text{Vector Field}</math>
| <math>\mathbb{K}^n \to ((\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K})</math>
|-
| <math>(X \times Y) \to Z</math>
| <math>\Uparrow</math>
| <math>(\mathbb{K}^n \times (\mathbb{K}^n \to \mathbb{K})) \to \mathbb{K}</math>
|-
| <math>(Y \times X) \to Z</math>
| <math>\Downarrow</math>
| <math>((\mathbb{K}^n \to \mathbb{K}) \times \mathbb{K}^n) \to \mathbb{K}</math>
|-
| <math>Y \to (X \to Z)</math>
| <math>\text{Derivation}</math>
| <math>(\mathbb{K}^n \to \mathbb{K}) \to (\mathbb{K}^n \to \mathbb{K})</math>
|}

<br>

===Reality at the Threshold of Logic===

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
But no science can rest entirely on measurement, and many scientific investigations are quite out of reach of that device. To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope.
| width="4%" |  
|-
| align="right" colspan="3" | — W.V. Quine, ''Mathematical Logic'', [Qui, 7]
|}

Table 5 accumulates an array of notation that I hope will not be too distracting. Some of it is rarely needed, but has been filled in for the sake of completeness. Its purpose is simple, to give literal expression to the visual intuitions that come with venn diagrams, and to help build a bridge between our qualitative and quantitative outlooks on dynamic systems.

<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:75%"
|+ style="height:30px" | <math>\text{Table 5.} ~~ \text{A Bridge Over Troubled Waters}</math>
|- style="height:40px; background:ghostwhite"
| align="center" | <math>\text{Linear Space}</math>
| align="center" | <math>\text{Liminal Space}</math>
| align="center" | <math>\text{Logical Space}</math>
|-
| <math>\begin{matrix}\mathcal{X} & = & \{ x_1, \ldots, x_n \}\end{matrix}</math>
| <math>\begin{matrix}\underline{\mathcal{X}} & = & \{ \underline{x}_1, \ldots, \underline{x}_n \}\end{matrix}</math>
| <math>\begin{matrix}\mathcal{A} & = & \{ a_1, \ldots, a_n \}\end{matrix}</math>
|-
|
<math>\begin{matrix}
X_i & = & \langle x_i \rangle
\\
& \cong & \mathbb{K}
\end{matrix}</math>
|
<math>\begin{matrix}
\underline{X}_i & = & \{ \texttt{(} \underline{x}_i \texttt{)}, \underline{x}_i \}
\\
& \cong & \mathbb{B}
\end{matrix}</math>
|
<math>\begin{matrix}
A_i & = & \{ \texttt{(} a_i \texttt{)}, a_i \}
\\
& \cong & \mathbb{B}
\end{matrix}</math>
|-
|
<math>\begin{matrix}
X
\\
= & \langle \mathcal{X} \rangle
\\
= & \langle x_1, \ldots, x_n \rangle
\\
= & X_1 \times \ldots \times X_n
\\
= & \prod_{i=1}^n X_i
\\
\cong & \mathbb{K}^n
\end{matrix}</math>
|
<math>\begin{matrix}
\underline{X}
\\
= & \langle \underline{\mathcal{X}} \rangle
\\
= & \langle \underline{x}_1, \ldots, \underline{x}_n \rangle
\\
= & \underline{X}_1 \times \ldots \times \underline{X}_n
\\
= & \prod_{i=1}^n \underline{X}_i
\\
\cong & \mathbb{B}^n
\end{matrix}</math>
|
<math>\begin{matrix}
A
\\
= & \langle \mathcal{A} \rangle
\\
= & \langle a_1, \ldots, a_n \rangle
\\
= & A_1 \times \ldots \times A_n
\\
= & \prod_{i=1}^n A_i
\\
\cong & \mathbb{B}^n
\end{matrix}</math>
|-
|
<math>\begin{matrix}
X^* & = & (\ell : X \to \mathbb{K})
\\
& \cong & \mathbb{K}^n
\end{matrix}</math>
|
<math>\begin{matrix}
\underline{X}^* & = & (\ell : \underline{X} \to \mathbb{B})
\\
& \cong & \mathbb{B}^n
\end{matrix}</math>
|
<math>\begin{matrix}
A^* & = & (\ell : A \to \mathbb{B})
\\
& \cong & \mathbb{B}^n
\end{matrix}</math>
|-
|
<math>\begin{matrix}
X^\uparrow & = & (X \to \mathbb{K})
\\
& \cong & (\mathbb{K}^n \to \mathbb{K})
\end{matrix}</math>
|
<math>\begin{matrix}
\underline{X}^\uparrow & = & (\underline{X} \to \mathbb{B})
\\
& \cong & (\mathbb{B}^n \to \mathbb{B})
\end{matrix}</math>
|
<math>\begin{matrix}
A^\uparrow & = & (A \to \mathbb{B})
\\
& \cong & (\mathbb{B}^n \to \mathbb{B})
\end{matrix}</math>
|-
|
<math>\begin{matrix}
X^\bullet
\\
= & [\mathcal{X}]
\\
= & [x_1, \ldots, x_n]
\\
= & (X, X^\uparrow)
\\
= & (X ~+\!\to \mathbb{K})
\\
= & (X, (X \to \mathbb{K}))
\\
\cong & (\mathbb{K}^n, (\mathbb{K}^n \to \mathbb{K}))
\\
= & (\mathbb{K}^n ~+\!\to \mathbb{K})
\\
= & [\mathbb{K}^n]
\end{matrix}</math>
|
<math>\begin{matrix}
\underline{X}^\bullet
\\
= & [\underline{\mathcal{X}}]
\\
= & [\underline{x}_1, \ldots, \underline{x}_n]
\\
= & (\underline{X}, \underline{X}^\uparrow)
\\
= & (\underline{X} ~+\!\to \mathbb{B})
\\
= & (\underline{X}, (\underline{X} \to \mathbb{B}))
\\
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))
\\
= & (\mathbb{B}^n ~+\!\to \mathbb{B})
\\
= & [\mathbb{B}^n]
\end{matrix}</math>
|
<math>\begin{matrix}
A^\bullet
\\
= & [\mathcal{A}]
\\
= & [a_1, \ldots, a_n]
\\
= & (A, A^\uparrow)
\\
= & (A ~+\!\to \mathbb{B})
\\
= & (A, (A \to \mathbb{B}))
\\
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))
\\
= & (\mathbb{B}^n ~+\!\to \mathbb{B})
\\
= & [\mathbb{B}^n]
\end{matrix}</math>
|}

<br>

The left side of the Table collects mostly standard notation for an <math>n</math>-dimensional vector space over a field <math>\mathbb{K}.</math> The right side of the table repeats the first elements of a notation that I sketched above, to be used in further developments of propositional calculus. (I plan to use this notation in the logical analysis of neural network systems.) The middle column of the table is designed as a transitional step from the case of an arbitrary field <math>\mathbb{K},</math> with a special interest in the continuous line <math>\mathbb{R},</math> to the qualitative and discrete situations that are instanced and typified by <math>\mathbb{B}.</math>

I now proceed to explain these concepts in more detail. The most important ideas developed in Table 5 are these:

* The idea of a universe of discourse, which includes both a space of ''points'' and a space of ''maps'' on those points.

* The idea of passing from a more complex universe to a simpler universe by a process of ''thresholding'' each dimension of variation down to a single bit of information.

For the sake of concreteness, let us suppose that we start with a continuous <math>n</math>-dimensional vector space like <math>X = \langle x_1, \ldots, x_n \rangle \cong \mathbb{R}^n.</math> The coordinate system <math>\mathcal{X} = \{x_i\}</math> is a set of maps <math>x_i : \mathbb{R}^n \to \mathbb{R},</math> also known as the ''coordinate projections''. Given a "dataset" of points <math>\mathbf{x}</math> in <math>\mathbb{R}^n,</math> there are numerous ways of sensibly reducing the data down to one bit for each dimension. One strategy that is general enough for our present purposes is as follows. For each <math>i</math> we choose an <math>n</math>-ary relation <math>L_i</math> on <math>\mathbb{R}^n,</math> that is, a subset of <math>\mathbb{R}^n,</math> and then we define the <math>i^\mathrm{th}</math> threshold map, or ''limen'' <math>\underline{x}_i</math> as follows:

{| align="center" cellpadding="8" width="90%"
| <math>\underline{x}_i : \mathbb{R}^n \to \mathbb{B}\ \text{such that:}</math>
|-
|
<math>\begin{matrix}
\underline{x}_i(\mathbf{x}) = 1 & \text{if} & \mathbf{x} \in L_i,
\\[4pt]
\underline{x}_i(\mathbf{x}) = 0 & \text{if} & \mathbf{x} \not\in L_i.
\end{matrix}</math>
|}

In other notations that are sometimes used, the operator <math>\chi (\ldots)</math> or the corner brackets <math>\lceil\ldots\rceil</math> can be used to denote a ''characteristic function'', that is, a mapping from statements to their truth values in <math>\mathbb{B}.</math> Finally, it is not uncommon to use the name of the relation itself as a predicate that maps <math>n</math>-tuples into truth values. Thus we have the following notational variants of the above definition:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
\underline{x}_i (\mathbf{x}) & = & \chi (\mathbf{x} \in L_i) & = & \lceil \mathbf{x} \in L_i \rceil & = & L_i (\mathbf{x}).
\end{matrix}</math>
|}

Notice that, as defined here, there need be no actual relation between the <math>n</math>-dimensional subsets <math>\{L_i\}</math> and the coordinate axes corresponding to <math>\{x_i\},</math> aside from the circumstance that the two sets have the same cardinality. In concrete cases, though, one usually has some reason for associating these "volumes" with these "lines", for instance, <math>L_i</math> is bounded by some hyperplane that intersects the <math>i^\text{th}</math> axis at a unique threshold value <math>r_i \in \mathbb{R}.</math> Often, the hyperplane is chosen normal to the axis. In recognition of this motive, let us make the following convention. When the set <math>L_i</math> has points on the <math>i^\text{th}</math> axis, that is, points of the form <math>(0, \ldots, 0, r_i, 0, \ldots, 0)</math> where only the <math>x_i</math> coordinate is possibly non-zero, we may pick any one of these coordinate values as a parametric index of the relation. In this case we say that the indexing is ''real'', otherwise the indexing is ''imaginary''. For a knowledge based system <math>X,</math> this should serve once again to mark the distinction between ''acquaintance'' and ''opinion''.

States of knowledge about the location of a system or about the distribution of a population of systems in a state space <math>X = \mathbb{R}^n</math> can now be expressed by taking the set <math>\underline{\mathcal{X}} = \{\underline{x}_i\}</math> as a basis of logical features. In picturesque terms, one may think of the underscore and the subscript as combining to form a subtextual spelling for the <math>i^\text{th}</math> threshold map. This can help to remind us that the ''threshold operator'' <math>(\underline{~})_i</math> acts on <math>\mathbf{x}</math> by setting up a kind of a “hurdle” for it. In this interpretation the coordinate proposition <math>\underline{x}_i</math> asserts that the representative point <math>\mathbf{x}</math> resides ''above'' the <math>i^\mathrm{th}</math> threshold.

Primitive assertions of the form <math>\underline{x}_i (\mathbf{x})</math> may then be negated and joined by means of propositional connectives in the usual ways to provide information about the state <math>\mathbf{x}</math> of a contemplated system or a statistical ensemble of systems. Parentheses <math>\texttt{(} \ldots \texttt{)}</math> may be used to indicate logical negation. Eventually one discovers the usefulness of the <math>k</math>-ary ''just one false'' operators of the form <math>\texttt{(} a_1 \texttt{,} \ldots \texttt{,} a_k \texttt{)},</math> as treated in earlier reports. This much tackle generates a space of points (cells, interpretations), <math>\underline{X} \cong \mathbb{B}^n,</math> and a space of functions (regions, propositions), <math>\underline{X}^\uparrow \cong (\mathbb{B}^n \to \mathbb{B}).</math> Together these form a new universe of discourse <math>\underline{X}^\bullet</math> of the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})),</math> which we may abbreviate as <math>\mathbb{B}^n\ +\!\to \mathbb{B}</math> or most succinctly as <math>[\mathbb{B}^n].</math>

The square brackets have been chosen to recall the rectangular frame of a venn diagram. In thinking about a universe of discourse it is a good idea to keep this picture in mind, graphically illustrating the links among the elementary cells <math>\underline{\mathbf{x}},</math> the defining features <math>\underline{x}_i,</math> and the potential shadings <math>f : \underline{X} \to \mathbb{B}</math> all at the same time, not to mention the arbitrariness of the way we choose to inscribe our distinctions in the medium of a continuous space.

Finally, let <math>X^*</math> denote the space of linear functions, <math>(\ell : X \to \mathbb{K}),</math> which has in the finite case the same dimensionality as <math>X,</math> and let the same notation be extended across the Table.

We have just gone through a lot of work, apparently doing nothing more substantial than spinning a complex spell of notational devices through a labyrinth of baffled spaces and baffling maps. The reason for doing this was to bind together and to constitute the intuitive concept of a universe of discourse into a coherent categorical object, the kind of thing, once grasped, that can be turned over in the mind and considered in all its manifold changes and facets. The effort invested in these preliminary measures is intended to pay off later, when we need to consider the state transformations and the time evolution of neural network systems.

===Tables of Propositional Forms===

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope. It provides explicit techniques for manipulating the most basic ingredients of discourse.
| width="4%" |  
|-
| align="right" colspan="3" | — W.V. Quine, ''Mathematical Logic'', [Qui, 7–8]
|}

To prepare for the next phase of discussion, Tables 6 and 7 collect and summarize all of the propositional forms on one and two variables. These propositional forms are represented over bases of boolean variables as complete sets of boolean-valued functions. Adjacent to their names and specifications are listed what are roughly the simplest expressions in the ''[[Cactus_Language_•_Overview|cactus language]]'', the particular syntax for propositional calculus that I use in formal and computational contexts. For the sake of orientation, the English paraphrases and the more common notations are listed in the last two columns. As simple and circumscribed as these low-dimensional universes may appear to be, a careful exploration of their differential extensions will involve us in complexities sufficient to demand our attention for some time to come.

Propositional forms on one variable correspond to boolean functions <math>f : \mathbb{B}^1 \to \mathbb{B}.</math> In Table 6 these functions are listed in a variant form of [[truth table]], one in which the axes of the usual arrangement are rotated through a right angle. Each function <math>f_i</math> is indexed by the string of values that it takes on the points of the universe <math>X^\bullet = [x] \cong \mathbb{B}^1.</math> The binary index generated in this way is converted to its decimal equivalent and these are used as conventional names for the <math>f_i,</math> as shown in the first column of the Table. In their own right the <math>2^1</math> points of the universe <math>X^\bullet</math> are coordinated as a space of type <math>\mathbb{B}^1,</math> this in light of the universe <math>X^\bullet</math> being a functional domain where the coordinate projection <math>x</math> takes on its values in <math>\mathbb{B}.</math>

<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 6.} ~~ \text{Propositional Forms on One Variable}</math>
|- style="height:40px; background:ghostwhite"
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math>
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math>
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math>
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math>
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math>
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math>
|- style="background:ghostwhite"
|  
| align="right" | <math>x\colon</math>
| <math>1~0</math>
|  
|  
|  
|-
| <math>f_0</math>
| <math>f_{00}</math>
| <math>0~0</math>
| <math>\texttt{(} ~ \texttt{)}</math>
| <math>\text{false}</math>
| <math>0</math>
|-
| <math>f_1</math>
| <math>f_{01}</math>
| <math>0~1</math>
| <math>\texttt{(} x \texttt{)}</math>
| <math>\text{not}~ x</math>
| <math>\lnot x</math>
|-
| <math>f_2</math>
| <math>f_{10}</math>
| <math>1~0</math>
| <math>x</math>
| <math>x</math>
| <math>x</math>
|-
| <math>f_3</math>
| <math>f_{11}</math>
| <math>1~1</math>
| <math>\texttt{((} ~ \texttt{))}</math>
| <math>\text{true}</math>
| <math>1</math>
|}

<br>

Propositional forms on two variables correspond to boolean functions <math>f : \mathbb{B}^2 \to \mathbb{B}.</math> In Table 7 each function <math>f_i</math> is indexed by the values that it takes on the points of the universe <math>X^\bullet = [x, y] \cong \mathbb{B}^2.</math> Converting the binary index thus generated to a decimal equivalent, we obtain the functional nicknames that are listed in the first column. The <math>2^2</math> points of the universe <math>X^\bullet</math> are coordinated as a space of type <math>\mathbb{B}^2,</math> as indicated under the heading of the Table, where the coordinate projections <math>x</math> and <math>y</math> run through the various combinations of their values in <math>\mathbb{B}.</math>

<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 7-a.} ~~ \text{Propositional Forms on Two Variables}</math>
|- style="height:40px; background:ghostwhite"
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math>
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math>
|- style="background:ghostwhite"
|  
| align="right" | <math>x\colon</math>
| <math>1~1~0~0</math>
|  
|  
|  
|- style="background:ghostwhite"
|  
| align="right" | <math>y\colon</math>
| <math>1~0~1~0</math>
|  
|  
|  
|-
| valign="bottom" |
<math>\begin{matrix}
f_{0}
\\[4pt]
f_{1}
\\[4pt]
f_{2}
\\[4pt]
f_{3}
\\[4pt]
f_{4}
\\[4pt]
f_{5}
\\[4pt]
f_{6}
\\[4pt]
f_{7}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
f_{0000}
\\[4pt]
f_{0001}
\\[4pt]
f_{0010}
\\[4pt]
f_{0011}
\\[4pt]
f_{0100}
\\[4pt]
f_{0101}
\\[4pt]
f_{0110}
\\[4pt]
f_{0111}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
0~0~0~0
\\[4pt]
0~0~0~1
\\[4pt]
0~0~1~0
\\[4pt]
0~0~1~1
\\[4pt]
0~1~0~0
\\[4pt]
0~1~0~1
\\[4pt]
0~1~1~0
\\[4pt]
0~1~1~1
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{(} ~ \texttt{)}
\\[4pt]
\texttt{(} x \texttt{)(} y \texttt{)}
\\[4pt]
\texttt{(} x \texttt{)} ~ y ~
\\[4pt]
\texttt{(} x \texttt{)}
\\[4pt]
~ x ~ \texttt{(} y \texttt{)}
\\[4pt]
\texttt{(} y \texttt{)}
\\[4pt]
\texttt{(} x \texttt{,} ~ y \texttt{)}
\\[4pt]
\texttt{(} x ~ y \texttt{)}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\text{false}
\\[4pt]
\text{neither}~ x ~\text{nor}~ y
\\[4pt]
y ~\text{without}~ x
\\[4pt]
\text{not}~ x
\\[4pt]
x ~\text{without}~ y
\\[4pt]
\text{not}~ y
\\[4pt]
x ~\text{not equal to}~ y
\\[4pt]
\text{not both}~ x ~\text{and}~ y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
0
\\[4pt]
\lnot x \land \lnot y
\\[4pt]
\lnot x \land y
\\[4pt]
\lnot x
\\[4pt]
x \land \lnot y
\\[4pt]
\lnot y
\\[4pt]
x \ne y
\\[4pt]
\lnot x \lor \lnot y
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
f_{8}
\\[4pt]
f_{9}
\\[4pt]
f_{10}
\\[4pt]
f_{11}
\\[4pt]
f_{12}
\\[4pt]
f_{13}
\\[4pt]
f_{14}
\\[4pt]
f_{15}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
f_{1000}
\\[4pt]
f_{1001}
\\[4pt]
f_{1010}
\\[4pt]
f_{1011}
\\[4pt]
f_{1100}
\\[4pt]
f_{1101}
\\[4pt]
f_{1110}
\\[4pt]
f_{1111}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
1~0~0~0
\\[4pt]
1~0~0~1
\\[4pt]
1~0~1~0
\\[4pt]
1~0~1~1
\\[4pt]
1~1~0~0
\\[4pt]
1~1~0~1
\\[4pt]
1~1~1~0
\\[4pt]
1~1~1~1
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
x ~ y
\\[4pt]
\texttt{((} x \texttt{,} ~ y \texttt{))}
\\[4pt]
y
\\[4pt]
\texttt{(} x ~ \texttt{(} y \texttt{))}
\\[4pt]
x
\\[4pt]
\texttt{((} x \texttt{)} ~ y \texttt{)}
\\[4pt]
\texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
\texttt{((} ~ \texttt{))}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
x ~\text{and}~ y
\\[4pt]
x ~\text{equal to}~ y
\\[4pt]
y
\\[4pt]
\text{not}~ x ~\text{without}~ y
\\[4pt]
x
\\[4pt]
\text{not}~ y ~\text{without}~ x
\\[4pt]
x ~\text{or}~ y
\\[4pt]
\text{true}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
x \land y
\\[4pt]
x = y
\\[4pt]
y
\\[4pt]
x \Rightarrow y
\\[4pt]
x
\\[4pt]
x \Leftarrow y
\\[4pt]
x \lor y
\\[4pt]
1
\end{matrix}</math>
|}

<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 7-b.} ~~ \text{Propositional Forms on Two Variables}</math>
|- style="height:40px; background:ghostwhite"
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math>
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math>
|- style="background:ghostwhite"
|  
| align="right" | <math>x\colon</math>
| <math>1~1~0~0</math>
|  
|  
|  
|- style="background:ghostwhite"
|  
| align="right" | <math>y\colon</math>
| <math>1~0~1~0</math>
|  
|  
|  
|-
| <math>f_{0}</math>
| <math>f_{0000}</math>
| <math>0~0~0~0</math>
| <math>\texttt{(} ~ \texttt{)}</math>
| <math>\text{false}</math>
| <math>0</math>
|-
| valign="bottom" |
<math>\begin{matrix}
f_{1}
\\[4pt]
f_{2}
\\[4pt]
f_{4}
\\[4pt]
f_{8}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
f_{0001}
\\[4pt]
f_{0010}
\\[4pt]
f_{0100}
\\[4pt]
f_{1000}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
0~0~0~1
\\[4pt]
0~0~1~0
\\[4pt]
0~1~0~0
\\[4pt]
1~0~0~0
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{(} x \texttt{)(} y \texttt{)}
\\[4pt]
\texttt{(} x \texttt{)} ~ y ~
\\[4pt]
~ x ~ \texttt{(} y \texttt{)}
\\[4pt]
~ x ~~ y ~
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\text{neither}~ x ~\text{nor}~ y
\\[4pt]
y ~\text{without}~ x
\\[4pt]
x ~\text{without}~ y
\\[4pt]
x ~\text{and}~ y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\lnot x \land \lnot y
\\[4pt]
\lnot x \land y
\\[4pt]
x \land \lnot y
\\[4pt]
x \land y
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
f_{3}
\\[4pt]
f_{12}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
f_{0011}
\\[4pt]
f_{1100}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
0~0~1~1
\\[4pt]
1~1~0~0
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\[4pt]
x
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\text{not}~ x
\\[4pt]
x
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\lnot x
\\[4pt]
x
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
f_{6}
\\[4pt]
f_{9}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
f_{0110}
\\[4pt]
f_{1001}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
0~1~1~0
\\[4pt]
1~0~0~1
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{(} x \texttt{,} y \texttt{)}
\\[4pt]
\texttt{((} x \texttt{,} y \texttt{))}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
x ~\text{not equal to}~ y
\\[4pt]
x ~\text{equal to}~ y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
x \ne y
\\[4pt]
x = y
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
f_{5}
\\[4pt]
f_{10}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
f_{0101}
\\[4pt]
f_{1010}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
0~1~0~1
\\[4pt]
1~0~1~0
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\[4pt]
y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\text{not}~ y
\\[4pt]
y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\lnot y
\\[4pt]
y
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
f_{7}
\\[4pt]
f_{11}
\\[4pt]
f_{13}
\\[4pt]
f_{14}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
f_{0111}
\\[4pt]
f_{1011}
\\[4pt]
f_{1101}
\\[4pt]
f_{1110}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
0~1~1~1
\\[4pt]
1~0~1~1
\\[4pt]
1~1~0~1
\\[4pt]
1~1~1~0
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{(} ~ x ~~ y ~ \texttt{)}
\\[4pt]
\texttt{(} ~ x ~ \texttt{(} y \texttt{))}
\\[4pt]
\texttt{((} x \texttt{)} ~ y ~ \texttt{)}
\\[4pt]
\texttt{((} x \texttt{)(} y \texttt{))}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\text{not both}~ x ~\text{and}~ y
\\[4pt]
\text{not}~ x ~\text{without}~ y
\\[4pt]
\text{not}~ y ~\text{without}~ x
\\[4pt]
x ~\text{or}~ y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\lnot x \lor \lnot y
\\[4pt]
x \Rightarrow y
\\[4pt]
x \Leftarrow y
\\[4pt]
x \lor y
\end{matrix}</math>
|-
| <math>f_{15}</math>
| <math>f_{1111}</math>
| <math>1~1~1~1</math>
| <math>\texttt{((} ~ \texttt{))}</math>
| <math>\text{true}</math>
| <math>1</math>
|}

<br>

==A Differential Extension of Propositional Calculus==

{| width="100%" cellpadding="0" cellspacing="0"
| width="40%" |  
| width="60%" |
Fire over water:<br>
The image of the condition before transition.<br>
Thus the superior man is careful<br>
In the differentiation of things,<br>
So that each finds its place.
|-
|  
| align="right" | — ''I Ching'', Hexagram 64, [Wil, 249]
|}

This much preparation is enough to begin introducing my subject, if I excuse myself from giving full arguments for my definitional choices until some later stage. I am trying to develop a ''differential theory of qualitative equations'' that parallels the application of differential geometry to dynamic systems. The idea of a tangent vector is key to this work and a major goal is to find the right logical analogues of tangent spaces, bundles, and functors. The strategy is taken of looking for the simplest versions of these constructions that can be discovered within the realm of propositional calculus, so long as they serve to fill out the general theme.

==={{anchor|Differential Propositions}}Differential Propositions : Qualitative Analogues of Differential Equations===

The differential extension of a universe of discourse <math>[\mathcal{A}]</math> is constructed by extending its initial alphabet <math>\mathfrak{A}</math> to include a set of symbols for <i>differential features</i>, or <i>basic changes</i> capable of occurring in <math>[\mathcal{A}].</math>  The added symbols are taken to denote primitive features of change, qualitative attributes of motion, or propositions about how items in the universe of discourse may change or move in relation to features noted in the original alphabet.

With that in mind we define the corresponding <i>differential alphabet</i> or <i>tangent alphabet</i> <math>\mathrm{d}\mathfrak{A}</math> <math>=</math> <math>\{ {}^{\backprime\backprime} \mathrm{d}a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} \mathrm{d}a_n {}^{\prime\prime} \},</math> in principle just an arbitrary alphabet of symbols, disjoint from the initial alphabet <math>\mathfrak{A}</math> <math>=</math> <math>\{ {}^{\backprime\backprime} a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} a_n {}^{\prime\prime}\}</math> and given the meanings just indicated.

In practice the precise interpretation of the symbols in <math>\mathrm{d}\mathfrak{A}</math> is conceived to be changeable from point to point of the underlying space <math>A.</math>  Indeed, for all we know, the state space <math>A</math> might well be the state space of a language interpreter, one concerned with the idiomatic meanings of the dialect generated by <math>\mathfrak{A}</math> and <math>\mathrm{d}\mathfrak{A}.</math>

{{anchor|Tangent Spaces}}
The <i>tangent space</i> to <math>A</math> at one of its points <math>x,</math> sometimes written <math>\mathrm{T}_x(A),</math> takes the form <math>\mathrm{d}A</math> <math>=</math> <math>\langle \mathrm{d}\mathcal{A} \rangle</math> <math>=</math> <math>\langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle.</math>  Strictly speaking, the name <i>cotangent space</i> is probably more correct for this construction but since we take up spaces and their duals in pairs to form our universes of discourse it allows our language to be pliable here.

Proceeding as we did with the base space <math>A,</math> the tangent space <math>\mathrm{d}A</math> at a point of <math>A</math> may be analyzed as the following product of distinct and independent factors.

{| align="center" cellpadding="8" width="90%"
| <math>\mathrm{d}A ~=~ \prod_{i=1}^n \mathrm{d}A_i ~=~ \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n.</math>
|}

Each factor <math>\mathrm{d}A_i</math> is a set consisting of two differential propositions, <math>\mathrm{d}A_i = \{ (\mathrm{d}a_i), \mathrm{d}a_i \},</math> where <math>\texttt{(} \mathrm{d}a_i \texttt{)}</math> is a proposition with the logical value of <math>\lnot\mathrm{d}a_i.</math>  Each component <math>\mathrm{d}A_i</math> has the type <math>\mathbb{B},</math> operating under the ordered correspondence <math>\{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \} \cong \{ 0, 1 \}.</math>  A measure of clarity is achieved, however, by acknowledging the differential usage with a superficially distinct type <math>\mathbb{D},</math> whose sense may be indicated as follows.

{| align="center" cellpadding="8" width="90%"
| <math>\mathbb{D} = \{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \} = \{ \text{same}, \text{different} \} = \{ \text{stay}, \text{change} \} = \{ \text{stop}, \text{step} \}.</math>
|}

Viewed within a coordinate representation, spaces of type <math>\mathbb{B}^n</math> and <math>\mathbb{D}^n</math> may appear to be identical sets of binary vectors, but taking a view at that level of abstraction would be like ignoring the qualitative units and the diverse dimensions that distinguish position and momentum, or the different roles of quantity and impulse.

===An Interlude on the Path===

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
There would have been no beginnings:  instead, speech would proceed from me, while I stood in its path – a slender gap – the point of its possible disappearance.
| width="4%" |  
|-
| align="right" colspan="3" | — Michel Foucault, ''The Discourse on Language'', [Fou, 215]
|}

A sense of the relation between <math>\mathbb{B}</math> and <math>\mathbb{D}</math> may be obtained by considering the ''path classifier'' (or the ''equivalence class of curves'') approach to tangent vectors.  Consider a universe <math>[\mathcal{X}].</math>  Given the boolean value system, a path in the space <math>X = \langle \mathcal{X} \rangle</math> is a map <math>q : \mathbb{B} \to X.</math>  In this context the set of paths <math>(\mathbb{B} \to X)</math> is isomorphic to the cartesian square <math>X^2 = X \times X,</math> or the set of ordered pairs chosen from <math>X.</math>

We may analyze <math>X^2 = \{ (u, v) : u, v \in X \}</math> into two parts, specifically, the ordered pairs <math>(u, v)</math> that lie on and off the diagonal:

{| align="center" cellpadding="8" width="90%"
| <math>\begin{matrix}X^2 & = & \{ (u, v) : u = v \} & \cup & \{ (u, v) : u \ne v \}.\end{matrix}</math>
|}

This partition may also be expressed in the following symbolic form:

{| align="center" cellpadding="8" width="90%"
| <math>\begin{matrix}X^2 & \cong & \operatorname{diag} (X) & + & 2 \binom{X}{2}.\end{matrix}</math>
|}

The separate terms of this formula are defined as follows:

{| align="center" cellpadding="8" width="90%"
| <math>\begin{matrix}\operatorname{diag} (X) & = & \{ (x, x) : x \in X \}.\end{matrix}</math>
|}

{| align="center" cellpadding="8" width="90%"
| <math>\begin{matrix}\binom{X}{k} & = & X ~\text{choose}~ k & = & \{ k\text{-sets from}~ X \}.\end{matrix}</math>
|}

Thus we have:

{| align="center" cellpadding="8" width="90%"
| <math>\begin{matrix}\binom{X}{2} & = & \{ \{ u, v \} : u, v \in X \}.\end{matrix}</math>
|}

We may now use the features in <math>\mathrm{d}\mathcal{X} = \{ \mathrm{d}x_i \} = \{ \mathrm{d}x_1, \ldots, \mathrm{d}x_n \}</math> to classify the paths of <math>(\mathbb{B} \to X)</math> by way of the pairs in <math>X^2.</math>  If <math>X \cong \mathbb{B}^n,</math> then a path <math>q</math> in <math>X</math> has the following form:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
q : (\mathbb{B} \to \mathbb{B}^n) & \cong & \mathbb{B}^n \times \mathbb{B}^n & \cong & \mathbb{B}^{2n} & \cong & (\mathbb{B}^2)^n.
\end{matrix}</math>
|}

Intuitively, we want to map this <math>(\mathbb{B}^2)^n</math> onto <math>\mathbb{D}^n</math> by mapping each component <math>\mathbb{B}^2</math> onto a copy of <math>\mathbb{D}.</math>  But in the presenting context <math>{}^{\backprime\backprime} \mathbb{D} {}^{\prime\prime}</math> is just a name associated with, or an incidental quality attributed to, coefficient values in <math>\mathbb{B}</math> when they are attached to features in <math>\mathrm{d}\mathcal{X}.</math>

Taking these intentions into account, define <math>\mathrm{d}x_i : X^2 \to \mathbb{B}</math> in the following manner:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{lcrcl}
\mathrm{d}x_i(u, v)
& = & \texttt{(} ~ x_i(u) & \texttt{,} & x_i(v) ~ \texttt{)}
\\
& = & x_i(u) & + & x_i(v)
\\
& = & x_i(v) & - & x_i(u).
\end{array}</math>
|}

In the above transcription, the operator bracket of the form <math>\texttt{(} \ldots \texttt{,} \ldots \texttt{)}</math> is a ''cactus lobe'', in general signifying that just one of the arguments listed is false.  In the case of two arguments this is the same thing as saying that the arguments are not equal.  The plus sign signifies boolean addition, in the sense of addition in <math>\mathrm{GF}(2),</math> and thus means the same thing in this context as the minus sign, in the sense of adding the additive inverse.

The above definition of <math>\mathrm{d}x_i : X^2 \to \mathbb{B}</math> is equivalent to defining <math>\mathrm{d}x_i : (\mathbb{B} \to X) \to \mathbb{B}</math> in the following way:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{lcrcl}
\mathrm{d}x_i (q)
& = & \texttt{(} ~ x_i(q_0) & \texttt{,} & x_i(q_1) ~ \texttt{)}
\\
& = & x_i(q_0) & + & x_i(q_1)
\\
& = & x_i(q_1) & - & x_i(q_0).
\end{array}</math>
|}

In this definition <math>q_b = q(b),</math> for each <math>b</math> in <math>\mathbb{B}.</math>  Thus, the proposition <math>\mathrm{d}x_i</math> is true of the path <math>q = (u, v)</math> exactly if the terms of <math>q,</math> the endpoints <math>u</math> and <math>v,</math> lie on different sides of the question <math>x_i.</math>

The language of features in <math>\langle \mathrm{d}\mathcal{X} \rangle,</math> indeed the whole calculus of propositions in <math>[\mathrm{d}\mathcal{X}],</math> may now be used to classify paths and sets of paths.  In other words, the paths can be taken as models of the propositions <math>g : \mathrm{d}X \to \mathbb{B}.</math>  For example, the paths corresponding to <math>\mathrm{diag}(X)</math> fall under the description <math>\texttt{(} \mathrm{d}x_1 \texttt{)} \cdots \texttt{(} \mathrm{d}x_n \texttt{)},</math> which says that nothing changes against the backdrop of the coordinate frame <math>\{ x_1, \ldots, x_n \}.</math>

Finally, a few words of explanation may be in order.  If this concept of a path appears to be described in a roundabout fashion, it is because I am trying to avoid using any assumption of vector space properties for the space <math>X</math> that contains its range.  In many ways the treatment is still unsatisfactory, but improvements will have to wait for the introduction of substitution operators acting on singular propositions.

==={{anchor|Extended Universe}}The Extended Universe of Discourse===

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
At the moment of speaking, I would like to have perceived a nameless voice, long preceding me, leaving me merely to enmesh myself in it, taking up its cadence, and to lodge myself, when no one was looking, in its interstices as if it had paused an instant, in suspense, to beckon to me.
| width="4%" |  
|-
| align="right" colspan="3" | — Michel Foucault, ''The Discourse on Language'', [Fou, 215]
|}

The <i>extended basis</i> <math>\mathrm{E}\mathcal{A}</math> of a universe of discourse <math>[\mathcal{A}]</math> is formed by taking the initial basis <math>\mathcal{A}</math> together with the differential basis <math>\mathrm{d}\mathcal{A}.</math>  Thus we have the following formula.

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{lclcl}
\mathrm{E}\mathcal{A}
& = & \mathcal{A} \cup \mathrm{d}\mathcal{A}
& = & \{ a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}.
\end{array}</math>
|}

This supplies enough material to construct the <i>differential extension</i> <math>\mathrm{E}A</math> of the space <math>A,</math> also called the <i>tangent bundle</i> of <math>A,</math> in the following fashion.

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{lcl}
\mathrm{E}A
& = & \langle \mathrm{E}\mathcal{A} \rangle
\\[4pt]
& = & \langle \mathcal{A} \cup \mathrm{d}\mathcal{A} \rangle
\\[4pt]
& = & \langle a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle,
\end{array}</math>
|}

and also

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{lcl}
\mathrm{E}A
& = & A \times \mathrm{d}A
\\[4pt]
& = & A_1 \times \ldots \times A_n \times \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n.
\end{array}</math>
|}

That gives <math>\mathrm{E}A</math> the type <math>\mathbb{B}^n \times \mathbb{D}^n.</math>

Finally, the extended universe <math>\mathrm{E}A^\bullet = [\mathrm{E}\mathcal{A}]</math> is the full collection of points and functions, or interpretations and propositions, based on the extended set of features <math>\mathrm{E}\mathcal{A},</math> a fact summed up in the following notation.

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{lclcl}
\mathrm{E}A^\bullet
& = & [\mathrm{E}\mathcal{A}]
& = & [a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n].
\end{array}</math>
|}

That gives the extended universe <math>\mathrm{E}A^\bullet</math> the following type.

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{lcl}
(\mathbb{B}^n \times \mathbb{D}^n\ +\!\to \mathbb{B})
& = & (\mathbb{B}^n \times \mathbb{D}^n, (\mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B})).
\end{array}</math>
|}

A proposition in the extended universe <math>[\mathrm{E}\mathcal{A}]</math> is called a <i>differential proposition</i> and forms the logical analogue of a system of differential equations, constraints, or relations in ordinary calculus.

With these constructions, the differential extension <math>\mathrm{E}A</math> and the space of differential propositions <math>(\mathrm{E}A \to \mathbb{B}),</math> we arrive at the launchpad of our space explorations.

Table 8 summarizes the notations needed to describe the first order differential extensions of propositional calculi in a systematic manner.

<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 8.} ~~ \text{Differential Extension : Basic Notation}</math>
|- style="height:40px; background:ghostwhite"
| <math>\text{Symbol}</math>
| <math>\text{Notation}</math>
| <math>\text{Description}</math>
| <math>\text{Type}</math>
|-
| <math>\mathrm{d}\mathfrak{A}</math>
| <math>\{ {}^{\backprime\backprime} \mathrm{d}a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} \mathrm{d}a_n {}^{\prime\prime} \}</math>
|
<math>\begin{matrix}
\text{Alphabet of}
\\[2pt]
\text{differential symbols}
\end{matrix}</math>
| <math>[n] = \mathbf{n}</math>
|-
| <math>\mathrm{d}\mathcal{A}</math>
| <math>\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}</math>
|
<math>\begin{matrix}
\text{Basis of}
\\[2pt]
\text{differential features}
\end{matrix}</math>
| <math>[n] = \mathbf{n}</math>
|-
| <math>\mathrm{d}A_i</math>
| <math>\{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \}</math>
| <math>\text{Differential dimension}~ i</math>
| <math>\mathbb{D}</math>
|-
| <math>\mathrm{d}A</math>
|
<math>\begin{matrix}
\langle \mathrm{d}\mathcal{A} \rangle
\\[2pt]
\langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle
\\[2pt]
\{ (\mathrm{d}a_1, \ldots, \mathrm{d}a_n) \}
\\[2pt]
\mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n
\\[2pt]
\textstyle \prod_i \mathrm{d}A_i
\end{matrix}</math>
|
<math>\begin{matrix}
\text{Tangent space at a point:}
\\[2pt]
\text{Set of changes, motions,}
\\[2pt]
\text{steps, tangent vectors}
\\[2pt]
\text{at a point}
\end{matrix}</math>
| <math>\mathbb{D}^n</math>
|-
| <math>\mathrm{d}A^*</math>
| <math>(\mathrm{hom} : \mathrm{d}A \to \mathbb{B})</math>
| <math>\text{Linear functions on}~ \mathrm{d}A</math>
| <math>(\mathbb{D}^n)^* \cong \mathbb{D}^n</math>
|-
| <math>\mathrm{d}A^\uparrow</math>
| <math>(\mathrm{d}A \to \mathbb{B})</math>
| <math>\text{Boolean functions on}~ \mathrm{d}A</math>
| <math>\mathbb{D}^n \to \mathbb{B}</math>
|-
| <math>\mathrm{d}A^\bullet</math>
|
<math>\begin{matrix}
[\mathrm{d}\mathcal{A}]
\\[2pt]
(\mathrm{d}A, \mathrm{d}A^\uparrow)
\\[2pt]
(\mathrm{d}A ~+\!\to \mathbb{B})
\\[2pt]
(\mathrm{d}A, (\mathrm{d}A \to \mathbb{B}))
\\[2pt]
[\mathrm{d}a_1, \ldots, \mathrm{d}a_n]
\end{matrix}</math>
|
<math>\begin{matrix}
\text{Tangent universe at a point of}~ A^\bullet,
\\[2pt]
\text{based on the tangent features}
\\[2pt]
\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}
\end{matrix}</math>
|
<math>\begin{matrix}
(\mathbb{D}^n, (\mathbb{D}^n \to \mathbb{B}))
\\[2pt]
(\mathbb{D}^n ~+\!\to \mathbb{B})
\\[2pt]
[\mathbb{D}^n]
\end{matrix}</math>
|}

<br>

The adjective <i>differential</i> or <i>tangent</i> is systematically attached to every construct based on the differential alphabet <math>\mathrm{d}\mathfrak{A},</math> taken by itself.  In like fashion, the adjective <i>extended</i> or the substantive <i>bundle</i> is systematically attached to any construct associated with the full complement of <math>{2n}</math> features.

It eventually becomes necessary to extend the initial alphabet even further, to allow for the discussion of higher order differential expressions. Table 9 provides a suggestion of how these further extensions can be carried out.

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:75%"
|+ style="height:30px" | <math>\text{Table 9.} ~~ \text{Higher Order Differential Features}</math>
|
<p><math>\begin{array}{lllll}
\mathrm{d}^0 \mathcal{A} & = & \{ a_1, \ldots, a_n \} & = & \mathcal{A}
\\
\mathrm{d}^1 \mathcal{A} & = & \{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \} & = & \mathrm{d}\mathcal{A}
\end{array}</math></p>

<p><math>\begin{array}{lll}
\mathrm{d}^k \mathcal{A} & = & \{ \mathrm{d}^k a_1, \ldots, \mathrm{d}^k a_n \}
\\
\mathrm{d}^* \mathcal{A} & = & \{ \mathrm{d}^0 \mathcal{A}, \ldots, \mathrm{d}^k \mathcal{A}, \ldots \}
\end{array}</math></p>
|
<math>\begin{array}{lll}
\mathrm{E}^0 \mathcal{A} & = & \mathrm{d}^0 \mathcal{A}
\\
\mathrm{E}^1 \mathcal{A} & = & \mathrm{d}^0 \mathcal{A} ~\cup~ \mathrm{d}^1 \mathcal{A}
\\
\mathrm{E}^k \mathcal{A} & = & \mathrm{d}^0 \mathcal{A} ~\cup~ \ldots ~\cup~ \mathrm{d}^k \mathcal{A}
\\
\mathrm{E}^\infty \mathcal{A} & = & \bigcup~ \mathrm{d}^* \mathcal{A}
\end{array}</math>
|}

<br>

===Intentional Propositions===

{| width="100%" cellpadding="0" cellspacing="0"
| width="40%" |  
| width="60%" |
Do you guess I have some intricate purpose?<br>
Well I have . . . . for the April rain has, and the mica on<br>
     the side of a rock has.
|-
|  
| align="right" | — Walt Whitman, ''Leaves of Grass'', [Whi, 45]
|}

In order to analyze the behavior of a system at successive moments in time, while staying within the limitations of propositional logic, it is necessary to create independent alphabets of logical features for each moment of time we contemplate using in our discussion.  Those moments have reference to typical instances and relative intervals, not actual or absolute times.

For example, to discuss <i>velocities</i> (first order rates of change) we need to consider points of time in pairs.  There are a number of natural ways of doing this.  Given an initial alphabet, we could use its symbols as a lexical basis to generate successive alphabets of compound symbols, say, with temporal markers appended as suffixes.

As a standard way of dealing with these situations, the following scheme of notation suggests a way of extending any alphabet of logical features through as many temporal moments as a particular order of analysis may demand.  The lexical operators <math>\mathrm{p}^k</math> and <math>\mathrm{Q}^k</math> are convenient in many contexts where the accumulation of prime symbols and union symbols would otherwise be cumbersome.

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:75%"
|+ style="height:30px" | <math>\text{Table 10.} ~~ \text{A Realm of Intentional Features}</math>
|
<p><math>\begin{array}{lllll}
\mathrm{p}^0 \mathcal{A} & = & \{ a_1, \ldots, a_n \} & = & \mathcal{A}
\\
\mathrm{p}^1 \mathcal{A} & = & \{ a_1^\prime, \ldots, a_n^\prime \} & = & \mathcal{A}^\prime
\\
\mathrm{p}^2 \mathcal{A} & = & \{ a_1^{\prime\prime}, \ldots, a_n^{\prime\prime} \} & = & \mathcal{A}^{\prime\prime}
\\
\cdots & & \cdots &
\end{array}</math></p>

<p><math>\begin{array}{lll}
\mathrm{p}^k \mathcal{A} & = & \{ \mathrm{p}^k a_1, \ldots, \mathrm{p}^k a_n \}
\end{array}</math></p>
|
<math>\begin{array}{lll}
\mathrm{Q}^0 \mathcal{A} & = & \mathcal{A}
\\
\mathrm{Q}^1 \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}'
\\
\mathrm{Q}^2 \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}' \cup \mathcal{A}''
\\
\cdots & & \cdots
\\
\mathrm{Q}^k \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}' \cup \ldots \cup \mathrm{p}^k \mathcal{A}
\end{array}</math>
|}

<br>

The resulting augmentations of our logical basis determine a series of discursive universes that may be called the ''intentional extension'' of propositional calculus. This extension follows a pattern analogous to the differential extension, which was developed in terms of the operators <math>\mathrm{d}^k</math> and <math>\mathrm{E}^k,</math> and there is a natural relation between these two extensions that bears further examination. In contexts displaying this pattern, where a sequence of domains stretches from an anchoring domain <math>X</math> through an indefinite number of higher reaches, a particular collection of domains based on <math>X</math> will be referred to as a ''realm'' of <math>X,</math> and when the succession exhibits a temporal aspect, as a ''reign'' of <math>X.</math>

For the purposes of this discussion, an ''intentional proposition'' is defined as a proposition in the universe of discourse <math>\mathrm{Q}X^\bullet = [\mathrm{Q}\mathcal{X}],</math> in other words, a map <math>q : \mathrm{Q}X \to \mathbb{B}.</math> The sense of this definition may be seen if we consider the following facts. First, the equivalence <math>\mathrm{Q}X = X \times X'</math> motivates the following chain of isomorphisms between spaces:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{lllcl}
(\mathrm{Q}X \to \mathbb{B})
& \cong & (X & \times & ~X' \to \mathbb{B})
\\[4pt]
& \cong & (X & \to & (X' \to \mathbb{B}))
\\[4pt]
& \cong & (X' & \to & (X~ \to \mathbb{B})).
\end{array}</math>
|}

Viewed in this light, an intentional proposition <math>q</math> may be rephrased as a map <math>q : X \times X' \to \mathbb{B},</math> which judges the juxtaposition of states in <math>X</math> from one moment to the next. Alternatively, <math>q</math> may be parsed in two stages in two different ways, as <math>q : X \to (X' \to \mathbb{B})</math> and as <math>q : X' \to (X \to \mathbb{B}),</math> which associate to each point of <math>X</math> or <math>X'</math> a proposition about states in <math>X'</math> or <math>X,</math> respectively. In this way, an intentional proposition embodies a type of value system, in effect, a proposal that places a value on a collection of ends-in-view, or a project that evaluates a set of goals as regarded from each point of view in the state space of a system.

In sum, the intentional proposition <math>q</math> indicates a method for the systematic selection of local goals. As a general form of description, a map of the type <math>q : \mathrm{Q}^i X \to \mathbb{B}</math> may be referred to as an "<math>i^\text{th}</math> order intentional proposition". Naturally, when we speak of intentional propositions without qualification, we usually mean first order intentions.

Many different realms of discourse have the same structure as the extensions that have been indicated here. From a strictly logical point of view, each new layer of terms is composed of independent logical variables that are no different in kind from those that go before, and each further course of logical atoms is treated like so many individual, but otherwise indifferent bricks by the prototype computer program that I use as a propositional interpreter. Thus, the names that I use to single out the differential and the intentional extensions, and the lexical paradigms that I follow to construct them, are meant to suggest the interpretations that I have in mind, but they can only hint at the extra meanings that human communicators may pack into their terms and inflections.

As applied here, the word ''intentional'' is drawn from common use and may have little bearing on its technical use in other, more properly philosophical, contexts. I am merely using the complex of intentional concepts — aims, ends, goals, objectives, purposes, and so on — metaphorically to flesh out and vividly to represent any situation where one needs to contemplate a system in multiple aspects of state and destination, that is, its being in certain states and at the same time acting as if headed through certain states. If confusion arises, more neutral words like ''conative'', ''contingent'', ''discretionary'', ''experimental'', ''kinetic'', ''progressive'', ''tentative'', or ''trial'' would probably serve as well.

==={{anchor|Easy Street}}Life on Easy Street===

{| width="100%" cellpadding="0" cellspacing="0"
| width="40%" |  
| width="60%" |
Failing to fetch me at first keep encouraged,<br>
Missing me one place search another,<br>
I stop some where waiting for you
|-
|  
| align="right" | — Walt Whitman, ''Leaves of Grass'', [Whi, 88]
|}

The finite character of the extended universe <math>[\mathrm{E}\mathcal{A}]</math> makes the task of solving differential propositions relatively straightforward.  The solution set of the differential proposition <math>q : \mathrm{E}A \to \mathbb{B}</math> is the set of models <math>q^{-1}(1)</math> in <math>\mathrm{E}A.</math>  Finding all models of <math>q,</math> the extended interpretations in <math>\mathrm{E}A</math> which satisfy <math>q,</math> can be carried out by a finite search.

Being in possession of complete algorithms for propositional calculus modeling, theorem checking, and theorem proving makes the analytic task fairly simple in principle, even if the question of efficiency in the face of arbitrary complexity remains another matter entirely.  The NP‑completeness of propositional satisfiability may weigh against the prospects of a single efficient algorithm capable of covering the whole space <math>[\mathrm{E}\mathcal{A}]</math> with equal facility but there appears to be much room for improvement in classifying special forms and in developing algorithms tailored to their practical processing.

In view of these constraints and contingencies, our focus shifts to the tasks of approximation and interpretation that support intuition, especially in dealing with the natural kinds of differential propositions that arise in applications, and in the effort to understand, in succinct and adaptive forms, their dynamic implications. In the absence of direct insights, these tasks are partially carried out by forging analogies with the familiar situations and customary routines of ordinary calculus. But the indirect approach, going by way of specious analogy and intuitive habit, forces us to remain on guard against the circumstance that occurs when the word ''forging'' takes on its shadier nuance, indicting the constant risk of a counterfeit in the proportion.

<br>

----
<div align="center">
• [[Differential Logic and Dynamic Systems • Overview|Overview]]
• [[Differential Logic and Dynamic Systems • Part 1|Part 1]]
• [[Differential Logic and Dynamic Systems • Part 2|Part 2]]
• [[Differential Logic and Dynamic Systems • Part 3|Part 3]]
• [[Differential Logic and Dynamic Systems • Part 4|Part 4]]
• [[Differential Logic and Dynamic Systems • Part 5|Part 5]]
• [[Differential Logic and Dynamic Systems • Appendices|Appendices]]
• [[Differential Logic and Dynamic Systems • References|References]]
• [[Differential Logic and Dynamic Systems • Document History|Document History]]
•
</div>
----

[[Category:Adaptive systems]]
[[Category:Artificial intelligence]]
[[Category:Boolean algebra]]
[[Category:Boolean functions]]
[[Category:Category theory]]
[[Category:Combinatorics]]
[[Category:Computation theory]]
[[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:Logic]]
[[Category:Logical graphs]]
[[Category:Neural networks]]
[[Category:Peirce, Charles Sanders]]
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[[Category:Systems theory]]
[[Category:Visualization]]

Differential Logic and Dynamic Systems • Overview

2026-02-02T16:14:25Z

Jon Awbrey: __NOTOC__<b>Author: Jon Awbrey</b>

Differential Logic and Dynamic Systems • Part 1

2026-02-02T16:12:20Z

Jon Awbrey: + Differential Logic and Dynamic Systems • Part 1

<b>Author: [[User:Jon Awbrey|Jon Awbrey]]</b>

----
<div align="center">
• [[Differential Logic and Dynamic Systems • Overview|Overview]]
• [[Differential Logic and Dynamic Systems • Part 1|Part 1]]
• [[Differential Logic and Dynamic Systems • Part 2|Part 2]]
• [[Differential Logic and Dynamic Systems • Part 3|Part 3]]
• [[Differential Logic and Dynamic Systems • Part 4|Part 4]]
• [[Differential Logic and Dynamic Systems • Part 5|Part 5]]
• [[Differential Logic and Dynamic Systems • Appendices|Appendices]]
• [[Differential Logic and Dynamic Systems • References|References]]
• [[Differential Logic and Dynamic Systems • Document History|Document History]]
•
</div>
----

{| align="center" cellpadding="10"
| [[File:Diff Log Dyn Sys • Tangent Functor Ferris Wheel 2.0.png|448px]]
|}

{| style="height:36px; width:100%"
| align="left" | ''Stand and unfold yourself.''
| align="right" | Hamlet: Francisco—1.1.2
|}

In modeling intelligent systems, whether we are trying to understand a natural system or engineer an artificial system, there has long been a tension or trade‑off between dynamic paradigms and symbolic paradigms.  Dynamic models take their cue from physics, using quantitative measures and differential equations to model the evolution of a system's state through time.  Symbolic models use logical methods to describe systems and their agents in qualitative terms, deriving logical consequences of a system's description or an agent's state of information.  Logic-based systems have tended to be static in character, largely because we have lacked a proper logical analogue of differential calculus.  The work laid out in this report is intended to address that lack.

This article develops a differential extension of propositional calculus and applies it to the analysis of dynamic systems whose states are described in qualitative logical terms.  The work pursued here is coordinated with a parallel application focusing on neural network systems but the dependencies are arranged to make the present article the main and the more self-contained work, to serve as a conceptual frame and a technical background for the network project.

==Review and Transition==

This note continues a previous discussion on the problem of dealing with change and diversity in logic-based intelligent systems.  It is useful to begin by summarizing essential material from previous reports.

Table 1 outlines a notation for propositional calculus based on two types of logical connectives, both of variable <math>k</math>-ary scope.

* A bracketed list of propositional expressions in the form <math>\texttt{(} e_1 \texttt{,} e_2 \texttt{,} \ldots \texttt{,} e_{k-1} \texttt{,} e_k \texttt{)}</math> indicates that exactly one of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> is false.

* A concatenation of propositional expressions in the form <math>e_1 ~ e_2 ~ \ldots ~ e_{k-1} ~ e_k</math> indicates that all of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> are true, in other words, that their [[logical conjunction]] is true.

All other propositional connectives can be obtained in a very efficient style of representation through combinations of these two forms.  Strictly speaking, the concatenation form is dispensable in light of the bracketed form but it is convenient to maintain it as an abbreviation of more complicated bracket expressions.

{| align="center" style="font-size:larger; text-align:center; width:100%"
| height="20px" | <math>\text{Table 1. Syntax and Semantics of a Calculus for Propositional Logic}</math>
|-
| [[File:Syntax and Semantics of a Calculus for Propositional Logic 4.0.png|600px]]
|}

Table 1 is the first of several “Rosetta Stones” we'll use in this discussion to translate between different languages for the same subject matters.  In this case the Table displays equivalent expressions for simple examples of propositional forms in four notations for propositional calculus.

* Column 1 shows the logical graphs used to represent a number of simple propositional forms.
* Column 2 shows the traverse strings corresponding to the logical graphs in Column 1.
* Column 3 interprets the graph and string by means of conventional verbal formulas.
* Column 4 translates the interpretation into a number of symbolic notations.

This treatment of propositional logic is derived from the work of C.S. Peirce [P1, P2], who gave this approach an extensive development in his graphical systems of predicate, relational, and modal logic [Rob]. More recently, these ideas were revived and supplemented in an alternative interpretation by George Spencer-Brown [SpB]. Both of these authors used other forms of enclosure where I use parentheses, but the structural topologies of expression and the functional varieties of interpretation are fundamentally the same.

While working with expressions solely in propositional calculus, it is easiest to use plain parentheses for logical connectives. In contexts where parentheses are needed for other purposes “teletype” parentheses <math>\texttt{(} \ldots \texttt{)}</math> or barred parentheses <math>(\!| \ldots |\!)</math> may be used for logical operators.

The briefest expression for logical truth is the empty word, usually denoted by <math>{}^{\backprime\backprime} \boldsymbol\varepsilon {}^{\prime\prime}</math> or <math>{}^{\backprime\backprime} \boldsymbol\lambda {}^{\prime\prime}</math> in formal languages, where it forms the identity element for concatenation. To make it visible in this text, it may be denoted by the equivalent expression <math>{}^{\backprime\backprime} \texttt{((} ~ \texttt{))} {}^{\prime\prime},</math> or, especially if operating in an algebraic context, by a simple <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}.</math> Also when working in an algebraic mode, the plus sign <math>{}^{\backprime\backprime} + {}^{\prime\prime}</math> may be used for [[exclusive disjunction]]. For example, we have the following paraphrases of algebraic expressions by bracket expressions:

{| align="center" cellpadding="6" style="text-align:center"
|
<math>\begin{matrix}
x + y ~=~ \texttt{(} x, y \texttt{)}
\\[6pt]
x + y + z ~=~ \texttt{((} x, y \texttt{)}, z \texttt{)} ~=~ \texttt{(} x, \texttt{(} y, z \texttt{))}
\end{matrix}</math>
|}

It is important to note that the last expressions are not equivalent to the triple bracket <math>\texttt{(} x, y, z \texttt{)}.</math>

<b>Note.</b> The usage that one often sees, of a plus sign "<math>+</math>" to represent inclusive disjunction, and the reference to this operation as ''boolean addition'', is a misnomer on at least two counts. Boole used the plus sign to represent exclusive disjunction (at any rate, an operation of aggregation restricted in its logical interpretation to cases where the represented sets are disjoint (Boole, 32)), as any mathematician with a sensitivity to the ring and field properties of algebra would do:

<blockquote>
The expression <math>x + y</math> seems indeed uninterpretable, unless it be assumed that the things represented by <math>x</math> and the things represented by <math>y</math> are entirely separate; that they embrace no individuals in common. (Boole, 66).
</blockquote>

It was only later that Peirce and Jevons treated inclusive disjunction as a fundamental operation, but these authors, with a respect for the algebraic properties that were already associated with the plus sign, used a variety of other symbols for inclusive disjunction (Sty, 177, 189). It seems to have been Schröder who later reassigned the plus sign to inclusive disjunction (Sty, 208). Additional information, discussion, and references can be found in (Boole) and (Sty, 177–263). Aside from these historical points, which never really count against a current practice that has gained a life of its own, this usage does have a further disadvantage of cutting or confounding the lines of communication between algebra and logic. For this reason, it will be avoided here.

==A Functional Conception of Propositional Calculus==

{| width="100%" cellpadding="0" cellspacing="0"
| width="40%" |  
| width="60%" |
Out of the dimness opposite equals advance . . . .<br>
     Always substance and increase,<br>
Always a knit of identity . . . . always distinction . . . .<br>
     always a breed of life.
|-
|  
| align="right" | — Walt Whitman, ''Leaves of Grass'', [Whi, 28]
|}

In the general case, we start with a set of logical features <math>\{a_1, \ldots, a_n\}</math> that represent properties of objects or propositions about the world. In concrete examples the features <math>\{a_i\}</math> commonly appear as capital letters from an ''alphabet'' like <math>\{A, B, C, \ldots\}</math> or as meaningful words from a linguistic ''vocabulary'' of codes. This language can be drawn from any sources, whether natural, technical, or artificial in character and interpretation. In the application to dynamic systems we tend to use the letters <math>\{x_1, \ldots, x_n\}</math> as our coordinate propositions, and to interpret them as denoting properties of a system's ''state'', that is, as propositions about its location in configuration space. Because I have to consider non-deterministic systems from the outset, I often use the word ''state'' in a loose sense, to denote the position or configuration component of a contemplated state vector, whether or not it ever gets a deterministic completion.

The set of logical features <math>\{a_1, \ldots, a_n\}</math> provides a basis for generating an <math>n</math>-dimensional ''[[universe of discourse]]'' that I denote as <math>[a_1, \ldots, a_n].</math> It is useful to consider each universe of discourse as a unified categorical object that incorporates both the set of points <math>\langle a_1, \ldots, a_n \rangle</math> and the set of propositions <math>f : \langle a_1, \ldots, a_n \rangle \to \mathbb{B}</math> that are implicit with the ordinary picture of a venn diagram on <math>n</math> features. Thus, we may regard the universe of discourse <math>[a_1, \ldots, a_n]</math> as an ordered pair having the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}),</math> and we may abbreviate this last type designation as <math>\mathbb{B}^n\ +\!\to \mathbb{B},</math> or even more succinctly as <math>[\mathbb{B}^n].</math> (Used this way, the angle brackets <math>\langle\ldots\rangle</math> are referred to as ''generator brackets''.)

Table 2 exhibits the scheme of notation I use to formalize the domain of propositional calculus, corresponding to the logical content of truth tables and venn diagrams. Although it overworks the square brackets a bit, I also use either one of the equivalent notations <math>[n]</math> or <math>\mathbf{n}</math> to denote the data type of a finite set on <math>n</math> elements.

<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 2.} ~~ \text{Propositional Calculus : Basic Notation}</math>
|- style="height:40px; background:ghostwhite"
| <math>\text{Symbol}</math>
| <math>\text{Notation}</math>
| <math>\text{Description}</math>
| <math>\text{Type}</math>
|-
| <math>\mathfrak{A}</math>
| <math>\{ {}^{\backprime\backprime} a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} a_n {}^{\prime\prime} \}</math>
| <math>\text{Alphabet}</math>
| <math>[n] = \mathbf{n}</math>
|-
| <math>\mathcal{A}</math>
| <math>\{ a_1, \ldots, a_n \}</math>
| <math>\text{Basis}</math>
| <math>[n] = \mathbf{n}</math>
|-
| <math>A_i</math>
| <math>\{ \texttt{(} a_i \texttt{)}, a_i \}</math>
| <math>\text{Dimension}~ i</math>
| <math>\mathbb{B}</math>
|-
| <math>A</math>
|
<math>\begin{matrix}
\langle \mathcal{A} \rangle
\\[2pt]
\langle a_1, \ldots, a_n \rangle
\\[2pt]
\{ (a_1, \ldots, a_n) \}
\\[2pt]
A_1 \times \ldots \times A_n
\\[2pt]
\textstyle \prod_{i=1}^n A_i
\end{matrix}</math>
|
<math>\begin{matrix}
\text{Set of cells},
\\[2pt]
\text{coordinate tuples},
\\[2pt]
\text{points, or vectors}
\\[2pt]
\text{in the universe}
\\[2pt]
\text{of discourse}
\end{matrix}</math>
| <math>\mathbb{B}^n</math>
|-
| <math>A^*</math>
| <math>(\mathrm{hom} : A \to \mathbb{B})</math>
| <math>\text{Linear functions}</math>
| <math>(\mathbb{B}^n)^* \cong \mathbb{B}^n</math>
|-
| <math>A^\uparrow</math>
| <math>(A \to \mathbb{B})</math>
| <math>\text{Boolean functions}</math>
| <math>\mathbb{B}^n \to \mathbb{B}</math>
|-
| <math>A^\bullet</math>
|
<math>\begin{matrix}
[\mathcal{A}]
\\[2pt]
(A, A^\uparrow)
\\[2pt]
(A ~+\!\to \mathbb{B})
\\[2pt]
(A, (A \to \mathbb{B}))
\\[2pt]
[a_1, \ldots, a_n]
\end{matrix}</math>
|
<math>\begin{matrix}
\text{Universe of discourse}
\\[2pt]
\text{based on the features}
\\[2pt]
\{ a_1, \ldots, a_n \}
\end{matrix}</math>
|
<math>\begin{matrix}
(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))
\\[2pt]
(\mathbb{B}^n ~+\!\to \mathbb{B})
\\[2pt]
[\mathbb{B}^n]
\end{matrix}</math>
|}

<br>

===Qualitative Logic and Quantitative Analogy===

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
''Logical'', however, is used in a third sense, which is at once more vital and more practical; to denote, namely, the systematic care, negative and positive, taken to safeguard reflection so that it may yield the best results under the given conditions.
| width="4%" |  
|-
| align="right" colspan="3" | — John Dewey, ''How We Think'', [Dew, 56]
|}

These concepts and notations may now be explained in greater detail. In order to begin as simply as possible, let us distinguish two levels of analysis and set out initially on the easier path. On the first level of analysis we take spaces like <math>\mathbb{B},</math> <math>\mathbb{B}^n,</math> and <math>(\mathbb{B}^n \to \mathbb{B})</math> at face value and treat them as the primary objects of interest. On the second level of analysis we use these spaces as coordinate charts for talking about points and functions in more fundamental spaces.

A pair of spaces, of types <math>\mathbb{B}^n</math> and <math>(\mathbb{B}^n \to \mathbb{B}),</math> give typical expression to everything we commonly associate with the ordinary picture of a venn diagram. The dimension, <math>n,</math> counts the number of “circles” or simple closed curves that are inscribed in the universe of discourse, corresponding to its relevant logical features or basic propositions. Elements of type <math>\mathbb{B}^n</math> correspond to what are often called propositional ''interpretations'' in logic, that is, the different assignments of truth values to sentence letters. Relative to a given universe of discourse, these interpretations are visualized as its ''cells'', in other words, the smallest enclosed areas or undivided regions of the venn diagram. The functions <math>f : \mathbb{B}^n \to \mathbb{B}</math> correspond to the different ways of shading the venn diagram to indicate arbitrary propositions, regions, or sets. Regions included under a shading indicate the ''models'', and regions excluded represent the ''non-models'' of a proposition. To recognize and formalize the natural cohesion of these two layers of concepts into a single universe of discourse, we introduce the type notations <math>[\mathbb{B}^n] = \mathbb{B}^n\ +\!\to \mathbb{B}</math> to stand for the pair of types <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})).</math> The resulting “stereotype” serves to frame the universe of discourse as a unified categorical object, and makes it subject to prescribed sets of evaluations and transformations (categorical morphisms or ''arrows'') that affect the universe of discourse as an integrated whole.

Most of the time we can serve the algebraic, geometric, and logical interests of our study without worrying about their occasional conflicts and incidental divergences. The conventions and definitions already set down will continue to cover most of the algebraic and functional aspects of our discussion, but to handle the logical and qualitative aspects we will need to add a few more. In general, abstract sets may be denoted by gothic, greek, or script capital variants of <math>A, B, C,</math> and so on, with their elements being denoted by a corresponding set of subscripted letters in plain lower case, for example, <math>\mathcal{A} = \{a_i\}.</math> Most of the time, a set such as <math>\mathcal{A} = \{a_i\}</math> will be employed as the ''alphabet'' of a [[formal language]]. These alphabet letters serve to name the logical features (properties or propositions) that generate a particular universe of discourse. When we want to discuss the particular features of a universe of discourse, beyond the abstract designation of a type like <math>(\mathbb{B}^n\ +\!\to \mathbb{B}),</math> then we may use the following notations. If <math>\mathcal{A} = \{a_1, \ldots, a_n\}</math> is an alphabet of logical features, then <math>A = \langle \mathcal{A} \rangle = \langle a_1, \ldots, a_n \rangle</math> is the set of interpretations, <math>A^\uparrow = (A \to \mathbb{B})</math> is the set of propositions, and <math>A^\bullet = [\mathcal{A}] = [a_1, \ldots, a_n]</math> is the combination of these interpretations and propositions into the universe of discourse that is based on the features <math>\{a_1, \ldots, a_n\}.</math>

As always, especially in concrete examples, these rules may be dropped whenever necessary, reverting to a free assortment of feature labels. However, when we need to talk about the logical aspects of a space that is already named as a vector space, it will be necessary to make special provisions. At any rate, these elaborations can be deferred until actually needed.

==={{anchor|Philosophy of Notation}}Philosophy of Notation : Formal Terms and Flexible Types===

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
Where number is irrelevant, regimented mathematical technique has hitherto tended to be lacking. Thus it is that the progress of natural science has depended so largely upon the discernment of measurable quantity of one sort or another.
| width="4%" |  
|-
| align="right" colspan="3" | — W.V. Quine, ''Mathematical Logic'', [Qui, 7]
|}

For much of our discussion propositions and boolean functions are treated as the same formal objects, or as different interpretations of the same formal calculus. This rule of interpretation has exceptions, though. There is a distinctively logical interest in the use of propositional calculus that is not exhausted by its functional interpretation. It is part of our task in this study to deal with these uniquely logical characteristics as they present themselves both in our subject matter and in our formal calculus. Just to provide a hint of what's at stake: In logic, as opposed to the more imaginative realms of mathematics, we consider it a good thing to always know what we are talking about. Where mathematics encourages tolerance for uninterpreted symbols as intermediate terms, logic exerts a keener effort to interpret directly each oblique carrier of meaning, no matter how slight, and to unfold the complicities of every indirection in the flow of information. Translated into functional terms, this means that we want to maintain a continual, immediate, and persistent sense of both the converse relation <math>f^{-1} \subseteq \mathbb{B} \times \mathbb{B}^n,</math> or what is the same thing, <math>f^{-1} : \mathbb{B} \to \mathcal{P}(\mathbb{B}^n),</math> and the ''fibers'' or inverse images <math>f^{-1}(0)</math> and <math>f^{-1}(1),</math> associated with each boolean function <math>f : \mathbb{B}^n \to \mathbb{B}</math> that we use. In practical terms, the desired implementation of a propositional interpreter should incorporate our intuitive recognition that the induced partition of the functional domain into level sets <math>f^{-1}(b),</math> for <math>b \in \mathbb{B},</math> is part and parcel of understanding the denotative uses of each propositional function <math>f.</math>

==={{anchor|Special Classes}}Special Classes of Propositions===

It is important to remember that the coordinate propositions <math>\{a_i\},</math> besides being projection maps <math>a_i : \mathbb{B}^n \to \mathbb{B},</math> are propositions on an equal footing with all others, even though employed as a basis in a particular moment. This set of <math>n</math> propositions may sometimes be referred to as the ''basic propositions'', the ''coordinate propositions'', or the ''simple propositions'' that found a universe of discourse. Either one of the equivalent notations, <math>\{a_i : \mathbb{B}^n \to \mathbb{B}\}</math> or <math>(\mathbb{B}^n \xrightarrow{i} \mathbb{B}),</math> may be used to indicate the adoption of the propositions <math>a_i</math> as a basis for describing a universe of discourse.

Among the <math>2^{2^n}</math> propositions in <math>[a_1, \ldots, a_n]</math> are several families of <math>2^n</math> propositions each that take on special forms with respect to the basis <math>\{ a_1, \ldots, a_n \}.</math> Three of these families are especially prominent in the present context, the ''linear'', the ''positive'', and the ''singular'' propositions. Each family is naturally parameterized by the coordinate <math>n</math>-tuples in <math>\mathbb{B}^n</math> and falls into <math>n + 1</math> ranks, with a binomial coefficient <math>\tbinom{n}{k}</math> giving the number of propositions that have rank or weight <math>k.</math>

<ul>
{{anchor|Linear Propositions}}<li>
<p>The ''linear propositions'', <math>\{ \ell : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{\ell} \mathbb{B}),</math> may be written as sums:</p>

{| align="center" cellpadding="8" width="90%"
|
<math>\sum_{i=1}^n e_i ~=~ e_1 + \ldots + e_n
~\text{where}~
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 0 \end{matrix}\right\}
~\text{for}~ i = 1 ~\text{to}~ n.</math>
|}</li>

{{anchor|Positive Propositions}}<li>
<p>The ''positive propositions'', <math>\{ p : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{p} \mathbb{B}),</math> may be written as products:</p>

{| align="center" cellpadding="8" width="90%"
|
<math>\prod_{i=1}^n e_i ~=~ e_1 \cdot \ldots \cdot e_n
~\text{where}~
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 1 \end{matrix}\right\}
~\text{for}~ i = 1 ~\text{to}~ n.</math>
|}</li>

{{anchor|Singular Propositions}}<li>
<p>The ''singular propositions'', <math>\{ \mathbf{x} : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{s} \mathbb{B}),</math> may be written as products:</p>

{| align="center" cellpadding="8" width="90%"
|
<math>\prod_{i=1}^n e_i ~=~ e_1 \cdot \ldots \cdot e_n
~\text{where}~
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = \texttt{(} a_i \texttt{)} \end{matrix}\right\}
~\text{for}~ i = 1 ~\text{to}~ n.</math>
|}</li>
</ul>

In each case the rank <math>k</math> ranges from <math>0</math> to <math>n</math> and counts the number of positive appearances of the coordinate propositions <math>a_1, \ldots, a_n</math> in the resulting expression. For example, for <math>{n = 3},</math> the linear proposition of rank <math>0</math> is <math>0,</math> the positive proposition of rank <math>0</math> is <math>1,</math> and the singular proposition of rank <math>0</math> is <math>\texttt{(} a_1 \texttt{)(} a_2 \texttt{)(} a_3\texttt{)}.</math>

The basic propositions <math>a_i : \mathbb{B}^n \to \mathbb{B}</math> are both linear and positive. So these two kinds of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions.

Linear propositions and positive propositions are generated by taking boolean sums and products, respectively, over selected subsets of basic propositions, so both families of propositions are parameterized by the powerset <math>\mathcal{P}(\mathcal{I}),</math> that is, the set of all subsets <math>J</math> of the basic index set <math>\mathcal{I} = \{1, \ldots, n\}.</math>

Let us define <math>\mathcal{A}_J</math> as the subset of <math>\mathcal{A}</math> that is given by <math>\{a_i : i \in J\}.</math> Then we may comprehend the action of the linear and the positive propositions in the following terms:

* The linear proposition <math>\ell_J : \mathbb{B}^n \to \mathbb{B}</math> evaluates each cell <math>\mathbf{x}</math> of <math>\mathbb{B}^n</math> by looking at the coefficients of <math>\mathbf{x}</math> with respect to the features that <math>\ell_J</math> "likes", namely those in <math>\mathcal{A}_J,</math> and then adds them up in <math>\mathbb{B}.</math> Thus, <math>\ell_J(\mathbf{x})</math> computes the parity of the number of features that <math>\mathbf{x}</math> has in <math>\mathcal{A}_J,</math> yielding one for odd and zero for even. Expressed in this idiom, <math>\ell_J(\mathbf{x}) = 1</math> says that <math>\mathbf{x}</math> seems ''odd'' (or ''oddly true'') to <math>\mathcal{A}_J,</math> whereas <math>\ell_J(\mathbf{x}) = 0</math> says that <math>\mathbf{x}</math> seems ''even'' (or ''evenly true'') to <math>\mathcal{A}_J,</math> so long as we recall that ''zero times'' is evenly often, too.

* The positive proposition <math>p_J : \mathbb{B}^n \to \mathbb{B}</math> evaluates each cell <math>\mathbf{x}</math> of <math>\mathbb{B}^n</math> by looking at the coefficients of <math>\mathbf{x}</math> with regard to the features that <math>p_J</math> "likes", namely those in <math>\mathcal{A}_J,</math> and then takes their product in <math>\mathbb{B}.</math> Thus, <math>p_J(\mathbf{x})</math> assesses the unanimity of the multitude of features that <math>\mathbf{x}</math> has in <math>\mathcal{A}_J,</math> yielding one for all and aught for else. In these consensual or contractual terms, <math>p_J(\mathbf{x}) = 1</math> means that <math>\mathbf{x}</math> is ''AOK'' or congruent with all of the conditions of <math>\mathcal{A}_J,</math> while <math>p_J(\mathbf{x}) = 0</math> means that <math>\mathbf{x}</math> defaults or dissents from some condition of <math>\mathcal{A}_J.</math>

===Basis Relativity and Type Ambiguity===

Finally, two things are important to keep in mind with regard to the simplicity, linearity, positivity, and singularity of propositions.

First, all of these properties are relative to a particular basis. For example, a singular proposition with respect to a basis <math>\mathcal{A}</math> will not remain singular if <math>\mathcal{A}</math> is extended by a number of new and independent features. Even if we stick to the original set of pairwise options <math>\{a_i\} \cup \{ \texttt{(} a_i \texttt{)} \}</math> to select a new basis, the sets of linear and positive propositions are determined by the choice of simple propositions, and this determination is tantamount to the conventional choice of a cell as origin.

Second, the singular propositions <math>\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B},</math> picking out as they do a single cell or a coordinate tuple <math>\mathbf{x}</math> of <math>\mathbb{B}^n,</math> become the carriers or the vehicles of a certain type-ambiguity that vacillates between the dual forms <math>\mathbb{B}^n</math> and <math>(\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B})</math> and infects the whole hierarchy of types built on them. In other words, the terms that signify the interpretations <math>\mathbf{x} : \mathbb{B}^n</math> and the singular propositions <math>\mathbf{x} : \mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B}</math> are fully equivalent in information, and this means that every token of the type <math>\mathbb{B}^n</math> can be reinterpreted as an appearance of the subtype <math>\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B}.</math> And vice versa, the two types can be exchanged with each other everywhere that they turn up. In practical terms, this allows the use of singular propositions as a way of denoting points, forming an alternative to coordinate tuples.

For example, relative to the universe of discourse <math>[a_1, a_2, a_3]</math> the singular proposition <math>a_1 a_2 a_3 : \mathbb{B}^3 \xrightarrow{s} \mathbb{B}</math> could be explicitly retyped as <math>a_1 a_2 a_3 : \mathbb{B}^3</math> to indicate the point <math>(1, 1, 1)</math> but in most cases the proper interpretation could be gathered from context. Both notations remain dependent on a particular basis, but the code that is generated under the singular option has the advantage in its self-commenting features, in other words, it constantly reminds us of its basis in the process of denoting points. When the time comes to put a multiplicity of different bases into play, and to search for objects and properties that remain invariant under the transformations between them, this infinitesimal potential advantage may well evolve into an overwhelming practical necessity.

===The Analogy Between Real and Boolean Types===

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
Measurement consists in correlating our subject matter with the series of real numbers; and such correlations are desirable because, once they are set up, all the well-worked theory of numerical mathematics lies ready at hand as a tool for our further reasoning.
| width="4%" |  
|-
| align="right" colspan="3" | — W.V. Quine, ''Mathematical Logic'', [Qui, 7]
|}

There are two further reasons why it useful to spend time on a careful treatment of types, and they both have to do with our being able to take full computational advantage of certain dimensions of flexibility in the types that apply to terms. First, the domains of differential geometry and logic programming are connected by analogies between real and boolean types of the same pattern. Second, the types involved in these patterns have important isomorphisms connecting them that apply on both the real and the boolean sides of the picture.

Amazingly enough, these isomorphisms are themselves schematized by the axioms and theorems of propositional logic. This fact is known as the ''propositions as types'' analogy or the Curry–Howard isomorphism [How]. In another formulation it says that terms are to types as proofs are to propositions. See [LaS, 42–46] and [SeH] for a good discussion and further references. To anticipate the bearing of these issues on our immediate topic, Table 3 sketches a partial overview of the Real to Boolean analogy that may serve to illustrate the paradigm in question.

<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 3.} ~~ \text{Analogy Between Real and Boolean Types}</math>
|- style="height:40px; background:ghostwhite"
| <math>\text{Real Domain} ~ \mathbb{R}</math>
| <math>\longleftrightarrow</math>
| <math>\text{Boolean Domain} ~ \mathbb{B}</math>
|-
| <math>\mathbb{R}^n</math>
| <math>\text{Basic Space}</math>
| <math>\mathbb{B}^n</math>
|-
| <math>\mathbb{R}^n \to \mathbb{R}</math>
| <math>\text{Function Space}</math>
| <math>\mathbb{B}^n \to \mathbb{B}</math>
|-
| <math>(\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R}</math>
| <math>\text{Tangent Vector}</math>
| <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math>
|-
| <math>\mathbb{R}^n \to ((\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R})</math>
| <math>\text{Vector Field}</math>
| <math>\mathbb{B}^n \to ((\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B})</math>
|-
| <math>(\mathbb{R}^n \times (\mathbb{R}^n \to \mathbb{R})) \to \mathbb{R}</math>
| '''<font size="4">"</font>'''
| <math>(\mathbb{B}^n \times (\mathbb{B}^n \to \mathbb{B})) \to \mathbb{B}</math>
|-
| <math>((\mathbb{R}^n \to \mathbb{R}) \times \mathbb{R}^n) \to \mathbb{R}</math>
| '''<font size="4">"</font>'''
| <math>((\mathbb{B}^n \to \mathbb{B}) \times \mathbb{B}^n) \to \mathbb{B}</math>
|-
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^n \to \mathbb{R})</math>
| <math>\text{Derivation}</math>
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^n \to \mathbb{B})</math>
|-
| <math>\mathbb{R}^n \to \mathbb{R}^m</math>
| <math>\begin{matrix}\text{Basic}\\[2pt]\text{Transformation}\end{matrix}</math>
| <math>\mathbb{B}^n \to \mathbb{B}^m</math>
|-
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^m \to \mathbb{R})</math>
| <math>\begin{matrix}\text{Function}\\[2pt]\text{Transformation}\end{matrix}</math>
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^m \to \mathbb{B})</math>
|}

<br>

The Table exhibits a sample of likely parallels between the real and boolean domains. The central column gives a selection of terminology that is borrowed from differential geometry and extended in its meaning to the logical side of the Table. These are the varieties of spaces that come up when we turn to analyzing the dynamics of processes that pursue their courses through the states of an arbitrary space <math>X.</math> Moreover, when it becomes necessary to approach situations of overwhelming dynamic complexity in a succession of qualitative reaches, then the methods of logic that are afforded by the boolean domains, with their declarative means of synthesis and deductive modes of analysis, supply a natural battery of tools for the task.

It is usually expedient to take these spaces two at a time, in dual pairs of the form <math>X</math> and <math>(X \to \mathbb{K}).</math> In general, one creates pairs of type schemas by replacing any space <math>X</math> with its dual <math>(X \to \mathbb{K}),</math> for example, pairing the type <math>X \to Y</math> with the type <math>(X \to \mathbb{K}) \to (Y \to \mathbb{K}),</math> and <math>X \times Y</math> with <math>(X \to \mathbb{K}) \times (Y \to \mathbb{K}).</math> The word ''dual'' is used here in its broader sense to mean all of the functionals, not just the linear ones. Given any function <math>f : X \to \mathbb{K},</math> the ''converse'' or inverse relation corresponding to <math>f</math> is denoted <math>f^{-1},</math> and the subsets of <math>X</math> that are defined by <math>f^{-1}(k),</math> taken over <math>k</math> in <math>\mathbb{K},</math> are called the ''fibers'' or the ''level sets'' of the function <math>f.</math>

===Theory of Control and Control of Theory===

{| width="100%" cellpadding="0" cellspacing="0"
| width="40%" |  
| width="60%" |
You will hardly know who I am or what I mean,<br>
But I shall be good health to you nevertheless,<br>
And filter and fibre your blood.
|-
|  
| align="right" | — Walt Whitman, ''Leaves of Grass'', [Whi, 88]
|}

In the boolean context a function <math>f : X \to \mathbb{B}</math> is tantamount to a ''proposition'' about elements of <math>X,</math> and the elements of <math>X</math> constitute the ''interpretations'' of that proposition. The fiber <math>f^{-1}(1)</math> comprises the set of ''models'' of <math>f,</math> or examples of elements in <math>X</math> satisfying the proposition <math>f.</math> The fiber <math>f^{-1}(0)</math> collects the complementary set of ''anti-models'', or the exceptions to the proposition <math>f</math> that exist in <math>X.</math> Of course, the space of functions <math>(X \to \mathbb{B})</math> is isomorphic to the set of all subsets of <math>X,</math> called the ''power set'' of <math>X,</math> and often denoted <math>\mathcal{P}(X)</math> or <math>2^X.</math>

The operation of replacing <math>X</math> by <math>(X \to \mathbb{B})</math> in a type schema corresponds to a certain shift of attitude towards the space <math>X,</math> in which one passes from a focus on the ostensibly individual elements of <math>X</math> to a concern with the states of information and uncertainty one possesses about objects and situations in <math>X.</math> The conceptual obstacles in the path of this transition can be smoothed over by using singular functions <math>(\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B})</math> as stepping stones. First of all, it's an easy step from an element <math>\mathbf{x}</math> of type <math>\mathbb{B}^n</math> to the equivalent information of a singular proposition <math>\mathbf{x} : X \xrightarrow{s} \mathbb{B}, </math> and then only a small jump of generalization remains to reach the type of an arbitrary proposition <math>f : X \to \mathbb{B},</math> perhaps understood to indicate a relaxed constraint on the singularity of points or a neighborhood circumscribing the original <math>\mathbf{x}.</math> This is frequently a useful transformation, communicating between the ''objective'' and the ''intentional'' perspectives, in spite perhaps of the open objection that this distinction is transient in the mean time and ultimately superficial.

It is hoped that this measure of flexibility, allowing us to stretch a point into a proposition, can be useful in the examination of inquiry driven systems, where the differences between empirical, intentional, and theoretical propositions constitute the discrepancies and the distributions that drive experimental activity. I can give this model of inquiry a cybernetic cast by realizing that theory change and theory evolution, as well as the choice and the evaluation of experiments, are actions that are taken by a system or its agent in response to the differences that are detected between observational contents and theoretical coverage.

All of the above notwithstanding, there are several points that distinguish these two tasks, namely, the ''theory of control'' and the ''control of theory'', features that are often obscured by too much precipitation in the quickness with which we understand their similarities. In the control of uncertainty through inquiry, some of the actuators that we need to be concerned with are axiom changers and theory modifiers, operators with the power to compile and to revise the theories that generate expectations and predictions, effectors that form and edit our grammars for the languages of observational data, and agencies that rework the proposed model to fit the actual sequences of events and the realized relationships of values that are observed in the environment. Moreover, when steps must be taken to carry out an experimental action, there must be something about the particular shape of our uncertainty that guides us in choosing what directions to explore, and this impression is more than likely influenced by previous accumulations of experience. Thus it must be anticipated that much of what goes into scientific progress, or any sustainable effort toward a goal of knowledge, is necessarily predicated on long term observation and modal expectations, not only on the more local or short term prediction and correction.

===Propositions as Types and Higher Order Types===

The types collected in Table 3 (repeated below) serve to illustrate the themes of ''higher order propositional expressions'' and the ''propositions as types'' (PAT) analogy.

<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 3.} ~~ \text{Analogy Between Real and Boolean Types}</math>
|- style="height:40px; background:ghostwhite"
| <math>\text{Real Domain} ~ \mathbb{R}</math>
| <math>\longleftrightarrow</math>
| <math>\text{Boolean Domain} ~ \mathbb{B}</math>
|-
| <math>\mathbb{R}^n</math>
| <math>\text{Basic Space}</math>
| <math>\mathbb{B}^n</math>
|-
| <math>\mathbb{R}^n \to \mathbb{R}</math>
| <math>\text{Function Space}</math>
| <math>\mathbb{B}^n \to \mathbb{B}</math>
|-
| <math>(\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R}</math>
| <math>\text{Tangent Vector}</math>
| <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math>
|-
| <math>\mathbb{R}^n \to ((\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R})</math>
| <math>\text{Vector Field}</math>
| <math>\mathbb{B}^n \to ((\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B})</math>
|-
| <math>(\mathbb{R}^n \times (\mathbb{R}^n \to \mathbb{R})) \to \mathbb{R}</math>
| '''<font size="4">"</font>'''
| <math>(\mathbb{B}^n \times (\mathbb{B}^n \to \mathbb{B})) \to \mathbb{B}</math>
|-
| <math>((\mathbb{R}^n \to \mathbb{R}) \times \mathbb{R}^n) \to \mathbb{R}</math>
| '''<font size="4">"</font>'''
| <math>((\mathbb{B}^n \to \mathbb{B}) \times \mathbb{B}^n) \to \mathbb{B}</math>
|-
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^n \to \mathbb{R})</math>
| <math>\text{Derivation}</math>
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^n \to \mathbb{B})</math>
|-
| <math>\mathbb{R}^n \to \mathbb{R}^m</math>
| <math>\begin{matrix}\text{Basic}\\[2pt]\text{Transformation}\end{matrix}</math>
| <math>\mathbb{B}^n \to \mathbb{B}^m</math>
|-
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^m \to \mathbb{R})</math>
| <math>\begin{matrix}\text{Function}\\[2pt]\text{Transformation}\end{matrix}</math>
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^m \to \mathbb{B})</math>
|}

<br>

First, observe that the type of a ''tangent vector at a point'', also known as a ''directional derivative'' at that point, has the form <math>(\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K},</math> where <math>\mathbb{K}</math> is the chosen ground field, in the present case either <math>\mathbb{R}</math> or <math>\mathbb{B}.</math> At a point in a space of type <math>\mathbb{K}^n,</math> a directional derivative operator <math>\vartheta</math> takes a function on that space, an <math>f</math> of type <math>(\mathbb{K}^n \to \mathbb{K}),</math> and maps it to a ground field value of type <math>\mathbb{K}.</math> This value is known as the ''derivative'' of <math>f</math> in the direction <math>\vartheta</math> [Che46, 76–77]. In the boolean case <math>\vartheta : (\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math> has the form of a proposition about propositions, in other words, a proposition of the next higher type.

Next, by way of illustrating the propositions as types idea, consider a proposition of the form <math>X \Rightarrow (Y \Rightarrow Z).</math> One knows from propositional calculus that this is logically equivalent to a proposition of the form <math>(X \land Y) \Rightarrow Z.</math> But this equivalence should remind us of the functional isomorphism that exists between a construction of the type <math>X \to (Y \to Z)</math> and a construction of the type <math>(X \times Y) \to Z.</math> The propositions as types analogy permits us to take a functional type like this and, under the right conditions, replace the functional arrows “<math>\to</math>” and products “<math>\times</math>” with the respective logical arrows “<math>\Rightarrow</math>” and products “<math>\land</math>”. Accordingly, viewing the result as a proposition, we can employ axioms and theorems of propositional calculus to suggest appropriate isomorphisms among the categorical and functional constructions.

Finally, examine the middle four rows of Table 3. These display a series of isomorphic types that stretch from the categories that are labeled ''Vector Field'' to those that are labeled ''Derivation''. A ''vector field'', also known as an ''infinitesimal transformation'', associates a tangent vector at a point with each point of a space. In symbols, a vector field is a function of the form <math>\textstyle \xi : X \to \bigcup_{x \in X} \xi_x</math> that assigns to each point <math>x</math> of the space <math>X</math> a tangent vector to <math>X</math> at that point, namely, the tangent vector <math>\xi_x</math> [Che46, 82–83]. If <math>X</math> is of the type <math>\mathbb{K}^n,</math> then <math>\xi</math> is of the type <math>\mathbb{K}^n \to ((\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K}).</math> This has the pattern <math>X \to (Y \to Z),</math> with <math>X = \mathbb{K}^n,</math> <math>Y = (\mathbb{K}^n \to \mathbb{K}),</math> and <math>Z = \mathbb{K}.</math>

Applying the propositions as types analogy, one can follow this pattern through a series of metamorphoses from the type of a vector field to the type of a derivation, as traced out in Table 4. Observe how the function <math>f : X \to \mathbb{K},</math> associated with the place of <math>Y</math> in the pattern, moves through its paces from the second to the first position. In this way, the vector field <math>\xi,</math> initially viewed as attaching each tangent vector <math>\xi_x</math> to the site <math>x</math> where it acts in <math>X,</math> now comes to be seen as acting on each scalar potential <math>f : X \to \mathbb{K}</math> like a generalized species of differentiation, producing another function <math>\xi f : X \to \mathbb{K}</math> of the same type.

<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 4.} ~~ \text{An Equivalence Based on the Propositions as Types Analogy}</math>
|- style="height:40px; background:ghostwhite"
| <math>\text{Pattern}</math>
| <math>\text{Construct}</math>
| <math>\text{Instance}</math>
|-
| <math>X \to (Y \to Z)</math>
| <math>\text{Vector Field}</math>
| <math>\mathbb{K}^n \to ((\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K})</math>
|-
| <math>(X \times Y) \to Z</math>
| <math>\Uparrow</math>
| <math>(\mathbb{K}^n \times (\mathbb{K}^n \to \mathbb{K})) \to \mathbb{K}</math>
|-
| <math>(Y \times X) \to Z</math>
| <math>\Downarrow</math>
| <math>((\mathbb{K}^n \to \mathbb{K}) \times \mathbb{K}^n) \to \mathbb{K}</math>
|-
| <math>Y \to (X \to Z)</math>
| <math>\text{Derivation}</math>
| <math>(\mathbb{K}^n \to \mathbb{K}) \to (\mathbb{K}^n \to \mathbb{K})</math>
|}

<br>

===Reality at the Threshold of Logic===

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
But no science can rest entirely on measurement, and many scientific investigations are quite out of reach of that device. To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope.
| width="4%" |  
|-
| align="right" colspan="3" | — W.V. Quine, ''Mathematical Logic'', [Qui, 7]
|}

Table 5 accumulates an array of notation that I hope will not be too distracting. Some of it is rarely needed, but has been filled in for the sake of completeness. Its purpose is simple, to give literal expression to the visual intuitions that come with venn diagrams, and to help build a bridge between our qualitative and quantitative outlooks on dynamic systems.

<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:75%"
|+ style="height:30px" | <math>\text{Table 5.} ~~ \text{A Bridge Over Troubled Waters}</math>
|- style="height:40px; background:ghostwhite"
| align="center" | <math>\text{Linear Space}</math>
| align="center" | <math>\text{Liminal Space}</math>
| align="center" | <math>\text{Logical Space}</math>
|-
| <math>\begin{matrix}\mathcal{X} & = & \{ x_1, \ldots, x_n \}\end{matrix}</math>
| <math>\begin{matrix}\underline{\mathcal{X}} & = & \{ \underline{x}_1, \ldots, \underline{x}_n \}\end{matrix}</math>
| <math>\begin{matrix}\mathcal{A} & = & \{ a_1, \ldots, a_n \}\end{matrix}</math>
|-
|
<math>\begin{matrix}
X_i & = & \langle x_i \rangle
\\
& \cong & \mathbb{K}
\end{matrix}</math>
|
<math>\begin{matrix}
\underline{X}_i & = & \{ \texttt{(} \underline{x}_i \texttt{)}, \underline{x}_i \}
\\
& \cong & \mathbb{B}
\end{matrix}</math>
|
<math>\begin{matrix}
A_i & = & \{ \texttt{(} a_i \texttt{)}, a_i \}
\\
& \cong & \mathbb{B}
\end{matrix}</math>
|-
|
<math>\begin{matrix}
X
\\
= & \langle \mathcal{X} \rangle
\\
= & \langle x_1, \ldots, x_n \rangle
\\
= & X_1 \times \ldots \times X_n
\\
= & \prod_{i=1}^n X_i
\\
\cong & \mathbb{K}^n
\end{matrix}</math>
|
<math>\begin{matrix}
\underline{X}
\\
= & \langle \underline{\mathcal{X}} \rangle
\\
= & \langle \underline{x}_1, \ldots, \underline{x}_n \rangle
\\
= & \underline{X}_1 \times \ldots \times \underline{X}_n
\\
= & \prod_{i=1}^n \underline{X}_i
\\
\cong & \mathbb{B}^n
\end{matrix}</math>
|
<math>\begin{matrix}
A
\\
= & \langle \mathcal{A} \rangle
\\
= & \langle a_1, \ldots, a_n \rangle
\\
= & A_1 \times \ldots \times A_n
\\
= & \prod_{i=1}^n A_i
\\
\cong & \mathbb{B}^n
\end{matrix}</math>
|-
|
<math>\begin{matrix}
X^* & = & (\ell : X \to \mathbb{K})
\\
& \cong & \mathbb{K}^n
\end{matrix}</math>
|
<math>\begin{matrix}
\underline{X}^* & = & (\ell : \underline{X} \to \mathbb{B})
\\
& \cong & \mathbb{B}^n
\end{matrix}</math>
|
<math>\begin{matrix}
A^* & = & (\ell : A \to \mathbb{B})
\\
& \cong & \mathbb{B}^n
\end{matrix}</math>
|-
|
<math>\begin{matrix}
X^\uparrow & = & (X \to \mathbb{K})
\\
& \cong & (\mathbb{K}^n \to \mathbb{K})
\end{matrix}</math>
|
<math>\begin{matrix}
\underline{X}^\uparrow & = & (\underline{X} \to \mathbb{B})
\\
& \cong & (\mathbb{B}^n \to \mathbb{B})
\end{matrix}</math>
|
<math>\begin{matrix}
A^\uparrow & = & (A \to \mathbb{B})
\\
& \cong & (\mathbb{B}^n \to \mathbb{B})
\end{matrix}</math>
|-
|
<math>\begin{matrix}
X^\bullet
\\
= & [\mathcal{X}]
\\
= & [x_1, \ldots, x_n]
\\
= & (X, X^\uparrow)
\\
= & (X ~+\!\to \mathbb{K})
\\
= & (X, (X \to \mathbb{K}))
\\
\cong & (\mathbb{K}^n, (\mathbb{K}^n \to \mathbb{K}))
\\
= & (\mathbb{K}^n ~+\!\to \mathbb{K})
\\
= & [\mathbb{K}^n]
\end{matrix}</math>
|
<math>\begin{matrix}
\underline{X}^\bullet
\\
= & [\underline{\mathcal{X}}]
\\
= & [\underline{x}_1, \ldots, \underline{x}_n]
\\
= & (\underline{X}, \underline{X}^\uparrow)
\\
= & (\underline{X} ~+\!\to \mathbb{B})
\\
= & (\underline{X}, (\underline{X} \to \mathbb{B}))
\\
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))
\\
= & (\mathbb{B}^n ~+\!\to \mathbb{B})
\\
= & [\mathbb{B}^n]
\end{matrix}</math>
|
<math>\begin{matrix}
A^\bullet
\\
= & [\mathcal{A}]
\\
= & [a_1, \ldots, a_n]
\\
= & (A, A^\uparrow)
\\
= & (A ~+\!\to \mathbb{B})
\\
= & (A, (A \to \mathbb{B}))
\\
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))
\\
= & (\mathbb{B}^n ~+\!\to \mathbb{B})
\\
= & [\mathbb{B}^n]
\end{matrix}</math>
|}

<br>

The left side of the Table collects mostly standard notation for an <math>n</math>-dimensional vector space over a field <math>\mathbb{K}.</math> The right side of the table repeats the first elements of a notation that I sketched above, to be used in further developments of propositional calculus. (I plan to use this notation in the logical analysis of neural network systems.) The middle column of the table is designed as a transitional step from the case of an arbitrary field <math>\mathbb{K},</math> with a special interest in the continuous line <math>\mathbb{R},</math> to the qualitative and discrete situations that are instanced and typified by <math>\mathbb{B}.</math>

I now proceed to explain these concepts in more detail. The most important ideas developed in Table 5 are these:

* The idea of a universe of discourse, which includes both a space of ''points'' and a space of ''maps'' on those points.

* The idea of passing from a more complex universe to a simpler universe by a process of ''thresholding'' each dimension of variation down to a single bit of information.

For the sake of concreteness, let us suppose that we start with a continuous <math>n</math>-dimensional vector space like <math>X = \langle x_1, \ldots, x_n \rangle \cong \mathbb{R}^n.</math> The coordinate system <math>\mathcal{X} = \{x_i\}</math> is a set of maps <math>x_i : \mathbb{R}^n \to \mathbb{R},</math> also known as the ''coordinate projections''. Given a "dataset" of points <math>\mathbf{x}</math> in <math>\mathbb{R}^n,</math> there are numerous ways of sensibly reducing the data down to one bit for each dimension. One strategy that is general enough for our present purposes is as follows. For each <math>i</math> we choose an <math>n</math>-ary relation <math>L_i</math> on <math>\mathbb{R}^n,</math> that is, a subset of <math>\mathbb{R}^n,</math> and then we define the <math>i^\mathrm{th}</math> threshold map, or ''limen'' <math>\underline{x}_i</math> as follows:

{| align="center" cellpadding="8" width="90%"
| <math>\underline{x}_i : \mathbb{R}^n \to \mathbb{B}\ \text{such that:}</math>
|-
|
<math>\begin{matrix}
\underline{x}_i(\mathbf{x}) = 1 & \text{if} & \mathbf{x} \in L_i,
\\[4pt]
\underline{x}_i(\mathbf{x}) = 0 & \text{if} & \mathbf{x} \not\in L_i.
\end{matrix}</math>
|}

In other notations that are sometimes used, the operator <math>\chi (\ldots)</math> or the corner brackets <math>\lceil\ldots\rceil</math> can be used to denote a ''characteristic function'', that is, a mapping from statements to their truth values in <math>\mathbb{B}.</math> Finally, it is not uncommon to use the name of the relation itself as a predicate that maps <math>n</math>-tuples into truth values. Thus we have the following notational variants of the above definition:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
\underline{x}_i (\mathbf{x}) & = & \chi (\mathbf{x} \in L_i) & = & \lceil \mathbf{x} \in L_i \rceil & = & L_i (\mathbf{x}).
\end{matrix}</math>
|}

Notice that, as defined here, there need be no actual relation between the <math>n</math>-dimensional subsets <math>\{L_i\}</math> and the coordinate axes corresponding to <math>\{x_i\},</math> aside from the circumstance that the two sets have the same cardinality. In concrete cases, though, one usually has some reason for associating these "volumes" with these "lines", for instance, <math>L_i</math> is bounded by some hyperplane that intersects the <math>i^\text{th}</math> axis at a unique threshold value <math>r_i \in \mathbb{R}.</math> Often, the hyperplane is chosen normal to the axis. In recognition of this motive, let us make the following convention. When the set <math>L_i</math> has points on the <math>i^\text{th}</math> axis, that is, points of the form <math>(0, \ldots, 0, r_i, 0, \ldots, 0)</math> where only the <math>x_i</math> coordinate is possibly non-zero, we may pick any one of these coordinate values as a parametric index of the relation. In this case we say that the indexing is ''real'', otherwise the indexing is ''imaginary''. For a knowledge based system <math>X,</math> this should serve once again to mark the distinction between ''acquaintance'' and ''opinion''.

States of knowledge about the location of a system or about the distribution of a population of systems in a state space <math>X = \mathbb{R}^n</math> can now be expressed by taking the set <math>\underline{\mathcal{X}} = \{\underline{x}_i\}</math> as a basis of logical features. In picturesque terms, one may think of the underscore and the subscript as combining to form a subtextual spelling for the <math>i^\text{th}</math> threshold map. This can help to remind us that the ''threshold operator'' <math>(\underline{~})_i</math> acts on <math>\mathbf{x}</math> by setting up a kind of a “hurdle” for it. In this interpretation the coordinate proposition <math>\underline{x}_i</math> asserts that the representative point <math>\mathbf{x}</math> resides ''above'' the <math>i^\mathrm{th}</math> threshold.

Primitive assertions of the form <math>\underline{x}_i (\mathbf{x})</math> may then be negated and joined by means of propositional connectives in the usual ways to provide information about the state <math>\mathbf{x}</math> of a contemplated system or a statistical ensemble of systems. Parentheses <math>\texttt{(} \ldots \texttt{)}</math> may be used to indicate logical negation. Eventually one discovers the usefulness of the <math>k</math>-ary ''just one false'' operators of the form <math>\texttt{(} a_1 \texttt{,} \ldots \texttt{,} a_k \texttt{)},</math> as treated in earlier reports. This much tackle generates a space of points (cells, interpretations), <math>\underline{X} \cong \mathbb{B}^n,</math> and a space of functions (regions, propositions), <math>\underline{X}^\uparrow \cong (\mathbb{B}^n \to \mathbb{B}).</math> Together these form a new universe of discourse <math>\underline{X}^\bullet</math> of the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})),</math> which we may abbreviate as <math>\mathbb{B}^n\ +\!\to \mathbb{B}</math> or most succinctly as <math>[\mathbb{B}^n].</math>

The square brackets have been chosen to recall the rectangular frame of a venn diagram. In thinking about a universe of discourse it is a good idea to keep this picture in mind, graphically illustrating the links among the elementary cells <math>\underline{\mathbf{x}},</math> the defining features <math>\underline{x}_i,</math> and the potential shadings <math>f : \underline{X} \to \mathbb{B}</math> all at the same time, not to mention the arbitrariness of the way we choose to inscribe our distinctions in the medium of a continuous space.

Finally, let <math>X^*</math> denote the space of linear functions, <math>(\ell : X \to \mathbb{K}),</math> which has in the finite case the same dimensionality as <math>X,</math> and let the same notation be extended across the Table.

We have just gone through a lot of work, apparently doing nothing more substantial than spinning a complex spell of notational devices through a labyrinth of baffled spaces and baffling maps. The reason for doing this was to bind together and to constitute the intuitive concept of a universe of discourse into a coherent categorical object, the kind of thing, once grasped, that can be turned over in the mind and considered in all its manifold changes and facets. The effort invested in these preliminary measures is intended to pay off later, when we need to consider the state transformations and the time evolution of neural network systems.

===Tables of Propositional Forms===

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope. It provides explicit techniques for manipulating the most basic ingredients of discourse.
| width="4%" |  
|-
| align="right" colspan="3" | — W.V. Quine, ''Mathematical Logic'', [Qui, 7–8]
|}

To prepare for the next phase of discussion, Tables 6 and 7 collect and summarize all of the propositional forms on one and two variables. These propositional forms are represented over bases of boolean variables as complete sets of boolean-valued functions. Adjacent to their names and specifications are listed what are roughly the simplest expressions in the ''[[Cactus_Language_•_Overview|cactus language]]'', the particular syntax for propositional calculus that I use in formal and computational contexts. For the sake of orientation, the English paraphrases and the more common notations are listed in the last two columns. As simple and circumscribed as these low-dimensional universes may appear to be, a careful exploration of their differential extensions will involve us in complexities sufficient to demand our attention for some time to come.

Propositional forms on one variable correspond to boolean functions <math>f : \mathbb{B}^1 \to \mathbb{B}.</math> In Table 6 these functions are listed in a variant form of [[truth table]], one in which the axes of the usual arrangement are rotated through a right angle. Each function <math>f_i</math> is indexed by the string of values that it takes on the points of the universe <math>X^\bullet = [x] \cong \mathbb{B}^1.</math> The binary index generated in this way is converted to its decimal equivalent and these are used as conventional names for the <math>f_i,</math> as shown in the first column of the Table. In their own right the <math>2^1</math> points of the universe <math>X^\bullet</math> are coordinated as a space of type <math>\mathbb{B}^1,</math> this in light of the universe <math>X^\bullet</math> being a functional domain where the coordinate projection <math>x</math> takes on its values in <math>\mathbb{B}.</math>

<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 6.} ~~ \text{Propositional Forms on One Variable}</math>
|- style="height:40px; background:ghostwhite"
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math>
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math>
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math>
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math>
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math>
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math>
|- style="background:ghostwhite"
|  
| align="right" | <math>x\colon</math>
| <math>1~0</math>
|  
|  
|  
|-
| <math>f_0</math>
| <math>f_{00}</math>
| <math>0~0</math>
| <math>\texttt{(} ~ \texttt{)}</math>
| <math>\text{false}</math>
| <math>0</math>
|-
| <math>f_1</math>
| <math>f_{01}</math>
| <math>0~1</math>
| <math>\texttt{(} x \texttt{)}</math>
| <math>\text{not}~ x</math>
| <math>\lnot x</math>
|-
| <math>f_2</math>
| <math>f_{10}</math>
| <math>1~0</math>
| <math>x</math>
| <math>x</math>
| <math>x</math>
|-
| <math>f_3</math>
| <math>f_{11}</math>
| <math>1~1</math>
| <math>\texttt{((} ~ \texttt{))}</math>
| <math>\text{true}</math>
| <math>1</math>
|}

<br>

Propositional forms on two variables correspond to boolean functions <math>f : \mathbb{B}^2 \to \mathbb{B}.</math> In Table 7 each function <math>f_i</math> is indexed by the values that it takes on the points of the universe <math>X^\bullet = [x, y] \cong \mathbb{B}^2.</math> Converting the binary index thus generated to a decimal equivalent, we obtain the functional nicknames that are listed in the first column. The <math>2^2</math> points of the universe <math>X^\bullet</math> are coordinated as a space of type <math>\mathbb{B}^2,</math> as indicated under the heading of the Table, where the coordinate projections <math>x</math> and <math>y</math> run through the various combinations of their values in <math>\mathbb{B}.</math>

<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 7-a.} ~~ \text{Propositional Forms on Two Variables}</math>
|- style="height:40px; background:ghostwhite"
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math>
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math>
|- style="background:ghostwhite"
|  
| align="right" | <math>x\colon</math>
| <math>1~1~0~0</math>
|  
|  
|  
|- style="background:ghostwhite"
|  
| align="right" | <math>y\colon</math>
| <math>1~0~1~0</math>
|  
|  
|  
|-
| valign="bottom" |
<math>\begin{matrix}
f_{0}
\\[4pt]
f_{1}
\\[4pt]
f_{2}
\\[4pt]
f_{3}
\\[4pt]
f_{4}
\\[4pt]
f_{5}
\\[4pt]
f_{6}
\\[4pt]
f_{7}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
f_{0000}
\\[4pt]
f_{0001}
\\[4pt]
f_{0010}
\\[4pt]
f_{0011}
\\[4pt]
f_{0100}
\\[4pt]
f_{0101}
\\[4pt]
f_{0110}
\\[4pt]
f_{0111}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
0~0~0~0
\\[4pt]
0~0~0~1
\\[4pt]
0~0~1~0
\\[4pt]
0~0~1~1
\\[4pt]
0~1~0~0
\\[4pt]
0~1~0~1
\\[4pt]
0~1~1~0
\\[4pt]
0~1~1~1
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{(} ~ \texttt{)}
\\[4pt]
\texttt{(} x \texttt{)(} y \texttt{)}
\\[4pt]
\texttt{(} x \texttt{)} ~ y ~
\\[4pt]
\texttt{(} x \texttt{)}
\\[4pt]
~ x ~ \texttt{(} y \texttt{)}
\\[4pt]
\texttt{(} y \texttt{)}
\\[4pt]
\texttt{(} x \texttt{,} ~ y \texttt{)}
\\[4pt]
\texttt{(} x ~ y \texttt{)}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\text{false}
\\[4pt]
\text{neither}~ x ~\text{nor}~ y
\\[4pt]
y ~\text{without}~ x
\\[4pt]
\text{not}~ x
\\[4pt]
x ~\text{without}~ y
\\[4pt]
\text{not}~ y
\\[4pt]
x ~\text{not equal to}~ y
\\[4pt]
\text{not both}~ x ~\text{and}~ y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
0
\\[4pt]
\lnot x \land \lnot y
\\[4pt]
\lnot x \land y
\\[4pt]
\lnot x
\\[4pt]
x \land \lnot y
\\[4pt]
\lnot y
\\[4pt]
x \ne y
\\[4pt]
\lnot x \lor \lnot y
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
f_{8}
\\[4pt]
f_{9}
\\[4pt]
f_{10}
\\[4pt]
f_{11}
\\[4pt]
f_{12}
\\[4pt]
f_{13}
\\[4pt]
f_{14}
\\[4pt]
f_{15}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
f_{1000}
\\[4pt]
f_{1001}
\\[4pt]
f_{1010}
\\[4pt]
f_{1011}
\\[4pt]
f_{1100}
\\[4pt]
f_{1101}
\\[4pt]
f_{1110}
\\[4pt]
f_{1111}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
1~0~0~0
\\[4pt]
1~0~0~1
\\[4pt]
1~0~1~0
\\[4pt]
1~0~1~1
\\[4pt]
1~1~0~0
\\[4pt]
1~1~0~1
\\[4pt]
1~1~1~0
\\[4pt]
1~1~1~1
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
x ~ y
\\[4pt]
\texttt{((} x \texttt{,} ~ y \texttt{))}
\\[4pt]
y
\\[4pt]
\texttt{(} x ~ \texttt{(} y \texttt{))}
\\[4pt]
x
\\[4pt]
\texttt{((} x \texttt{)} ~ y \texttt{)}
\\[4pt]
\texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
\texttt{((} ~ \texttt{))}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
x ~\text{and}~ y
\\[4pt]
x ~\text{equal to}~ y
\\[4pt]
y
\\[4pt]
\text{not}~ x ~\text{without}~ y
\\[4pt]
x
\\[4pt]
\text{not}~ y ~\text{without}~ x
\\[4pt]
x ~\text{or}~ y
\\[4pt]
\text{true}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
x \land y
\\[4pt]
x = y
\\[4pt]
y
\\[4pt]
x \Rightarrow y
\\[4pt]
x
\\[4pt]
x \Leftarrow y
\\[4pt]
x \lor y
\\[4pt]
1
\end{matrix}</math>
|}

<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 7-b.} ~~ \text{Propositional Forms on Two Variables}</math>
|- style="height:40px; background:ghostwhite"
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math>
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math>
|- style="background:ghostwhite"
|  
| align="right" | <math>x\colon</math>
| <math>1~1~0~0</math>
|  
|  
|  
|- style="background:ghostwhite"
|  
| align="right" | <math>y\colon</math>
| <math>1~0~1~0</math>
|  
|  
|  
|-
| <math>f_{0}</math>
| <math>f_{0000}</math>
| <math>0~0~0~0</math>
| <math>\texttt{(} ~ \texttt{)}</math>
| <math>\text{false}</math>
| <math>0</math>
|-
| valign="bottom" |
<math>\begin{matrix}
f_{1}
\\[4pt]
f_{2}
\\[4pt]
f_{4}
\\[4pt]
f_{8}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
f_{0001}
\\[4pt]
f_{0010}
\\[4pt]
f_{0100}
\\[4pt]
f_{1000}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
0~0~0~1
\\[4pt]
0~0~1~0
\\[4pt]
0~1~0~0
\\[4pt]
1~0~0~0
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{(} x \texttt{)(} y \texttt{)}
\\[4pt]
\texttt{(} x \texttt{)} ~ y ~
\\[4pt]
~ x ~ \texttt{(} y \texttt{)}
\\[4pt]
~ x ~~ y ~
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\text{neither}~ x ~\text{nor}~ y
\\[4pt]
y ~\text{without}~ x
\\[4pt]
x ~\text{without}~ y
\\[4pt]
x ~\text{and}~ y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\lnot x \land \lnot y
\\[4pt]
\lnot x \land y
\\[4pt]
x \land \lnot y
\\[4pt]
x \land y
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
f_{3}
\\[4pt]
f_{12}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
f_{0011}
\\[4pt]
f_{1100}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
0~0~1~1
\\[4pt]
1~1~0~0
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\[4pt]
x
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\text{not}~ x
\\[4pt]
x
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\lnot x
\\[4pt]
x
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
f_{6}
\\[4pt]
f_{9}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
f_{0110}
\\[4pt]
f_{1001}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
0~1~1~0
\\[4pt]
1~0~0~1
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{(} x \texttt{,} y \texttt{)}
\\[4pt]
\texttt{((} x \texttt{,} y \texttt{))}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
x ~\text{not equal to}~ y
\\[4pt]
x ~\text{equal to}~ y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
x \ne y
\\[4pt]
x = y
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
f_{5}
\\[4pt]
f_{10}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
f_{0101}
\\[4pt]
f_{1010}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
0~1~0~1
\\[4pt]
1~0~1~0
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\[4pt]
y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\text{not}~ y
\\[4pt]
y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\lnot y
\\[4pt]
y
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
f_{7}
\\[4pt]
f_{11}
\\[4pt]
f_{13}
\\[4pt]
f_{14}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
f_{0111}
\\[4pt]
f_{1011}
\\[4pt]
f_{1101}
\\[4pt]
f_{1110}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
0~1~1~1
\\[4pt]
1~0~1~1
\\[4pt]
1~1~0~1
\\[4pt]
1~1~1~0
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{(} ~ x ~~ y ~ \texttt{)}
\\[4pt]
\texttt{(} ~ x ~ \texttt{(} y \texttt{))}
\\[4pt]
\texttt{((} x \texttt{)} ~ y ~ \texttt{)}
\\[4pt]
\texttt{((} x \texttt{)(} y \texttt{))}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\text{not both}~ x ~\text{and}~ y
\\[4pt]
\text{not}~ x ~\text{without}~ y
\\[4pt]
\text{not}~ y ~\text{without}~ x
\\[4pt]
x ~\text{or}~ y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\lnot x \lor \lnot y
\\[4pt]
x \Rightarrow y
\\[4pt]
x \Leftarrow y
\\[4pt]
x \lor y
\end{matrix}</math>
|-
| <math>f_{15}</math>
| <math>f_{1111}</math>
| <math>1~1~1~1</math>
| <math>\texttt{((} ~ \texttt{))}</math>
| <math>\text{true}</math>
| <math>1</math>
|}

<br>

==A Differential Extension of Propositional Calculus==

{| width="100%" cellpadding="0" cellspacing="0"
| width="40%" |  
| width="60%" |
Fire over water:<br>
The image of the condition before transition.<br>
Thus the superior man is careful<br>
In the differentiation of things,<br>
So that each finds its place.
|-
|  
| align="right" | — ''I Ching'', Hexagram 64, [Wil, 249]
|}

This much preparation is enough to begin introducing my subject, if I excuse myself from giving full arguments for my definitional choices until some later stage. I am trying to develop a ''differential theory of qualitative equations'' that parallels the application of differential geometry to dynamic systems. The idea of a tangent vector is key to this work and a major goal is to find the right logical analogues of tangent spaces, bundles, and functors. The strategy is taken of looking for the simplest versions of these constructions that can be discovered within the realm of propositional calculus, so long as they serve to fill out the general theme.

==={{anchor|Differential Propositions}}Differential Propositions : Qualitative Analogues of Differential Equations===

The differential extension of a universe of discourse <math>[\mathcal{A}]</math> is constructed by extending its initial alphabet <math>\mathfrak{A}</math> to include a set of symbols for <i>differential features</i>, or <i>basic changes</i> capable of occurring in <math>[\mathcal{A}].</math>  The added symbols are taken to denote primitive features of change, qualitative attributes of motion, or propositions about how items in the universe of discourse may change or move in relation to features noted in the original alphabet.

With that in mind we define the corresponding <i>differential alphabet</i> or <i>tangent alphabet</i> <math>\mathrm{d}\mathfrak{A}</math> <math>=</math> <math>\{ {}^{\backprime\backprime} \mathrm{d}a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} \mathrm{d}a_n {}^{\prime\prime} \},</math> in principle just an arbitrary alphabet of symbols, disjoint from the initial alphabet <math>\mathfrak{A}</math> <math>=</math> <math>\{ {}^{\backprime\backprime} a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} a_n {}^{\prime\prime}\}</math> and given the meanings just indicated.

In practice the precise interpretation of the symbols in <math>\mathrm{d}\mathfrak{A}</math> is conceived to be changeable from point to point of the underlying space <math>A.</math>  Indeed, for all we know, the state space <math>A</math> might well be the state space of a language interpreter, one concerned with the idiomatic meanings of the dialect generated by <math>\mathfrak{A}</math> and <math>\mathrm{d}\mathfrak{A}.</math>

{{anchor|Tangent Spaces}}
The <i>tangent space</i> to <math>A</math> at one of its points <math>x,</math> sometimes written <math>\mathrm{T}_x(A),</math> takes the form <math>\mathrm{d}A</math> <math>=</math> <math>\langle \mathrm{d}\mathcal{A} \rangle</math> <math>=</math> <math>\langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle.</math>  Strictly speaking, the name <i>cotangent space</i> is probably more correct for this construction but since we take up spaces and their duals in pairs to form our universes of discourse it allows our language to be pliable here.

Proceeding as we did with the base space <math>A,</math> the tangent space <math>\mathrm{d}A</math> at a point of <math>A</math> may be analyzed as the following product of distinct and independent factors.

{| align="center" cellpadding="8" width="90%"
| <math>\mathrm{d}A ~=~ \prod_{i=1}^n \mathrm{d}A_i ~=~ \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n.</math>
|}

Each factor <math>\mathrm{d}A_i</math> is a set consisting of two differential propositions, <math>\mathrm{d}A_i = \{ (\mathrm{d}a_i), \mathrm{d}a_i \},</math> where <math>\texttt{(} \mathrm{d}a_i \texttt{)}</math> is a proposition with the logical value of <math>\lnot\mathrm{d}a_i.</math>  Each component <math>\mathrm{d}A_i</math> has the type <math>\mathbb{B},</math> operating under the ordered correspondence <math>\{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \} \cong \{ 0, 1 \}.</math>  A measure of clarity is achieved, however, by acknowledging the differential usage with a superficially distinct type <math>\mathbb{D},</math> whose sense may be indicated as follows.

{| align="center" cellpadding="8" width="90%"
| <math>\mathbb{D} = \{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \} = \{ \text{same}, \text{different} \} = \{ \text{stay}, \text{change} \} = \{ \text{stop}, \text{step} \}.</math>
|}

Viewed within a coordinate representation, spaces of type <math>\mathbb{B}^n</math> and <math>\mathbb{D}^n</math> may appear to be identical sets of binary vectors, but taking a view at that level of abstraction would be like ignoring the qualitative units and the diverse dimensions that distinguish position and momentum, or the different roles of quantity and impulse.

===An Interlude on the Path===

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
There would have been no beginnings:  instead, speech would proceed from me, while I stood in its path – a slender gap – the point of its possible disappearance.
| width="4%" |  
|-
| align="right" colspan="3" | — Michel Foucault, ''The Discourse on Language'', [Fou, 215]
|}

A sense of the relation between <math>\mathbb{B}</math> and <math>\mathbb{D}</math> may be obtained by considering the ''path classifier'' (or the ''equivalence class of curves'') approach to tangent vectors.  Consider a universe <math>[\mathcal{X}].</math>  Given the boolean value system, a path in the space <math>X = \langle \mathcal{X} \rangle</math> is a map <math>q : \mathbb{B} \to X.</math>  In this context the set of paths <math>(\mathbb{B} \to X)</math> is isomorphic to the cartesian square <math>X^2 = X \times X,</math> or the set of ordered pairs chosen from <math>X.</math>

We may analyze <math>X^2 = \{ (u, v) : u, v \in X \}</math> into two parts, specifically, the ordered pairs <math>(u, v)</math> that lie on and off the diagonal:

{| align="center" cellpadding="8" width="90%"
| <math>\begin{matrix}X^2 & = & \{ (u, v) : u = v \} & \cup & \{ (u, v) : u \ne v \}.\end{matrix}</math>
|}

This partition may also be expressed in the following symbolic form:

{| align="center" cellpadding="8" width="90%"
| <math>\begin{matrix}X^2 & \cong & \operatorname{diag} (X) & + & 2 \binom{X}{2}.\end{matrix}</math>
|}

The separate terms of this formula are defined as follows:

{| align="center" cellpadding="8" width="90%"
| <math>\begin{matrix}\operatorname{diag} (X) & = & \{ (x, x) : x \in X \}.\end{matrix}</math>
|}

{| align="center" cellpadding="8" width="90%"
| <math>\begin{matrix}\binom{X}{k} & = & X ~\text{choose}~ k & = & \{ k\text{-sets from}~ X \}.\end{matrix}</math>
|}

Thus we have:

{| align="center" cellpadding="8" width="90%"
| <math>\begin{matrix}\binom{X}{2} & = & \{ \{ u, v \} : u, v \in X \}.\end{matrix}</math>
|}

We may now use the features in <math>\mathrm{d}\mathcal{X} = \{ \mathrm{d}x_i \} = \{ \mathrm{d}x_1, \ldots, \mathrm{d}x_n \}</math> to classify the paths of <math>(\mathbb{B} \to X)</math> by way of the pairs in <math>X^2.</math>  If <math>X \cong \mathbb{B}^n,</math> then a path <math>q</math> in <math>X</math> has the following form:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
q : (\mathbb{B} \to \mathbb{B}^n) & \cong & \mathbb{B}^n \times \mathbb{B}^n & \cong & \mathbb{B}^{2n} & \cong & (\mathbb{B}^2)^n.
\end{matrix}</math>
|}

Intuitively, we want to map this <math>(\mathbb{B}^2)^n</math> onto <math>\mathbb{D}^n</math> by mapping each component <math>\mathbb{B}^2</math> onto a copy of <math>\mathbb{D}.</math>  But in the presenting context <math>{}^{\backprime\backprime} \mathbb{D} {}^{\prime\prime}</math> is just a name associated with, or an incidental quality attributed to, coefficient values in <math>\mathbb{B}</math> when they are attached to features in <math>\mathrm{d}\mathcal{X}.</math>

Taking these intentions into account, define <math>\mathrm{d}x_i : X^2 \to \mathbb{B}</math> in the following manner:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{lcrcl}
\mathrm{d}x_i(u, v)
& = & \texttt{(} ~ x_i(u) & \texttt{,} & x_i(v) ~ \texttt{)}
\\
& = & x_i(u) & + & x_i(v)
\\
& = & x_i(v) & - & x_i(u).
\end{array}</math>
|}

In the above transcription, the operator bracket of the form <math>\texttt{(} \ldots \texttt{,} \ldots \texttt{)}</math> is a ''cactus lobe'', in general signifying that just one of the arguments listed is false.  In the case of two arguments this is the same thing as saying that the arguments are not equal.  The plus sign signifies boolean addition, in the sense of addition in <math>\mathrm{GF}(2),</math> and thus means the same thing in this context as the minus sign, in the sense of adding the additive inverse.

The above definition of <math>\mathrm{d}x_i : X^2 \to \mathbb{B}</math> is equivalent to defining <math>\mathrm{d}x_i : (\mathbb{B} \to X) \to \mathbb{B}</math> in the following way:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{lcrcl}
\mathrm{d}x_i (q)
& = & \texttt{(} ~ x_i(q_0) & \texttt{,} & x_i(q_1) ~ \texttt{)}
\\
& = & x_i(q_0) & + & x_i(q_1)
\\
& = & x_i(q_1) & - & x_i(q_0).
\end{array}</math>
|}

In this definition <math>q_b = q(b),</math> for each <math>b</math> in <math>\mathbb{B}.</math>  Thus, the proposition <math>\mathrm{d}x_i</math> is true of the path <math>q = (u, v)</math> exactly if the terms of <math>q,</math> the endpoints <math>u</math> and <math>v,</math> lie on different sides of the question <math>x_i.</math>

The language of features in <math>\langle \mathrm{d}\mathcal{X} \rangle,</math> indeed the whole calculus of propositions in <math>[\mathrm{d}\mathcal{X}],</math> may now be used to classify paths and sets of paths.  In other words, the paths can be taken as models of the propositions <math>g : \mathrm{d}X \to \mathbb{B}.</math>  For example, the paths corresponding to <math>\mathrm{diag}(X)</math> fall under the description <math>\texttt{(} \mathrm{d}x_1 \texttt{)} \cdots \texttt{(} \mathrm{d}x_n \texttt{)},</math> which says that nothing changes against the backdrop of the coordinate frame <math>\{ x_1, \ldots, x_n \}.</math>

Finally, a few words of explanation may be in order.  If this concept of a path appears to be described in a roundabout fashion, it is because I am trying to avoid using any assumption of vector space properties for the space <math>X</math> that contains its range.  In many ways the treatment is still unsatisfactory, but improvements will have to wait for the introduction of substitution operators acting on singular propositions.

==={{anchor|Extended Universe}}The Extended Universe of Discourse===

{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |  
| width="92%" |
At the moment of speaking, I would like to have perceived a nameless voice, long preceding me, leaving me merely to enmesh myself in it, taking up its cadence, and to lodge myself, when no one was looking, in its interstices as if it had paused an instant, in suspense, to beckon to me.
| width="4%" |  
|-
| align="right" colspan="3" | — Michel Foucault, ''The Discourse on Language'', [Fou, 215]
|}

The <i>extended basis</i> <math>\mathrm{E}\mathcal{A}</math> of a universe of discourse <math>[\mathcal{A}]</math> is formed by taking the initial basis <math>\mathcal{A}</math> together with the differential basis <math>\mathrm{d}\mathcal{A}.</math>  Thus we have the following formula.

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{lclcl}
\mathrm{E}\mathcal{A}
& = & \mathcal{A} \cup \mathrm{d}\mathcal{A}
& = & \{ a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}.
\end{array}</math>
|}

This supplies enough material to construct the <i>differential extension</i> <math>\mathrm{E}A</math> of the space <math>A,</math> also called the <i>tangent bundle</i> of <math>A,</math> in the following fashion.

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{lcl}
\mathrm{E}A
& = & \langle \mathrm{E}\mathcal{A} \rangle
\\[4pt]
& = & \langle \mathcal{A} \cup \mathrm{d}\mathcal{A} \rangle
\\[4pt]
& = & \langle a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle,
\end{array}</math>
|}

and also

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{lcl}
\mathrm{E}A
& = & A \times \mathrm{d}A
\\[4pt]
& = & A_1 \times \ldots \times A_n \times \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n.
\end{array}</math>
|}

That gives <math>\mathrm{E}A</math> the type <math>\mathbb{B}^n \times \mathbb{D}^n.</math>

Finally, the extended universe <math>\mathrm{E}A^\bullet = [\mathrm{E}\mathcal{A}]</math> is the full collection of points and functions, or interpretations and propositions, based on the extended set of features <math>\mathrm{E}\mathcal{A},</math> a fact summed up in the following notation.

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{lclcl}
\mathrm{E}A^\bullet
& = & [\mathrm{E}\mathcal{A}]
& = & [a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n].
\end{array}</math>
|}

That gives the extended universe <math>\mathrm{E}A^\bullet</math> the following type.

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{lcl}
(\mathbb{B}^n \times \mathbb{D}^n\ +\!\to \mathbb{B})
& = & (\mathbb{B}^n \times \mathbb{D}^n, (\mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B})).
\end{array}</math>
|}

A proposition in the extended universe <math>[\mathrm{E}\mathcal{A}]</math> is called a <i>differential proposition</i> and forms the logical analogue of a system of differential equations, constraints, or relations in ordinary calculus.

With these constructions, the differential extension <math>\mathrm{E}A</math> and the space of differential propositions <math>(\mathrm{E}A \to \mathbb{B}),</math> we arrive at the launchpad of our space explorations.

Table 8 summarizes the notations needed to describe the first order differential extensions of propositional calculi in a systematic manner.

<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 8.} ~~ \text{Differential Extension : Basic Notation}</math>
|- style="height:40px; background:ghostwhite"
| <math>\text{Symbol}</math>
| <math>\text{Notation}</math>
| <math>\text{Description}</math>
| <math>\text{Type}</math>
|-
| <math>\mathrm{d}\mathfrak{A}</math>
| <math>\{ {}^{\backprime\backprime} \mathrm{d}a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} \mathrm{d}a_n {}^{\prime\prime} \}</math>
|
<math>\begin{matrix}
\text{Alphabet of}
\\[2pt]
\text{differential symbols}
\end{matrix}</math>
| <math>[n] = \mathbf{n}</math>
|-
| <math>\mathrm{d}\mathcal{A}</math>
| <math>\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}</math>
|
<math>\begin{matrix}
\text{Basis of}
\\[2pt]
\text{differential features}
\end{matrix}</math>
| <math>[n] = \mathbf{n}</math>
|-
| <math>\mathrm{d}A_i</math>
| <math>\{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \}</math>
| <math>\text{Differential dimension}~ i</math>
| <math>\mathbb{D}</math>
|-
| <math>\mathrm{d}A</math>
|
<math>\begin{matrix}
\langle \mathrm{d}\mathcal{A} \rangle
\\[2pt]
\langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle
\\[2pt]
\{ (\mathrm{d}a_1, \ldots, \mathrm{d}a_n) \}
\\[2pt]
\mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n
\\[2pt]
\textstyle \prod_i \mathrm{d}A_i
\end{matrix}</math>
|
<math>\begin{matrix}
\text{Tangent space at a point:}
\\[2pt]
\text{Set of changes, motions,}
\\[2pt]
\text{steps, tangent vectors}
\\[2pt]
\text{at a point}
\end{matrix}</math>
| <math>\mathbb{D}^n</math>
|-
| <math>\mathrm{d}A^*</math>
| <math>(\mathrm{hom} : \mathrm{d}A \to \mathbb{B})</math>
| <math>\text{Linear functions on}~ \mathrm{d}A</math>
| <math>(\mathbb{D}^n)^* \cong \mathbb{D}^n</math>
|-
| <math>\mathrm{d}A^\uparrow</math>
| <math>(\mathrm{d}A \to \mathbb{B})</math>
| <math>\text{Boolean functions on}~ \mathrm{d}A</math>
| <math>\mathbb{D}^n \to \mathbb{B}</math>
|-
| <math>\mathrm{d}A^\bullet</math>
|
<math>\begin{matrix}
[\mathrm{d}\mathcal{A}]
\\[2pt]
(\mathrm{d}A, \mathrm{d}A^\uparrow)
\\[2pt]
(\mathrm{d}A ~+\!\to \mathbb{B})
\\[2pt]
(\mathrm{d}A, (\mathrm{d}A \to \mathbb{B}))
\\[2pt]
[\mathrm{d}a_1, \ldots, \mathrm{d}a_n]
\end{matrix}</math>
|
<math>\begin{matrix}
\text{Tangent universe at a point of}~ A^\bullet,
\\[2pt]
\text{based on the tangent features}
\\[2pt]
\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}
\end{matrix}</math>
|
<math>\begin{matrix}
(\mathbb{D}^n, (\mathbb{D}^n \to \mathbb{B}))
\\[2pt]
(\mathbb{D}^n ~+\!\to \mathbb{B})
\\[2pt]
[\mathbb{D}^n]
\end{matrix}</math>
|}

<br>

The adjective <i>differential</i> or <i>tangent</i> is systematically attached to every construct based on the differential alphabet <math>\mathrm{d}\mathfrak{A},</math> taken by itself.  In like fashion, the adjective <i>extended</i> or the substantive <i>bundle</i> is systematically attached to any construct associated with the full complement of <math>{2n}</math> features.

It eventually becomes necessary to extend the initial alphabet even further, to allow for the discussion of higher order differential expressions. Table 9 provides a suggestion of how these further extensions can be carried out.

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:75%"
|+ style="height:30px" | <math>\text{Table 9.} ~~ \text{Higher Order Differential Features}</math>
|
<p><math>\begin{array}{lllll}
\mathrm{d}^0 \mathcal{A} & = & \{ a_1, \ldots, a_n \} & = & \mathcal{A}
\\
\mathrm{d}^1 \mathcal{A} & = & \{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \} & = & \mathrm{d}\mathcal{A}
\end{array}</math></p>

<p><math>\begin{array}{lll}
\mathrm{d}^k \mathcal{A} & = & \{ \mathrm{d}^k a_1, \ldots, \mathrm{d}^k a_n \}
\\
\mathrm{d}^* \mathcal{A} & = & \{ \mathrm{d}^0 \mathcal{A}, \ldots, \mathrm{d}^k \mathcal{A}, \ldots \}
\end{array}</math></p>
|
<math>\begin{array}{lll}
\mathrm{E}^0 \mathcal{A} & = & \mathrm{d}^0 \mathcal{A}
\\
\mathrm{E}^1 \mathcal{A} & = & \mathrm{d}^0 \mathcal{A} ~\cup~ \mathrm{d}^1 \mathcal{A}
\\
\mathrm{E}^k \mathcal{A} & = & \mathrm{d}^0 \mathcal{A} ~\cup~ \ldots ~\cup~ \mathrm{d}^k \mathcal{A}
\\
\mathrm{E}^\infty \mathcal{A} & = & \bigcup~ \mathrm{d}^* \mathcal{A}
\end{array}</math>
|}

<br>

===Intentional Propositions===

{| width="100%" cellpadding="0" cellspacing="0"
| width="40%" |  
| width="60%" |
Do you guess I have some intricate purpose?<br>
Well I have . . . . for the April rain has, and the mica on<br>
     the side of a rock has.
|-
|  
| align="right" | — Walt Whitman, ''Leaves of Grass'', [Whi, 45]
|}

In order to analyze the behavior of a system at successive moments in time, while staying within the limitations of propositional logic, it is necessary to create independent alphabets of logical features for each moment of time we contemplate using in our discussion.  Those moments have reference to typical instances and relative intervals, not actual or absolute times.

For example, to discuss <i>velocities</i> (first order rates of change) we need to consider points of time in pairs.  There are a number of natural ways of doing this.  Given an initial alphabet, we could use its symbols as a lexical basis to generate successive alphabets of compound symbols, say, with temporal markers appended as suffixes.

As a standard way of dealing with these situations, the following scheme of notation suggests a way of extending any alphabet of logical features through as many temporal moments as a particular order of analysis may demand.  The lexical operators <math>\mathrm{p}^k</math> and <math>\mathrm{Q}^k</math> are convenient in many contexts where the accumulation of prime symbols and union symbols would otherwise be cumbersome.

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:75%"
|+ style="height:30px" | <math>\text{Table 10.} ~~ \text{A Realm of Intentional Features}</math>
|
<p><math>\begin{array}{lllll}
\mathrm{p}^0 \mathcal{A} & = & \{ a_1, \ldots, a_n \} & = & \mathcal{A}
\\
\mathrm{p}^1 \mathcal{A} & = & \{ a_1^\prime, \ldots, a_n^\prime \} & = & \mathcal{A}^\prime
\\
\mathrm{p}^2 \mathcal{A} & = & \{ a_1^{\prime\prime}, \ldots, a_n^{\prime\prime} \} & = & \mathcal{A}^{\prime\prime}
\\
\cdots & & \cdots &
\end{array}</math></p>

<p><math>\begin{array}{lll}
\mathrm{p}^k \mathcal{A} & = & \{ \mathrm{p}^k a_1, \ldots, \mathrm{p}^k a_n \}
\end{array}</math></p>
|
<math>\begin{array}{lll}
\mathrm{Q}^0 \mathcal{A} & = & \mathcal{A}
\\
\mathrm{Q}^1 \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}'
\\
\mathrm{Q}^2 \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}' \cup \mathcal{A}''
\\
\cdots & & \cdots
\\
\mathrm{Q}^k \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}' \cup \ldots \cup \mathrm{p}^k \mathcal{A}
\end{array}</math>
|}

<br>

The resulting augmentations of our logical basis determine a series of discursive universes that may be called the ''intentional extension'' of propositional calculus. This extension follows a pattern analogous to the differential extension, which was developed in terms of the operators <math>\mathrm{d}^k</math> and <math>\mathrm{E}^k,</math> and there is a natural relation between these two extensions that bears further examination. In contexts displaying this pattern, where a sequence of domains stretches from an anchoring domain <math>X</math> through an indefinite number of higher reaches, a particular collection of domains based on <math>X</math> will be referred to as a ''realm'' of <math>X,</math> and when the succession exhibits a temporal aspect, as a ''reign'' of <math>X.</math>

For the purposes of this discussion, an ''intentional proposition'' is defined as a proposition in the universe of discourse <math>\mathrm{Q}X^\bullet = [\mathrm{Q}\mathcal{X}],</math> in other words, a map <math>q : \mathrm{Q}X \to \mathbb{B}.</math> The sense of this definition may be seen if we consider the following facts. First, the equivalence <math>\mathrm{Q}X = X \times X'</math> motivates the following chain of isomorphisms between spaces:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{lllcl}
(\mathrm{Q}X \to \mathbb{B})
& \cong & (X & \times & ~X' \to \mathbb{B})
\\[4pt]
& \cong & (X & \to & (X' \to \mathbb{B}))
\\[4pt]
& \cong & (X' & \to & (X~ \to \mathbb{B})).
\end{array}</math>
|}

Viewed in this light, an intentional proposition <math>q</math> may be rephrased as a map <math>q : X \times X' \to \mathbb{B},</math> which judges the juxtaposition of states in <math>X</math> from one moment to the next. Alternatively, <math>q</math> may be parsed in two stages in two different ways, as <math>q : X \to (X' \to \mathbb{B})</math> and as <math>q : X' \to (X \to \mathbb{B}),</math> which associate to each point of <math>X</math> or <math>X'</math> a proposition about states in <math>X'</math> or <math>X,</math> respectively. In this way, an intentional proposition embodies a type of value system, in effect, a proposal that places a value on a collection of ends-in-view, or a project that evaluates a set of goals as regarded from each point of view in the state space of a system.

In sum, the intentional proposition <math>q</math> indicates a method for the systematic selection of local goals. As a general form of description, a map of the type <math>q : \mathrm{Q}^i X \to \mathbb{B}</math> may be referred to as an "<math>i^\text{th}</math> order intentional proposition". Naturally, when we speak of intentional propositions without qualification, we usually mean first order intentions.

Many different realms of discourse have the same structure as the extensions that have been indicated here. From a strictly logical point of view, each new layer of terms is composed of independent logical variables that are no different in kind from those that go before, and each further course of logical atoms is treated like so many individual, but otherwise indifferent bricks by the prototype computer program that I use as a propositional interpreter. Thus, the names that I use to single out the differential and the intentional extensions, and the lexical paradigms that I follow to construct them, are meant to suggest the interpretations that I have in mind, but they can only hint at the extra meanings that human communicators may pack into their terms and inflections.

As applied here, the word ''intentional'' is drawn from common use and may have little bearing on its technical use in other, more properly philosophical, contexts. I am merely using the complex of intentional concepts — aims, ends, goals, objectives, purposes, and so on — metaphorically to flesh out and vividly to represent any situation where one needs to contemplate a system in multiple aspects of state and destination, that is, its being in certain states and at the same time acting as if headed through certain states. If confusion arises, more neutral words like ''conative'', ''contingent'', ''discretionary'', ''experimental'', ''kinetic'', ''progressive'', ''tentative'', or ''trial'' would probably serve as well.

==={{anchor|Easy Street}}Life on Easy Street===

{| width="100%" cellpadding="0" cellspacing="0"
| width="40%" |  
| width="60%" |
Failing to fetch me at first keep encouraged,<br>
Missing me one place search another,<br>
I stop some where waiting for you
|-
|  
| align="right" | — Walt Whitman, ''Leaves of Grass'', [Whi, 88]
|}

The finite character of the extended universe <math>[\mathrm{E}\mathcal{A}]</math> makes the task of solving differential propositions relatively straightforward.  The solution set of the differential proposition <math>q : \mathrm{E}A \to \mathbb{B}</math> is the set of models <math>q^{-1}(1)</math> in <math>\mathrm{E}A.</math>  Finding all models of <math>q,</math> the extended interpretations in <math>\mathrm{E}A</math> which satisfy <math>q,</math> can be carried out by a finite search.

Being in possession of complete algorithms for propositional calculus modeling, theorem checking, and theorem proving makes the analytic task fairly simple in principle, even if the question of efficiency in the face of arbitrary complexity remains another matter entirely.  The NP‑completeness of propositional satisfiability may weigh against the prospects of a single efficient algorithm capable of covering the whole space <math>[\mathrm{E}\mathcal{A}]</math> with equal facility but there appears to be much room for improvement in classifying special forms and in developing algorithms tailored to their practical processing.

In view of these constraints and contingencies, our focus shifts to the tasks of approximation and interpretation that support intuition, especially in dealing with the natural kinds of differential propositions that arise in applications, and in the effort to understand, in succinct and adaptive forms, their dynamic implications. In the absence of direct insights, these tasks are partially carried out by forging analogies with the familiar situations and customary routines of ordinary calculus. But the indirect approach, going by way of specious analogy and intuitive habit, forces us to remain on guard against the circumstance that occurs when the word ''forging'' takes on its shadier nuance, indicting the constant risk of a counterfeit in the proportion.

<br>

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<div align="center">
• [[Differential Logic and Dynamic Systems • Overview|Overview]]
• [[Differential Logic and Dynamic Systems • Part 1|Part 1]]
• [[Differential Logic and Dynamic Systems • Part 2|Part 2]]
• [[Differential Logic and Dynamic Systems • Part 3|Part 3]]
• [[Differential Logic and Dynamic Systems • Part 4|Part 4]]
• [[Differential Logic and Dynamic Systems • Part 5|Part 5]]
• [[Differential Logic and Dynamic Systems • Appendices|Appendices]]
• [[Differential Logic and Dynamic Systems • References|References]]
• [[Differential Logic and Dynamic Systems • Document History|Document History]]
•
</div>
----

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2026-01-23T17:00:02Z