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====Note 7====
 
====Note 7====
   −
<pre>
+
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>\operatorname{E}</math> and <math>\operatorname{D}</math> at once whelm over its discrete and finite powers to grasp them.  But here, in the fully serene idylls of ZOL, we find ourselves fit with the compass of a wit that is all we'd ever wish to explore their effects with care.
−
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 E and D at once
  −
whelm over its discrete and finite powers to grasp them.  But here, in
  −
the fully serene idylls of ZOL, we find ourselves fit with the compass
  −
of a wit that is all we'd ever wish to explore their effects with care.
      
So let us do just that.
 
So let us do just that.
   −
I will first rationalize the novel grouping of propositional forms
+
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&nbsp;A3.
−
in the last set of Tables, as that will extend a gentle invitation
+
 
−
to the mathematical subject of "group theory", and demonstrate its
+
<br>
−
relevance to differential logic in a strikingly apt and useful way.
+
 
−
The data for that account is contained in Table 3.
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
 +
|+ <math>\text{Table A3.}~~\operatorname{E}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}x, \operatorname{d}y \}</math>
 +
|- style="background:#f0f0ff"
 +
| width="10%" | &nbsp;
 +
| width="18%" | <math>f\!</math>
 +
| width="18%" |
 +
<p><math>\operatorname{T}_{11} f</math></p>
 +
<p><math>\operatorname{E}f|_{\operatorname{d}x~\operatorname{d}y}</math></p>
 +
| width="18%" |
 +
<p><math>\operatorname{T}_{10} f</math></p>
 +
<p><math>\operatorname{E}f|_{\operatorname{d}x(\operatorname{d}y)}</math></p>
 +
| width="18%" |
 +
<p><math>\operatorname{T}_{01} f</math></p>
 +
<p><math>\operatorname{E}f|_{(\operatorname{d}x)\operatorname{d}y}</math></p>
 +
| width="18%" |
 +
<p><math>\operatorname{T}_{00} f</math></p>
 +
<p><math>\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}</math></p>
 +
|-
 +
| <math>f_0\!</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_1
 +
\\[4pt]
 +
f_2
 +
\\[4pt]
 +
f_4
 +
\\[4pt]
 +
f_8
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(x)(y)
 +
\\[4pt]
 +
(x)~y~
 +
\\[4pt]
 +
~x~(y)
 +
\\[4pt]
 +
~x~~y~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~x~~y~
 +
\\[4pt]
 +
~x~(y)
 +
\\[4pt]
 +
(x)~y~
 +
\\[4pt]
 +
(x)(y)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~x~(y)
 +
\\[4pt]
 +
~x~~y~
 +
\\[4pt]
 +
(x)(y)
 +
\\[4pt]
 +
(x)~y~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(x)~y~
 +
\\[4pt]
 +
(x)(y)
 +
\\[4pt]
 +
~x~~y~
 +
\\[4pt]
 +
~x~(y)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(x)(y)
 +
\\[4pt]
 +
(x)~y~
 +
\\[4pt]
 +
~x~(y)
 +
\\[4pt]
 +
~x~~y~
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_3
 +
\\[4pt]
 +
f_{12}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(x)
 +
\\[4pt]
 +
~x~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~x~
 +
\\[4pt]
 +
(x)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~x~
 +
\\[4pt]
 +
(x)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(x)
 +
\\[4pt]
 +
~x~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(x)
 +
\\[4pt]
 +
~x~
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_6
 +
\\[4pt]
 +
f_9
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(x,~y)~
 +
\\[4pt]
 +
((x,~y))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(x,~y)~
 +
\\[4pt]
 +
((x,~y))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((x,~y))
 +
\\[4pt]
 +
~(x,~y)~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((x,~y))
 +
\\[4pt]
 +
~(x,~y)~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(x,~y)~
 +
\\[4pt]
 +
((x,~y))
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_5
 +
\\[4pt]
 +
f_{10}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(y)
 +
\\[4pt]
 +
~y~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~y~
 +
\\[4pt]
 +
(y)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(y)
 +
\\[4pt]
 +
~y~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~y~
 +
\\[4pt]
 +
(y)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(y)
 +
\\[4pt]
 +
~y~
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_7
 +
\\[4pt]
 +
f_{11}
 +
\\[4pt]
 +
f_{13}
 +
\\[4pt]
 +
f_{14}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~x~~y~)
 +
\\[4pt]
 +
(~x~(y))
 +
\\[4pt]
 +
((x)~y~)
 +
\\[4pt]
 +
((x)(y))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((x)(y))
 +
\\[4pt]
 +
((x)~y~)
 +
\\[4pt]
 +
(~x~(y))
 +
\\[4pt]
 +
(~x~~y~)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((x)~y~)
 +
\\[4pt]
 +
((x)(y))
 +
\\[4pt]
 +
(~x~~y~)
 +
\\[4pt]
 +
(~x~(y))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~x~(y))
 +
\\[4pt]
 +
(~x~~y~)
 +
\\[4pt]
 +
((x)(y))
 +
\\[4pt]
 +
((x)~y~)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~x~~y~)
 +
\\[4pt]
 +
(~x~(y))
 +
\\[4pt]
 +
((x)~y~)
 +
\\[4pt]
 +
((x)(y))
 +
\end{matrix}</math>
 +
|-
 +
| <math>f_{15}\!</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
|- style="background:#f0f0ff"
 +
| colspan="2" | <math>\text{Fixed Point Total}\!</math>
 +
| <math>4\!</math>
 +
| <math>4\!</math>
 +
| <math>4\!</math>
 +
| <math>16\!</math>
 +
|}
   −
Table 3.  Ef Expanded Over Differential Features {dx, dy}
+
<br>
−
o------o------------o------------o------------o------------o------------o
  −
|      |            |            |            |            |            |
  −
|      |    f      |  T_11 f  |  T_10 f  |  T_01 f  |  T_00 f  |
  −
|      |            |            |            |            |            |
  −
|      |            | Ef| dx·dy  | Ef| dx(dy) | Ef| (dx)dy | Ef|(dx)(dy)|
  −
|      |            |            |            |            |            |
  −
o------o------------o------------o------------o------------o------------o
  −
|      |            |            |            |            |            |
  −
| f_0  |    ()    |    ()    |    ()    |    ()    |    ()    |
  −
|      |            |            |            |            |            |
  −
o------o------------o------------o------------o------------o------------o
  −
|      |            |            |            |            |            |
  −
| f_1  |  (x)(y)  |    x  y    |    x (y)  |  (x) y    |  (x)(y)  |
  −
|      |            |            |            |            |            |
  −
| f_2  |  (x) y    |    x (y)  |    x  y    |  (x)(y)  |  (x) y    |
  −
|      |            |            |            |            |            |
  −
| f_4  |    x (y)  |  (x) y    |  (x)(y)  |    x  y    |    x (y)  |
  −
|      |            |            |            |            |            |
  −
| f_8  |    x  y    |  (x)(y)  |  (x) y    |    x (y)  |    x  y    |
  −
|      |            |            |            |            |            |
  −
o------o------------o------------o------------o------------o------------o
  −
|      |            |            |            |            |            |
  −
| f_3  |  (x)      |    x      |    x      |  (x)      |  (x)      |
  −
|      |            |            |            |            |            |
  −
| f_12 |    x      |  (x)      |  (x)      |    x      |    x      |
  −
|      |            |            |            |            |            |
  −
o------o------------o------------o------------o------------o------------o
  −
|      |            |            |            |            |            |
  −
| f_6  |  (x, y)  |  (x, y)  |  ((x, y))  |  ((x, y))  |  (x, y)  |
  −
|      |            |            |            |            |            |
  −
| f_9  |  ((x, y))  |  ((x, y))  |  (x, y)  |  (x, y)  |  ((x, y))  |
  −
|      |            |            |            |            |            |
  −
o------o------------o------------o------------o------------o------------o
  −
|      |            |            |            |            |            |
  −
| f_5  |      (y)  |      y    |      (y)  |      y    |      (y)  |
  −
|      |            |            |            |            |            |
  −
| f_10 |      y    |      (y)  |      y    |      (y)  |      y    |
  −
|      |            |            |            |            |            |
  −
o------o------------o------------o------------o------------o------------o
  −
|      |            |            |            |            |            |
  −
| f_7  |  (x  y)  |  ((x)(y))  |  ((x) y)  |  (x (y))  |  (x  y)  |
  −
|      |            |            |            |            |            |
  −
| f_11 |  (x (y))  |  ((x) y)  |  ((x)(y))  |  (x  y)  |  (x (y))  |
  −
|      |            |            |            |            |            |
  −
| f_13 |  ((x) y)  |  (x (y))  |  (x  y)  |  ((x)(y))  |  ((x) y)  |
  −
|      |            |            |            |            |            |
  −
| f_14 |  ((x)(y))  |  (x  y)  |  (x (y))  |  ((x) y)  |  ((x)(y))  |
  −
|      |            |            |            |            |            |
  −
o------o------------o------------o------------o------------o------------o
  −
|      |            |            |            |            |            |
  −
| f_15 |    (())    |    (())    |    (())    |    (())    |    (())    |
  −
|      |            |            |            |            |            |
  −
o------o------------o------------o------------o------------o------------o
  −
|                  |            |            |            |            |
  −
| Fixed Point Total |      4    |      4    |      4    |    16    |
  −
|                  |            |            |            |            |
  −
o-------------------o------------o------------o------------o------------o
      +
<pre>
 
The shift operator E can be understood as enacting a substitution operation
 
The shift operator E can be understood as enacting a substitution operation
 
on the proposition that is given as its argument.  In our immediate example,
 
on the proposition that is given as its argument.  In our immediate example,
Jon Awbrey
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