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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.
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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
12,089

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