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Revision as of 18:20, 30 November 2015
, 18:20, 30 November 2015try first part of ==Operational Representation==
We may understand the enlarged proposition <math>\mathrm{E}f\!</math> as telling us all the different ways to reach a model of the proposition <math>f\!</math> from each point of the universe <math>X.\!</math>
We may understand the enlarged proposition <math>\mathrm{E}f\!</math> as telling us all the different ways to reach a model of the proposition <math>f\!</math> from each point of the universe <math>X.\!</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 A3.
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ <math>\text{Table A3.}~~\mathrm{E}f ~\text{Expanded over Differential Features}~ \{ \mathrm{d}p, \mathrm{d}q \}\!</math>
|- style="background:#f0f0ff"
| width="10%" |
| width="18%" | <math>f\!</math>
| width="18%" |
<p><math>\mathrm{T}_{11} f\!</math></p>
<p><math>\mathrm{E}f|_{\mathrm{d}p~\mathrm{d}q}\!</math></p>
| width="18%" |
<p><math>\mathrm{T}_{10} f\!</math></p>
<p><math>\mathrm{E}f|_{\mathrm{d}p(\mathrm{d}q)}\!</math></p>
| width="18%" |
<p><math>\mathrm{T}_{01} f\!</math></p>
<p><math>\mathrm{E}f|_{(\mathrm{d}p)\mathrm{d}q}\!</math></p>
| width="18%" |
<p><math>\mathrm{T}_{00} f\!</math></p>
<p><math>\mathrm{E}f|_{(\mathrm{d}p)(\mathrm{d}q)}\!</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}
(p)(q)
\\[4pt]
(p)~q~
\\[4pt]
~p~(q)
\\[4pt]
~p~~q~
\end{matrix}\!</math>
|
<math>\begin{matrix}
~p~~q~
\\[4pt]
~p~(q)
\\[4pt]
(p)~q~
\\[4pt]
(p)(q)
\end{matrix}\!</math>
|
<math>\begin{matrix}
~p~(q)
\\[4pt]
~p~~q~
\\[4pt]
(p)(q)
\\[4pt]
(p)~q~
\end{matrix}\!</math>
|
<math>\begin{matrix}
(p)~q~
\\[4pt]
(p)(q)
\\[4pt]
~p~~q~
\\[4pt]
~p~(q)
\end{matrix}\!</math>
|
<math>\begin{matrix}
(p)(q)
\\[4pt]
(p)~q~
\\[4pt]
~p~(q)
\\[4pt]
~p~~q~
\end{matrix}\!</math>
|-
|
<math>\begin{matrix}
f_3
\\[4pt]
f_{12}
\end{matrix}\!</math>
|
<math>\begin{matrix}
(p)
\\[4pt]
~p~
\end{matrix}\!</math>
|
<math>\begin{matrix}
~p~
\\[4pt]
(p)
\end{matrix}\!</math>
|
<math>\begin{matrix}
~p~
\\[4pt]
(p)
\end{matrix}\!</math>
|
<math>\begin{matrix}
(p)
\\[4pt]
~p~
\end{matrix}\!</math>
|
<math>\begin{matrix}
(p)
\\[4pt]
~p~
\end{matrix}\!</math>
|-
|
<math>\begin{matrix}
f_6
\\[4pt]
f_9
\end{matrix}\!</math>
|
<math>\begin{matrix}
~(p,~q)~
\\[4pt]
((p,~q))
\end{matrix}\!</math>
|
<math>\begin{matrix}
~(p,~q)~
\\[4pt]
((p,~q))
\end{matrix}\!</math>
|
<math>\begin{matrix}
((p,~q))
\\[4pt]
~(p,~q)~
\end{matrix}\!</math>
|
<math>\begin{matrix}
((p,~q))
\\[4pt]
~(p,~q)~
\end{matrix}\!</math>
|
<math>\begin{matrix}
~(p,~q)~
\\[4pt]
((p,~q))
\end{matrix}\!</math>
|-
|
<math>\begin{matrix}
f_5
\\[4pt]
f_{10}
\end{matrix}\!</math>
|
<math>\begin{matrix}
(q)
\\[4pt]
~q~
\end{matrix}\!</math>
|
<math>\begin{matrix}
~q~
\\[4pt]
(q)
\end{matrix}\!</math>
|
<math>\begin{matrix}
(q)
\\[4pt]
~q~
\end{matrix}\!</math>
|
<math>\begin{matrix}
~q~
\\[4pt]
(q)
\end{matrix}\!</math>
|
<math>\begin{matrix}
(q)
\\[4pt]
~q~
\end{matrix}\!</math>
|-
|
<math>\begin{matrix}
f_7
\\[4pt]
f_{11}
\\[4pt]
f_{13}
\\[4pt]
f_{14}
\end{matrix}\!</math>
|
<math>\begin{matrix}
(~p~~q~)
\\[4pt]
(~p~(q))
\\[4pt]
((p)~q~)
\\[4pt]
((p)(q))
\end{matrix}\!</math>
|
<math>\begin{matrix}
((p)(q))
\\[4pt]
((p)~q~)
\\[4pt]
(~p~(q))
\\[4pt]
(~p~~q~)
\end{matrix}\!</math>
|
<math>\begin{matrix}
((p)~q~)
\\[4pt]
((p)(q))
\\[4pt]
(~p~~q~)
\\[4pt]
(~p~(q))
\end{matrix}\!</math>
|
<math>\begin{matrix}
(~p~(q))
\\[4pt]
(~p~~q~)
\\[4pt]
((p)(q))
\\[4pt]
((p)~q~)
\end{matrix}\!</math>
|
<math>\begin{matrix}
(~p~~q~)
\\[4pt]
(~p~(q))
\\[4pt]
((p)~q~)
\\[4pt]
((p)(q))
\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>
|}
<br>
==Propositional Forms on Two Variables==
==Propositional Forms on Two Variables==
