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Directory:Jon Awbrey/Papers/Differential Logic : Introduction (view source)
Revision as of 02:48, 21 June 2009
, 02:48, 21 June 2009→Note 3: spacing
following formula:
following formula:
{| align="center" cellspacing="10" width="90%"
{| align="center" cellspacing="10" style="text-align:center"
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{E}f(p, q, \operatorname{d}p, \operatorname{d}q)
\operatorname{E}f(p, q, \operatorname{d}p, \operatorname{d}q)
In the example <math>f(p, q) = pq,\!</math> the enlargement <math>\operatorname{E}f</math> is computed as follows:
In the example <math>f(p, q) = pq,\!</math> the enlargement <math>\operatorname{E}f</math> is computed as follows:
{| align="center" cellspacing="10" width="90%"
{| align="center" cellspacing="10" style="text-align:center"
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{E}f(p, q, \operatorname{d}p, \operatorname{d}q)
\operatorname{E}f(p, q, \operatorname{d}p, \operatorname{d}q)
\texttt{(} p, \operatorname{d}p \texttt{)(} q, \operatorname{d}q \texttt{)}
\texttt{(} p, \operatorname{d}p \texttt{)(} q, \operatorname{d}q \texttt{)}
\end{matrix}</math>
\end{matrix}</math>
|}
|-
| [[Image:Cactus Graph Ef = (P,dP)(Q,dQ).jpg|500px]]
| [[Image:Cactus Graph Ef = (P,dP)(Q,dQ).jpg|500px]]
|}
|}
Given the proposition <math>f(p, q)\!</math> over <math>X = P \times Q,</math> the ''(first order) difference'' of <math>f\!</math> is the proposition <math>\operatorname{D}f</math> over <math>\operatorname{E}X</math> that is defined by the formula <math>\operatorname{D}f = \operatorname{E}f - f,</math> or, written out in full:
Given the proposition <math>f(p, q)\!</math> over <math>X = P \times Q,</math> the ''(first order) difference'' of <math>f\!</math> is the proposition <math>\operatorname{D}f</math> over <math>\operatorname{E}X</math> that is defined by the formula <math>\operatorname{D}f = \operatorname{E}f - f,</math> or, written out in full:
{| align="center" cellspacing="10" width="90%"
{| align="center" cellspacing="10" style="text-align:center"
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{D}f(p, q, \operatorname{d}p, \operatorname{d}q)
\operatorname{D}f(p, q, \operatorname{d}p, \operatorname{d}q)
In the example <math>f(p, q) = pq,\!</math> the difference <math>\operatorname{D}f</math> is computed as follows:
In the example <math>f(p, q) = pq,\!</math> the difference <math>\operatorname{D}f</math> is computed as follows:
{| align="center" cellspacing="10" width="90%"
{| align="center" cellspacing="10" style="text-align:center"
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{D}f(p, q, \operatorname{d}p, \operatorname{d}q)
\operatorname{D}f(p, q, \operatorname{d}p, \operatorname{d}q)
\texttt{((} p, \operatorname{d}p \texttt{)(} q, \operatorname{d}q \texttt{)}, pq \texttt{)}
\texttt{((} p, \operatorname{d}p \texttt{)(} q, \operatorname{d}q \texttt{)}, pq \texttt{)}
\end{matrix}</math>
\end{matrix}</math>
|}
|-
| [[Image:Cactus Graph Df = ((P,dP)(Q,dQ),PQ).jpg|500px]]
| [[Image:Cactus Graph Df = ((P,dP)(Q,dQ),PQ).jpg|500px]]
|}
|}
We did not yet go through the trouble to interpret this (first order) ''difference of conjunction'' fully, but were happy simply to evaluate it with respect to a single location in the universe of discourse, namely, at the point picked out by the singular proposition <math>pq,\!</math> that is, at the place where <math>p = 1\!</math> and <math>q = 1.\!</math> This evaluation is written in the form <math>\operatorname{D}f|_{pq}</math> or <math>\operatorname{D}f|_{(1, 1)},</math> and we arrived at the locally applicable law that is stated and illustrated as follows:
We did not yet go through the trouble to interpret this (first order) ''difference of conjunction'' fully, but were happy simply to evaluate it with respect to a single location in the universe of discourse, namely, at the point picked out by the singular proposition <math>pq,\!</math> that is, at the place where <math>p = 1\!</math> and <math>q = 1.\!</math> This evaluation is written in the form <math>\operatorname{D}f|_{pq}</math> or <math>\operatorname{D}f|_{(1, 1)},</math> and we arrived at the locally applicable law that is stated and illustrated as follows:
{| align="center" cellspacing="10" width="90%"
{| align="center" cellspacing="10" style="text-align:center"
|
<math>f(p, q) ~=~ pq ~=~ p ~\operatorname{and}~ q \quad \Rightarrow \quad \operatorname{D}f|_{pq} ~=~ \texttt{((} \operatorname{dp} \texttt{)(} \operatorname{d}q \texttt{))} ~=~ \operatorname{d}p ~\operatorname{or}~ \operatorname{d}q</math>
<math>f(p, q) ~=~ pq ~=~ p ~\operatorname{and}~ q \quad \Rightarrow \quad \operatorname{D}f|_{pq} ~=~ \texttt{((} \operatorname{dp} \texttt{)(} \operatorname{d}q \texttt{))} ~=~ \operatorname{d}p ~\operatorname{or}~ \operatorname{d}q</math>
|-
|-
| align="center" |
| [[Image:Venn Diagram PQ Difference Conj At Conj.jpg|500px]]
[[Image:Venn Diagram PQ Difference Conj At Conj.jpg|500px]]
|-
|-
| align="center" |
| [[Image:Cactus Graph PQ Difference Conj At Conj.jpg|500px]]
[[Image:Cactus Graph PQ Difference Conj At Conj.jpg|500px]]
|}
|}
The picture shows the analysis of the inclusive disjunction <math>\texttt{((} \operatorname{d}p \texttt{)(} \operatorname{d}q \texttt{))}</math> into the following exclusive disjunction:
The picture shows the analysis of the inclusive disjunction <math>\texttt{((} \operatorname{d}p \texttt{)(} \operatorname{d}q \texttt{))}</math> into the following exclusive disjunction:
{| align="center" cellspacing="10" width="90%"
{| align="center" cellspacing="10" style="text-align:center"
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{d}p ~\texttt{(} \operatorname{d}q \texttt{)}
\operatorname{d}p ~\texttt{(} \operatorname{d}q \texttt{)}