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Directory:Jon Awbrey/Papers/Dynamics And Logic (view source)
Revision as of 14:58, 13 June 2009
, 14:58, 13 June 2009→Note 6: markup
==Note 6==
==Note 6==
The ''enlargement'' or ''shift'' operator <math>\operatorname{E}</math> exhibits a wealth of interesting and useful properties in its own right, so it pays to examine a few of the more salient features that play out on the surface of our initial example, <math>f(p, q) = pq.\!</math>
<pre>
<pre>
To begin we need to formulate a suitably generic
To begin we need to formulate a suitably generic
definition of the extended universe of discourse:
definition of the extended universe of discourse:
the value of the enlarged proposition Ef at each of the
the value of the enlarged proposition Ef at each of the
points in the initial domain of discourse X = !P! x !Q!.
points in the initial domain of discourse X = !P! x !Q!.
</pre>
{| align="center" cellpadding="6" width="90%"
| align="center" |
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| |
| Ef = (p, dp) (q, dq) |
| Ef = (p, dp) (q, dq) |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
|-
| align="center" |
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| |
| Ef|pq = (dp) (dq) |
| Ef|pq = (dp) (dq) |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
|-
| align="center" |
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| |
| Ef|p(q) = (dp) dq |
| Ef|p(q) = (dp) dq |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
|-
| align="center" |
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| |
| Ef|(p)q = dp (dq) |
| Ef|(p)q = dp (dq) |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
|-
| align="center" |
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| |
| Ef|(p)(q) = dp dq |
| Ef|(p)(q) = dp dq |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
|}
<pre>
Given the kind of data that arises from this form of analysis,
Given the kind of data that arises from this form of analysis,
we can now fold the disjoined ingredients back into a boolean
we can now fold the disjoined ingredients back into a boolean
a digraph picture, where the "no change" element (dp)(dq)
a digraph picture, where the "no change" element (dp)(dq)
is drawn as a loop at the point p q.
is drawn as a loop at the point p q.
</pre>
{| align="center" cellpadding="10"
| [[Image:Directed Graph PQ Enlargement Conj.jpg|500px]]
|}
{| align="center" cellpadding="10"
| f = p q |
|
<math>\begin{array}{rcccccc}
f
& = & p & \cdot & q
\\[4pt]
\operatorname{E}f
& = & p & \cdot & q & \cdot & (\operatorname{d}p)(\operatorname{d}q)
\\[4pt]
& + & p & \cdot & (q) & \cdot & (\operatorname{d}p)~\operatorname{d}q~
\\[4pt]
& + & (p) & \cdot & q & \cdot & ~\operatorname{d}p~(\operatorname{d}q)
\\[4pt]
& + & (p) & \cdot & (q) & \cdot & ~\operatorname{d}p~~\operatorname{d}q~\end{array}</math>
|}
| | |
We may understand the enlarged proposition Ef
We may understand the enlarged proposition <math>\operatorname{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>
as telling us all the different ways to reach
a model of f from any point of the universe X.
</pre>
==Note 7==
==Note 7==