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Directory:Jon Awbrey/Papers/Dynamics And Logic (view source)
Revision as of 03:04, 13 June 2009
, 03:04, 13 June 2009→Note 4: markup
Sticking with the example f<p, q> = pq, let us compute the
Sticking with the example f<p, q> = pq, let us compute the
value of the difference proposition Df at all of the points.
value of the difference proposition Df at all of the points.
</pre>
{| align="center" cellpadding="6" width="90%"
| align="center" |
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| |
| Df = ((p, dp)(q, dq), pq) |
| Df = ((p, dp)(q, dq), pq) |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
|-
| align="center" |
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| |
| Df|pq = ((dp) (dq)) |
| Df|pq = ((dp) (dq)) |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
|-
| align="center" |
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| |
| Df|p(q) = (dp) dq |
| Df|p(q) = (dp) dq |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
|-
| align="center" |
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| |
| Df|(p)q = dp (dq) |
| Df|(p)q = dp (dq) |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
|-
| align="center" |
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| |
| Df|(p)(q) = dp dq |
| Df|(p)(q) = dp dq |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
|}
The easy way to visualize the values of these graphical
The easy way to visualize the values of these graphical expressions is just to notice the following equivalents:
expressions is just to notice the following equivalents:
{| align="center" cellpadding="6" width="90%"
| align="center" |
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| |
| (e, , ... , , ) = (e) |
| (e, , ... , , ) = (e) |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
|-
| align="center" |
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| |
| (e_1, ..., e_k, ()) = e_1 ... e_k |
| (e_1, ..., e_k, ()) = e_1 ... e_k |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
|}
Laying out the arrows on the augmented venn diagram,
Laying out the arrows on the augmented venn diagram, one gets a picture of a ''differential vector field''.
one gets a picture of a "differential vector field".
{| align="center" cellpadding="10"
| [[Image:Venn Diagram PQ Difference Conj.jpg|500px]]
|}
| \ \%|%/ / |
| \ \|/ / |
<pre>
This just amounts to a depiction of the points,
This just amounts to a depiction of the points,
truth-value assignments, or interpretations in
truth-value assignments, or interpretations in