[Inquiry] Re: Dynamics And Logic

[Jon Awbrey]

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DAL.  Note 3

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Given the proposition f<p, q> over the space X = !P! x !Q!,
the (first order) "enlargement" of f is the proposition Ef
over the differential extension EX that is defined by the
following formula:

   Ef<p, q, dp, dq>

   =  f<p + dp, q + dq>

   =  f<(p, dp), (q, dq)>

In the example f<p, q> = pq, the enlargement Ef is given by:

   Ef<p, q, dp, dq>

   =  [p + dp][q + dq]

   =  (p, dp)(q, dq)

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` p `dp q `dq ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o---o o---o ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` | | `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ | | / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\| |/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` @=@ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| Ef =` ` ` ` ` ` (p, dp) (q, dq) ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

Given the proposition f<p, q> over X = !P! x !Q!, the
(first order) "difference" of f is the proposition Df
over EX that is defined by the formula Df = Ef - f, or,
written out in full:

   Df<p, q, dp, dq>

   =  f<p + dp, q + dq> - f<p, q>

   =  (f<(p, dp), (q, dq)>, f<p, q>)

In the example f<p, q> = pq, the difference Df is given by:

   Df<p, q, dp, dq>

   =  [p + dp][q + dq] - pq

   =  ((p, dp)(q, dq), pq)

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` p `dp q `dq ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` o---o o---o ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\ `| | `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ | | / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\| |/` ` ` ` `p q` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` o=o-----------o ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| Df =` ` ` ` ` `((p, dp)(q, dq), pq) ` ` ` ` ` ` |
o-------------------------------------------------o

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 pq,
in as much as if to say, at the place where p = 1 and q = 1.
This evaluation is written in the form Df|pq or Df|<1, 1>,
and we arrived at the locally applicable law that states
that f = pq = p and q => Df|pq = ((dp)(dq)) = dp or dq.

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `o-----------o` `o-----------o` ` ` ` ` |
| ` ` ` ` / ` ` ` ` ` ` \ / ` ` ` ` ` ` \ ` ` ` ` |
| ` ` ` `/` ` ` `p` ` ` `o` ` ` `q` ` ` `\` ` ` ` |
| ` ` ` / ` ` ` ` ` ` ` /%\ ` ` ` ` ` ` ` \ ` ` ` |
| ` ` `/` ` ` ` ` ` ` `/%%%\` ` ` ` ` ` ` `\` ` ` |
| ` ` o ` ` ` ` ` ` ` o%%%%%o ` ` ` ` ` ` ` o ` ` |
| ` ` | ` ` ` ` ` ` ` |%%%%%| ` ` ` ` ` ` ` | ` ` |
| ` ` | ` ` `dq (dp)` |%%%%%| `dp (dq)` ` ` | ` ` |
| ` ` | ` o<----------|--o--|---------->o ` | ` ` |
| ` ` | ` ` ` ` ` ` ` |%%|%%| ` ` ` ` ` ` ` | ` ` |
| ` ` | ` ` ` ` ` ` ` |%%|%%| ` ` ` ` ` ` ` | ` ` |
| ` ` o ` ` ` ` ` ` ` o%%|%%o ` ` ` ` ` ` ` o ` ` |
| ` ` `\` ` ` ` ` ` ` `\%|%/` ` ` ` ` ` ` `/` ` ` |
| ` ` ` \ ` ` ` ` ` ` ` \|/ ` ` ` ` ` ` ` / ` ` ` |
| ` ` ` `\` ` ` ` ` ` ` `|` ` ` ` ` ` ` `/` ` ` ` |
| ` ` ` ` \ ` ` ` ` ` ` /|\ ` ` ` ` ` ` / ` ` ` ` |
| ` ` ` ` `o-----------o | o-----------o` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `dp|dq` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `v` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` dp` `dq ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `o` `o` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| Df|pq = ` ` ` ` ` ((dp) (dq)) ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

The picture illustrates the analysis of the inclusive
disjunction ((dp)(dq)) into the exclusive disjunction:
dp(dq) + (dp)dq + dp dq, a differential proposition that
may be interpreted to say "change p or change q or both".
And this can be recognized as just what you need to do if
you happen to find yourself in the center cell and require
a complete and detailed description of ways to escape it.

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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