[Inquiry] Re: Dynamics And Logic

[Jon Awbrey]

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DAL.  Note 5

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We have been studying the action of the difference operator D,
also known as the "localization operator", on the proposition
f : !P! x !Q! -> B that is commonly called the conjunction pq.
We categorized Df as a (first order) differential proposition,
a proposition of the type Df : !P! x !Q! x d!P! x d!Q! -> B.

Abstracting from the augmented venn diagram that shows how the
models or the satisfying interpretations of Df distribute over
the (first order) extended space EX = !P! x !Q! x d!P! x d!Q!,
we can represent Df in the form of a digraph or directed graph,
one whose points are labeled with the elements of X = !P! x !Q!
and whose arcs are labeled with the elements of dX = d!P! x d!Q!.

o-------------------------------------------------o
| `f = ` ` ` ` ` ` ` ` `p q ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| Df =` ` ` ` ` ` ` p `q` ((dp)(dq))` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` + ` ` ` p (q) `(dp) dq` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` + ` ` `(p) q` ` dp (dq) ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` + ` ` `(p)(q) ` dp` dq` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` p q ` ` ` ` ` ` ` ` ` ` ` |
| `p (q) o<------------->o<------------->o (p) q` |
| ` ` ` ` ` ` (dp) dq ` `^` ` dp (dq) ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` dp | dq ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `v` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` (p) (q) ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

Any proposition worth its salt has many equivalent ways to view it,
any one of which may reveal some unsuspected aspect of its meaning.
We will encounter more and more of these variant readings as we go.

Jon Awbrey

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