[Inquiry] Re: Dynamics And Logic

[Jon Awbrey]

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DAL.  Note 24

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Now that we've introduced the field picture for thinking about
propositions and their analytic series, a very pleasing way of
picturing the relationship among a proposition f : X -> B, its
enlargement or shift map Ef : EX -> B, and its difference map
Df : EX -> B can now be drawn.

To illustrate this possibility, let's return to the differential
analysis of the conjunctive proposition f<p, q> = pq, giving the
development a slightly different twist at the appropriate point.

Figure 24-1 shows the proposition pq once again, which we now view
as a scalar field, in effect, a potential "plateau" of elevation 1
over the shaded region, with an elevation of 0 everywhere else.

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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` X ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `o-------------------o` `o-------------------o` ` ` ` ` ` |
| ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` |
| ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` `\` ` ` ` ` |
| ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` /%\ ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` |
| ` ` ` `/` ` ` ` ` ` ` ` ` ` ` `/%%%\` ` ` ` ` ` ` ` ` ` ` `\` ` ` ` |
| ` ` ` / ` ` ` ` ` ` ` ` ` ` ` /%%%%%\ ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` |
| ` ` `/` ` ` ` ` ` ` ` ` ` ` `/%%%%%%%\` ` ` ` ` ` `` ` ` ` ` \` ` ` |
| ` ` / ` ` ` ` ` ` ` ` ` ` ` /%%%%%%%%%\ ` ` ` ` ` ` ` ` ` ` ` \ ` ` |
| ` `o` ` ` ` ` ` ` ` ` ` ` `o%%%%%%%%%%%o` ` ` ` ` ` ` ` ` ` ` `o` ` |
| ` `|` ` ` ` ` ` ` ` ` ` ` `|%%%%%%%%%%%|` ` ` ` ` ` ` ` ` ` ` `|` ` |
| ` `|` ` ` ` ` ` ` ` ` ` ` `|%%%%%%%%%%%|` ` ` ` ` ` ` ` ` ` ` `|` ` |
| ` `|` ` ` ` ` ` ` ` ` ` ` `|%%%%%%%%%%%|` ` ` ` ` ` ` ` ` ` ` `|` ` |
| ` `|` ` ` ` ` P ` ` ` ` ` `|%%%%%%%%%%%|` ` ` ` ` ` Q ` ` ` ` `|` ` |
| ` `|` ` ` ` ` ` ` ` ` ` ` `|%%%%%%%%%%%| ` ` ` ` ` ` ` ` ` ` ` |` ` |
| ` `|` ` ` ` ` ` ` ` ` ` ` `|%%%%%%%%%%%| ` ` ` ` ` ` ` ` ` ` ` |` ` |
| ` `|` ` ` ` ` ` ` ` ` ` ` `|%%%%%%%%%%%| ` ` ` ` ` ` ` ` ` ` ` |` ` |
| ` `o` ` ` ` ` ` ` ` ` ` ` `o%%%%%%%%%%%o ` ` ` ` ` ` ` ` ` ` ` o` ` |
| ` ` \ ` ` ` ` ` ` ` ` ` ` ` \%%%%%%%%%/` ` ` ` ` ` ` ` ` ` ` `/ ` ` |
| ` ` `\ ` ` ` ` ` ` ` ` ` ` ` \%%%%%%%/ ` ` ` ` ` ` ` ` ` ` ` /` ` ` |
| ` ` ` \` ` ` ` ` ` ` ` ` ` ` `\%%%%%/` ` ` ` ` ` ` ` ` ` ` `/ ` ` ` |
| ` ` ` `\ ` ` ` ` ` ` ` ` ` ` ` \%%%/ ` ` ` ` ` ` ` ` ` ` ` /` ` ` ` |
| ` ` ` ` \` ` ` ` ` ` ` ` ` ` ` `\%/` ` ` ` ` ` ` ` ` ` ` `/ ` ` ` ` |
| ` ` ` ` `\ ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` /` ` ` ` ` |
| ` ` ` ` ` \` ` ` ` ` ` ` ` ` ` `/`\` ` ` ` ` ` ` ` ` ` `/ ` ` ` ` ` |
| ` ` ` ` ` `o-------------------o ` o-------------------o` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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Figure 24-1.  Proposition pq : X -> B

Given any proposition f : X -> B, the "tacit extension" of f to EX
is notated !e!f : EX -> B and defined by the equation !e!f = f, so
it's really just the same proposition living in a bigger universe.

Tacit extensions formalize the intuitive idea that a new function
is related to an old function in such a way that it obeys the same
constraints on the old variables, with a "don't care" condition on
the new variables.

Figure 24-2 illustrates the "tacit extension" of the proposition
or scalar field f = pq : X -> B to give the extended proposition
or differential field that we notate as !e!f = !e![pq] : EX -> B.

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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` X ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `o-------------------o` `o-------------------o` ` ` ` ` ` |
| ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` |
| ` ` ` ` `/` P ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` Q `\` ` ` ` ` |
| ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` /`\ ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` |
| ` ` ` `/` ` ` ` ` ` ` ` ` ` ` `/```\` ` ` ` ` ` ` ` ` ` ` `\` ` ` ` |
| ` ` ` / ` ` ` ` ` ` ` ` ` ` ` /`````\ ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` |
| ` ` `/` ` ` ` ` ` ` ` ` ` ` `/```````\` ` ` ` ` ` `` ` ` ` ` \` ` ` |
| ` ` / ` ` ` ` ` ` ` ` ` ` ` /`````````\ ` ` ` ` ` ` ` ` ` ` ` \ ` ` |
| ` `o` ` ` ` ` ` ` ` ` ` ` `o`(dp)`(dq)`o` ` ` ` ` ` ` ` ` ` ` `o` ` |
| ` `|` ` ` ` ` ` ` ` ` ` ` `|``o-->--o``|` ` ` ` ` ` ` ` ` ` ` `|` ` |
| ` `|` ` ` ` ` ` ` ` ` ` ` `|```\```/```|` ` ` ` ` ` ` ` ` ` ` `|` ` |
| ` `|` ` ` ` ` ` `(dp) dq` `|``` \`/````|` `dp (dq)` ` ` ` ` ` `|` ` |
| ` `|` ` ` ` ` o<-----------------o----------------->o ` ` ` ` `|` ` |
| ` `|` ` ` ` ` ` ` ` ` ` ` `|`````|`````|` ` ` ` ` ` ` ` ` ` ` `|` ` |
| ` `|` ` ` ` ` ` ` ` ` ` ` `|`````|`````|` ` ` ` ` ` ` ` ` ` ` `|` ` |
| ` `|` ` ` ` ` ` ` ` ` ` ` `|`````|`````|` ` ` ` ` ` ` ` ` ` ` `|` ` |
| ` `o` ` ` ` ` ` ` ` ` ` ` `o`````|`````o` ` ` ` ` ` ` ` ` ` ` `o` ` |
| ` ` \ ` ` ` ` ` ` ` ` ` ` ` \````|````/ ` ` ` ` ` ` ` ` ` ` ` / ` ` |
| ` ` `\` ` ` ` ` ` ` ` ` ` ` `\```|```/` ` ` ` ` ` ` ` ` ` ` `/` ` ` |
| ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` \``|``/ ` ` ` ` ` ` ` ` ` ` ` / ` ` ` |
| ` ` ` `\` ` ` ` ` ` ` ` ` ` ` `\`|`/` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` |
| ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` \|/ ` ` ` ` ` ` ` ` ` ` ` / ` ` ` ` |
| ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` |
| ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` /|\ ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` |
| ` ` ` ` ` `o-------------------o | o-------------------o` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` dp | dq ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `v` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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Figure 24-2.  Tacit Extension !e![pq] : EX -> B

Thus we have a pictorial way of visualizing the following data:

   !e![pq]

    =

    p q . dp dq

    +

    p q . dp (dq)

    +

    p q . (dp) dq

    +

    p q . (dp)(dq)

Jon Awbrey

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