[Inquiry] Re: Dynamics And Logic

[Jon Awbrey]

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DAL.  Note 22

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It would be good to summarize, in rough but intuitive terms,
the outlook on differential logic that we have reached so far.

We've been considering a class of operators on universes
of discourse, each of which takes us from considering one
universe of discourse, X%, to considering a larger universe
of discourse, EX%.

Each of these operators, in broad terms having the form
W : X% -> EX%, acts on each proposition f : X -> B of the
source universe X% to produce a proposition Wf : EX -> B
of the target universe EX%.

The two main operators that we have worked with up to this
point are the enlargement or shift operator E : X% -> EX%
and the difference operator D : X% -> EX%.

E and D take a proposition in X%, that is, a proposition f : X -> B
that is said to be "about" the subject matter of X, and produce the
extended propositions Ef, Df : EX -> B, which may be interpreted as
being about specified collections of changes that might occur in X.

Here we have need of visual representations,
some array of concrete pictures to anchor our
more earthy intuitions and to help us keep our
wits about us before we try to climb any higher
into the ever more rarefied air of abstractions.

One good picture comes to us by way of the "field" concept.
Given a space X, a "field" of a specified type Y over X is
formed by assigning to each point of X an object of type Y.
If that sounds like the same thing as a function from X to
the space of things of type Y, it is, but it does seems to
help to vary the mental pictures and the figures of speech
that naturally spring to mind within these kinds of fields.

In the field picture, a proposition f : X -> B becomes
a "scalar" field, that is, a field of values in B, or
a "field of model indications" (FOMI).

Let us take a moment to view an old proposition
in this new light, for example, the conjunction
pq : X -> B that is depicted in Figure 22-a.

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `o-------------o` `o-------------o` ` ` ` |
| ` ` ` / ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` \ ` ` ` |
| ` ` `/` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` `\` ` ` |
| ` ` / ` ` ` ` ` ` ` ` /%\ ` ` ` ` ` ` ` ` \ ` ` |
| ` `/` ` ` ` ` ` ` ` `/%%%\` ` ` ` ` ` ` ` `\` ` |
| ` o ` ` ` ` ` ` ` ` o%%%%%o ` ` ` ` ` ` ` ` o ` |
| ` | ` ` ` ` ` ` ` ` |%%%%%| ` ` ` ` ` ` ` ` | ` |
| ` | ` ` ` `P` ` ` ` |%%%%%| ` ` ` `Q` ` ` ` | ` |
| ` | ` ` ` ` ` ` ` ` |%%%%%| ` ` ` ` ` ` ` ` | ` |
| ` o ` ` ` ` ` ` ` ` o%%%%%o ` ` ` ` ` ` ` ` o ` |
| ` `\` ` ` ` ` ` ` ` `\%%%/` ` ` ` ` ` ` ` `/` ` |
| ` ` \ ` ` ` ` ` ` ` ` \%/ ` ` ` ` ` ` ` ` / ` ` |
| ` ` `\` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` `/` ` ` |
| ` ` ` \ ` ` ` ` ` ` ` / \ ` ` ` ` ` ` ` / ` ` ` |
| ` ` ` `o-------------o` `o-------------o` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| `f =` ` ` ` ` ` ` ` ` p q ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
Figure 22-a.  Conjunction pq : X -> B

Each of the operators E, D : X% -> EX% takes us from considering
propositions f : X -> B, here viewed as "scalar fields" over X,
to considering the corresponding "differential fields" over X,
analogous to what are usually called "vector fields" over X.

The structure of these differential fields can be described this way.
To each point of X there is attached an object of the following type:
a proposition about changes in X, that is, a proposition g : dX -> B.
In this frame, if X% is the universe that is generated by the set of
coordinate propositions {p, q}, then dX% is the differential universe
that is generated by the set of differential propositions {dp, dq}.
These differential propositions may be interpreted as indicating
"change in p" and "change in q", respectively.

A differential operator W, of the first order sort that we have
been considering, takes a proposition f : X -> B and gives back
a differential proposition Wf: EX -> B.

In the field view, we see the proposition f : X -> B as a scalar field
and we see the differential proposition Wf: EX -> B as a vector field,
specifically, a field of propositions about contemplated changes in X.

The field of changes produced by E on pq is shown in Figure 22-b.

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `o-------------o` `o-------------o` ` ` ` |
| ` ` ` / ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` \ ` ` ` |
| ` ` `/` ` ` ` P ` ` ` `o` ` ` ` Q ` ` ` `\` ` ` |
| ` ` / ` ` ` ` ` ` ` ` /%\ ` ` ` ` ` ` ` ` \ ` ` |
| ` `/` ` ` ` ` ` ` ` `/%%%\` ` ` ` ` ` ` ` `\` ` |
| ` o ` ` ` ` ` ` ` ` o.->-.o ` ` ` ` ` ` ` ` o ` |
| ` | ` `p(q)(dp)dq ` |%\%/%| `(p)q dp(dq)` ` | ` |
| ` | o---------------|->o<-|---------------o | ` |
| ` | ` ` ` ` ` ` ` ` |%%^%%| ` ` ` ` ` ` ` ` | ` |
| ` o ` ` ` ` ` ` ` ` o%%|%%o ` ` ` ` ` ` ` ` o ` |
| ` `\` ` ` ` ` ` ` ` `\%|%/` ` ` ` ` ` ` ` `/` ` |
| ` ` \ ` ` ` ` ` ` ` ` \|/ ` ` ` ` ` ` ` ` / ` ` |
| ` ` `\` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` `/` ` ` |
| ` ` ` \ ` ` ` ` ` ` ` /|\ ` ` ` ` ` ` ` / ` ` ` |
| ` ` ` `o-------------o | o-------------o` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `(p)(q) dp dq ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| `f =` ` ` ` ` ` ` ` ` p q ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| Ef =` ` ` ` ` ` ` p `q` `(dp)(dq) ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` + ` ` ` p (q) `(dp) dq` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` + ` ` `(p) q` ` dp (dq) ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` + ` ` `(p)(q) ` dp` dq` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
Figure 22-b.  Enlargement E[pq] : EX -> B

The differential field E[pq] specifies the changes
that need to be made from each point of X in order
to reach one of the models of the proposition pq,
that is, in order to satisfy the proposition pq.

The field of changes produced by D on pq is shown in Figure 22-c.

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `o-------------o` `o-------------o` ` ` ` |
| ` ` ` / ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` \ ` ` ` |
| ` ` `/` ` ` ` P ` ` ` `o` ` ` ` Q ` ` ` `\` ` ` |
| ` ` / ` ` ` ` ` ` ` ` /%\ ` ` ` ` ` ` ` ` \ ` ` |
| ` `/` ` ` ` ` ` ` ` `/%%%\` ` ` ` ` ` ` ` `\` ` |
| ` o ` ` ` ` ` ` ` ` o%%%%%o ` ` ` ` ` ` ` ` o ` |
| ` | ` ` ` (dp)dq` ` |%%%%%| ` `dp(dq) ` ` ` | ` |
| ` | o<--------------|->o<-|-------------->o | ` |
| ` | ` ` ` ` ` ` ` ` |%%^%%| ` ` ` ` ` ` ` ` | ` |
| ` o ` ` ` ` ` ` ` ` o%%|%%o ` ` ` ` ` ` ` ` o ` |
| ` `\` ` ` ` ` ` ` ` `\%|%/` ` ` ` ` ` ` ` `/` ` |
| ` ` \ ` ` ` ` ` ` ` ` \|/ ` ` ` ` ` ` ` ` / ` ` |
| ` ` `\` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` `/` ` ` |
| ` ` ` \ ` ` ` ` ` ` ` /|\ ` ` ` ` ` ` ` / ` ` ` |
| ` ` ` `o-------------o | o-------------o` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `v` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `dp dq` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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
Figure 22-c.  Difference D[pq] : EX -> B

The differential field D[pq] specifies the changes
that need to be made from each point of X in order
to feel a change in the felt value of the field pq.

Jon Awbrey

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