ONT Re: Differential Logic & Dynamic Systems
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Dif Log Dyn Sys Group --
I think that we finally have enough of the preliminary
set-ups and warm-ups out of the way that we can begin
to tackle the differential analysis proper of our
sample proposition q = (( u v )( u w )( v w )).
When X is the type of space that is generated by {u, v, w},
let dX be the type of space that is generated by (du, dv, dw},
and let X x dX be the type of space that is generated by the
extended set of boolean basis elements {u, v, w, du, dv, dw}.
For convenience, define a notation "EX" so that EX = X x dX.
Even though the differential variables are in some abstract
sense no different than other boolean variables, it usually
helps to mark their distinctive roles and their differential
interpretation by means of the distinguishing domain name "dB".
Using these designations of logical spaces, the propositions
over them can be assigned both abstract and concrete types.
For instance, consider the proposition q<u, v, w>, as before,
and then consider its tacit extension q<u, v, w, du, dv, dw>,
the latter of which may be indicated more explicitly as "eq".
1. Proposition q is abstractly typed as q : B^3 -> B.
Proposition q is concretely typed as q : X -> B.
2. Proposition eq is abstractly typed as eq : B^3 x dB^3 -> B.
Proposition eq is concretely typed as eq : X x dX -> B.
Succinctly, eq : EX -> B.
We now return to our consideration of the effects
of various differential operators on propositions.
This time around we have enough exact terminology
that we shall be able to explain what is actually
going on here in a rather more articulate fashion.
The first transformation of the source proposition q that we may
wish to stop and examine, though it is not unusual to skip right
over this stage of analysis, frequently regarding it as a purely
intermediary stage, holding scarcely even so much as the passing
interest, is the work of the "enlargement" or "shift" operator E.
Applying the operator E to the operand proposition q yields:
| u v u w v w
| o o o o o o
| / \ / \ / \ / \ / \ / \
| o---o---o o---o---o o---o---o
| du \ dv du | dw dv / dw
| \ | /
| \ | /
| \ | /
| \ | /
| \ | /
| \ | /
| \ | /
| \|/
| o
| |
| |
| |
| Eq = @
|
| (( ( u , du )( v , dv )
| )( ( u , du )( w , dw )
| )( ( v , dv )( w , dw )
| ))
The enlarged proposition Eq is a minimally interpretable as
as a function on the six variables of {u, v, w, du, dv, dw}.
In other words, Eq : EX -> B, or Eq : X x dX -> B.
Conjoining a query on the center cell, c = uvw, yields:
| u v u w v w
| o o o o o o
| / \ / \ / \ / \ / \ / \
| o---o---o o---o---o o---o---o
| du \ dv du | dw dv / dw
| \ | /
| \ | /
| \ | /
| \ | /
| \ | /
| \ | /
| \ | /
| \|/
| o
| |
| |
| |
| Eq c = @ u v w
|
| (( ( u , du )( v , dv )
| )( ( u , du )( w , dw )
| )( ( v , dv )( w , dw )
| ))
|
| u v w
The models of this last expression tell us which combinations of
feature changes among the set {du, dv, dw} will take us from our
present interpretation, the center cell expressed by "u v w", to
a true value under the given proposition (( u v )( u w )( v w )).
The models of Eq.c can be described in the usual ways as follows:
1. The points of the space EX with the coordinate descriptions:
<u, v, w, du, dv, dw> =
<1, 1, 1, 0, 0, 0>,
<1, 1, 1, 0, 0, 1>,
<1, 1, 1, 0, 1, 0>,
<1, 1, 1, 1, 0, 0>.
2. The points of the space EX with the conjunctive expressions:
u v w (du)(dv)(dw),
u v w (du)(dv) dw ,
u v w (du) dv (dw),
u v w du (dv)(dw).
In summary, Eq.c informs us that we can get from c to a model of q by
making the following changes in our position with respect to u, v, w,
to wit: change none or just one among u, v, w.
I think that it would be worth our time to diagram the models
of the "enlarged" or "shifted" proposition, Eq, at least, the
selection of them that we find issuing from the center cell c.
Figure 3 is an extended venn diagram for the proposition Eq.c,
where the shaded area gives the models of q and the stars "*"
mark the terminal points of the requisite feature alterations.
o-----------------------------------o
| X |
| o-----------------------o |
| | U | |
| | o o | |
| | /`\ /`\ | |
| | /```\ /```\ | |
| | /`````.`````\ | |
| | /`````/`\`````\ | |
| | /``*-dw-*-dv-*``\ | |
| | /`````/``|``\`````\ | |
| o----------du-----------o |
| / /````|````\ \ |
| o o`````*`````o o |
| \ \`````````/ / |
| \ V \```````/ W / |
| \ \`````/ / |
| \ \```/ / |
| \ \`/ / |
| \ . / |
| \ / \ / |
| \ / \ / |
| o o |
| |
o-----------------------------------o
Figure 3. Extended Venn Diagram For Eq.c
| Notational Intermezzo
|
| I am using the scrip-device "$A$" for "script A".
|
| One more piece of notation will save us a few bytes
| in the length of many of our schematic formulations:
|
| Let $X$ = {x_1, ..., x_k} be a finite class of variables --
| whose names I list, according to the usual custom, without
| what seems to my semiotic consciousness like the necessary
| quotation marks around their particular characters, though
| not without not a little trepidation, or without a worried
| cognizance that I may be obligated to reinsert them all to
| their rightful places at a subsequent stage of development --
| with regard to which we may now define the following items:
|
| The "(first order) differential alphabet",
|
| d$X$ = {dx_1, ..., dx_k}.
|
| The "(first order) extended alphabet",
|
| E$X$ = $X$ |_| d$X$ = {{x_1, ..., x_k, dx_1, ..., dx_k}.
Before we continue with the differential analysis
of the source proposition q, we need to pause and
take another look at just how it shapes up in the
light of the extended universe EX, in other words,
to examine in utter detail its tacit extension eq.
The models of eq in EX can be comprehended as follows:
1. Working in the "summary coefficient" form of representation,
if the coordinate list x is a model of q in X, then one can
construct a coordinate list ex as a model for eq in EX just
by appending any combination of values for the differential
variables in d$X$.
For example, to focus once again on the center cell c,
which happens to be a model of the proposition q in X,
one can extend c in eight different ways into EX, and
thus get eight models of the tacit extension eq in EX.
Though it may seem an utter triviality to write these
out, I will do it for the sake of seeing the patterns.
The models of eq in EX that are tacit extensions of c:
<u, v, w, du, dv, dw> =
<1, 1, 1, 0, 0, 0>,
<1, 1, 1, 0, 0, 1>,
<1, 1, 1, 0, 1, 0>,
<1, 1, 1, 0, 1, 1>,
<1, 1, 1, 1, 0, 0>,
<1, 1, 1, 1, 0, 1>,
<1, 1, 1, 1, 1, 0>,
<1, 1, 1, 1, 1, 1>.
2. Working in the "conjunctive product" form of representation,
if the conjunct symbol x is a model of q in X, then one can
construct a conjunct symbol ex as a model for eq in EX just
by appending any combination of values for the differential
variables in d$X$.
The models of eq in EX that are tacit extensions of c:
u v w (du)(dv)(dw),
u v w (du)(dv) dw ,
u v w (du) dv (dw),
u v w (du) dv dw ,
u v w du (dv)(dw),
u v w du (dv) dw ,
u v w du dv (dw),
u v w du dv dw .
In short, eq.c just enumerates all of the possible changes in EX
that "derive from", "issue from", or "stem from" the cell c in X.
Okay, that was pretty tedious, and I know that it all appears
to be totally trivial, which is precisely why we usually just
leave it "tacit" in the first place, but hard experience, and
a real acquaintance with the confusion that can beset us when
we do not render these implicit grounds explicit, have taught
me that it will ultimately be necessary to get clear about it.
Jon Awbrey
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