[Inquiry] Re: Differential Logic
Jon Awbrey
jawbrey at oakland.edu
Tue May 13 14:48:07 CDT 2003
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
DLOG. Note D20
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Example 1. A Square Rigging
| Urge and urge and urge,
| Always the procreant urge of the world.
|
| Walt Whitman, 'Leaves of Grass', [Whi, 28]
By way of example, suppose that we are given the initial condition A = dA and
the law d^2.A = (A). Then, since "A = dA" <=> "A dA or (A)(dA)", we may infer
two possible trajectories, as displayed in Table 11. In either of these cases,
the state A (dA)(d^2.A) is a stable attractor or a terminal condition for both
starting points.
Table 11. A Pair of Commodious Trajectories
o---------o-------------------o-------------------o
| Time | Trajectory 1 | Trajectory 2 |
o---------o-------------------o-------------------o
| | | |
| 0 | A dA (d^2.A) | (A) (dA) d^2.A |
| | | |
| 1 | (A) dA d^2.A | (A) dA d^2.A |
| | | |
| 2 | A (dA) (d^2.A) | A (dA) (d^2.A) |
| | | |
| 3 | A (dA) (d^2.A) | A (dA) (d^2.A) |
| | | |
| 4 | " " " | " " " |
| | | |
o---------o-------------------o-------------------o
Because the initial space X = <|A|> is one-dimensional, we can easily fit
the second order extension E^2.X = <|A, dA, d^2.A|> within the compass of
a single venn diagram, charting the couple of converging trajectories as
shown in Figure 12.
o-------------------------------------------------o
| E^2.X |
| |
| o-------------o |
| / \ |
| / A \ |
| / \ |
| / ->- \ |
| o / \ o |
| | \ / | |
| | -o- | |
| | ^ | |
| o---o---------o | o---------o---o |
| / \ \|/ / \ |
| / \ o | / \ |
| / \ | /|\ / \ |
| / \ | / | \ / \ |
| o o-|-o--|--o---o o |
| | | | | | | |
| | ---->o<----o | |
| | | | | |
| o dA o o d^2.A o |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| o-------------o o-------------o |
| |
| |
o-------------------------------------------------o
Figure 12. The Anchor
If we eliminate from view the regions of E^2.X that are ruled out
by the dynamic law d^2.A = (A), then what remains is the quotient
structure that is shown in Figure 13. This picture makes it easy
to see that the dynamically allowable portion of the universe is
partitioned between the properties A and d^2.A. As it happens,
this fact might have been expressed "right off the bat" by an
equivalent formulation of the differential law, one that uses
the exclusive disjunction to state the law as (A, d^2.A).
o-------------------------------------------------o
| |
| ->- |
| / \ |
| \ / |
| o-------------o -o- |
| / \ ^ |
| / dA \/ A |
| / /\ |
| / / \ |
| o o / o |
| | \ / | |
| | \ / | |
o------------|-------\-------/-------|------------o
| | \ / | |
| | \ / | |
| o v / o |
| \ o / |
| \ ^ / |
| \ | / d^2.A |
| \ | / |
| o------|------o |
| | |
| | |
| o |
| |
o-------------------------------------------------o
Figure 13. The Tiller
What we have achieved in this example is to give a differential description of
a simple dynamic process. In effect, we did this by embedding a directed graph,
which can be taken to represent the state transitions of a finite automaton, in
a dynamically allotted quotient structure that is created from a boolean lattice
or an n-cube by nullifying all of the regions that the dynamics outlaws. With
growth in the dimensions of our contemplated universes, it becomes essential,
both for human comprehension and for computer implementation, that the dynamic
structures of interest to us be represented not actually, by acquaintance, but
virtually, by description. In our present study, we are using the language of
propositional calculus to express the relevant descriptions, and to comprehend
the structure that is implicit in the subsets of a n-cube without necessarily
being forced to actualize all of its points.
One of the reasons for engaging in this kind of extremely reduced, but explicitly
controlled case study is to throw light on the general study of languages, formal
and natural, in their full array of syntactic, semantic, and pragmatic aspects.
Propositional calculus is one of the last points of departure where we can view
these three aspects interacting in a non-trivial way without being immediately
and totally overwhelmed by the complexity they generate. Often this complexity
causes investigators of formal and natural languages to adopt the strategy of
focusing on a single aspect and to abandon all hope of understanding the whole,
whether it's the still living natural language or the dynamics of inquiry that
lies crystallized in formal logic.
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