ONT Re: Differential Logic
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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.
From the perspective that I find most useful here, a language is a syntactic
system that is designed or evolved in part to express a set of descriptions.
When the explicit symbols of a language have extensions in its object world
that are actually infinite, or when the implicit categories and generative
devices of a linguistic theory have extensions in its subject matter that
are potentially infinite, then the finite characters of terms, statements,
arguments, grammars, logics, and rhetorics force an excess of intension
to reside in all these symbols and functions, across the spectrum from
the object language to the metalinguistic uses. In the aphorism from
W. von Humboldt that Chomsky often cites, for example, in [Cho86, 30]
and [Cho93, 49], language requires "the infinite use of finite means".
This is necessarily true when the extensions are infinite, when the
referential symbols and grammatical categories of a language possess
infinite sets of models and instances. But it also voices a practical
truth when the extensions, though finite at every stage, tend to grow
at exponential rates.
This consequence of dealing with extensions that are "practically infinite"
becomes crucial when one tries to build neural network systems that learn,
since the learning competence of any intelligent system is limited to the
objects and domains that it is able to represent. If we want to design
systems that operate intelligently with the full deck of propositions
dealt by intact universes of discourse, then we must supply them with
succinct representations and efficient transformations in this domain.
Furthermore, in the project of constructing inquiry driven systems,
we find ourselves forced to contemplate the level of generality
that is embodied in propositions, because the dynamic evolution
of these systems is driven by the measurable discrepancies that
occur among their expectations, intentions, and observations,
and because each of these subsystems or components of knowledge
constitutes a propositional modality that can take on the fully
generic character of an empirical summary or an axiomatic theory.
A compression scheme by any other name is a symbolic representation,
and this is what the differential extension of propositional calculus,
through all of its many universes of discourse, is intended to supply.
Why is this particular program of mental calisthenics worth carrying out
in general? By providing a uniform logical medium for describing dynamic
systems we can make the task of understanding complex systems much easier,
both in looking for invariant representations of individual cases and in
finding points of comparison among diverse structures that would otherwise
appear as isolated systems. All of this goes to facilitate the search for
compact knowledge and to adapt what is learned from individual cases to
the general realm.
Jon Awbrey
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