ONT Re: Differential Logic
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DLOG. Note D19
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A One-Dimensional Universe
| There was never any more inception than there is now,
| Nor any more youth or age than there is now;
| And will never be any more perfection than there is now,
| Nor any more heaven or hell than there is now.
|
| Walt Whitman, Leaves of Grass, [Whi, 28]
Let !X! = {x_1} = {A} be an alphabet that represents one boolean variable or
a single logical feature. In this example I am using the capital letter "A"
in a more usual informal way, to name a feature and not a space, at variance
with my formerly stated formal conventions. At any rate, the basis element
A = x_1 may be interpreted as a simple proposition or a coordinate projection
A = x_1 : B^1 -:> B. The space X = <|A|> = {(A), A} of points (cells, vectors,
interpretations) has cardinality 2^n = 2^1 = 2 and is isomorphic to B = {0, 1}.
Moreover, X may be identified with the set of singular propositions {x : B ::> B}.
The space of linear propositions X* = {hom : B ++> B} = {0, A} is algebraically
dual to X and also has cardinality 2. Here, "0" is interpreted as denoting the
constant function 0 : B -> B, amounting to the linear proposition of rank 0,
while A is the linear proposition of rank 1. Last but not least we have the
positive propositions {pos : B oo> B} = {A, 1}, of rank 1 and 0, respectively,
where "1" is understood as denoting the constant function 1 : B -> B. In sum,
there are 2^2^n = 2^2^1 = 4 propositions altogether in the universe of discourse,
comprising the set X^ = {f : X -> B} = {0, (A), A, 1} ~=~ (B -> B).
The first order differential extension of !X! is E!X! = {x_1, dx_1} = {A, dA}.
If the feature "A" is understood as applying to some object or state, then the
feature "dA" may be interpreted as an attribute of the same object or state that
says that it is changing "significantly" with respect to the property A, or that
it has an "escape velocity" with respect to the state A. In practice, differential
features acquire their logical meaning through a class of "temporal inference rules".
For example, relative to a frame of observation that is left implicit for now,
one is permitted to make the following sorts of inference: from the fact that
A and dA are true at a given moment one may infer that (A) will be true in the
next moment of observation. Altogether in the present instance, there is the
fourfold scheme of inference that is shown below:
o-------------------------------------------------o
| |
| From (A) & (dA) infer (A) next. |
| |
| From (A) & dA infer A next. |
| |
| From A & (dA) infer A next. |
| |
| From A & dA infer (A) next. |
| |
o-------------------------------------------------o
It might be thought that we need to bring in an independent time variable
at this point, but an insight of fundamental importance appears to be that
the idea of process is more basic than the notion of time. A time variable
is actually a reference to a "clock", that is, a canonical or a convenient
process that is established or accepted as a standard of measurement, but
in essence no different than any other process. This raises the question
of how different subsystems in a more global process can be brought into
comparison, and what it means for one process to serve the function of
a local standard for others. But these inquiries only wrap up puzzles
in further riddles, and are obviously too involved to be handled at
our current level of approximation.
| The clock indicates the moment . . . . but what does
| eternity indicate?
|
| Walt Whitman, 'Leaves of Grass', [Whi, 79]
Observe that the secular inference rules, used by themselves,
involve a loss of information, since nothing in them can tell
us whether the momenta {(dA), dA} are preserved or changed in
the next instance. In order to know this, we would have to
determine d^2.A, and so on, pursuing an infinite regress.
Ultimately, in order to rest with a finitely determinate
system, it is necessary to make an infinite assumption,
for example, that d^k.A = 0 for all k greater than some
fixed value M. Another way to escape the regress is
through the provision of a dynamic law, in typical
form making higher order differentials dependent
on lower degrees and estates.
Jon Awbrey
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