Changes

==Note 6==

==Note 6==

One more example may serve to suggest just how much dynamic complexity can be built on a universe of discourse that has but a single logical feature at its base.

One more example may serve to suggest just how much dynamic

 

complexity can be built on a universe of discourse that has

But first, let me introduce a few more elements of general notation that I'll be using to describe finite dimensional universes of discourse and the qualitative dynamics that we envision occurring in them.

but a single logical feature at its base.

But first, let me introduce a few more elements of general

Let <math>\mathcal{X} = \{ x_1, \ldots, x_n \}</math> be the ''alphabet'' of logical ''features'' or ''variables'' that we use to describe the n-dimensional universe of discourse <math>X^\circ = [\mathcal{X}] = [ x_1, \ldots, x_n ].</math>  Picturesquely viewed, one may think of a venn diagram with n overlapping "circles" that are labeled with the feature names in the set <math>\mathcal{X}.</math>  Staying with this picture, one visualizes the universe of discourse <math>X^\circ = [\mathcal{X}]</math> as having two layers:  (1) the set <math>X = \langle \mathcal{X} \rangle = \langle x_1, \dots, x_n \rangle</math> of ''points'' or ''cells'' &mdash; in another sense of the word than when we speak of ''cellular automata'' &mdash; (2) the set <math>X^\uparrow = (X \to \mathbb{B})</math> of ''propositions'', boolean-valued functions, or maps from <math>X\!</math> to <math>\mathbb{B}.</math>

notation that I'll be using to describe finite dimensional

universes of discourse and the qualitative dynamics that

we envision occurring in them.

Let !X! = {x_1, ..., x_n} be the "alphabet" of logical "features"

Thus, we may speak of the universe of discourse <math>X^\circ</math> as being an ordered pair <math>(X, X^\uparrow),</math> with <math>2^n\!</math> points in the underlying space <math>X\!</math> and <math>2^{2^n}</math> propositions in the function space <math>X^\uparrow.</math>

or "variables" that we use to describe the n-dimensional universe

one may think of a venn diagram with n overlapping "circles" that

this picture, one visualizes the universe of discourse X% = [!X!]

as having two layers:  (1) the set X = <|!X!|> = <|x_1, ..., x_n|>

we speak of "cellular automata" -- (2) the set X^ = (X -> B) of

Thus, we may speak of the universe of discourse X% as being

an ordered pair <X, X^>, with 2^n points in the underlying

space X and 2^(2^n) propositions in the function space X^.

A more complete table setting out these notations can be found here:

A more complete table setting out these notations can be found here:

DLOG D2.  http://stderr.org/pipermail/inquiry/2003-May/000480.html

:* [http://stderr.org/pipermail/inquiry/2003-May/000480.html DLOG D2]

Now, to the Example.

Now, to the Example.

Once again, let us begin with a 1-feature alphabet !X! = {x_1} = {x}.

Once again, let us begin with a 1-feature alphabet !X! = {x_1} = {x}.

In the discussion that follows I will consider a class of trajectories

In the discussion that follows I will consider a class of trajectories