[Inquiry] Re: Theme One Program

[Jon Awbrey]

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TOP.  Expository Note 15

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3.3.  Logical Cacti (concl.)

The foregoing discussion has given a fairly thorough description of
the abstract graphs and the concrete data structures that fall under
the name of "painted and rooted cacti".  As far as their applications
to organizing lexical and literal data, the function of the "paints"
is fairly clear:  They are more or less straightforward references
to the characters and the phrases of a two-level formal language.
This all appears to be about as concrete as anything can be and
quite free of any degree of interpretive play or flexibility.

But there is an aspect of the prefix-sharing strategy that gives
each word in a lexical cactus and each phrase in a literal cactus
a double meaning.  For one thing, it stands for itself, as always;
For another thing, it stands for the family of words or the family
of phrases, respectively, in the two-level formal language that has
that sequence of characters or that sequence of words as its prefix.

For example, consider the two-level language that is given as follows:

L_1 = {"a", "all", "an", "angry", "ape", "apes", "are", "ate",
       "bees", "big", "bold", "bug", "buzz"}

L_2 = {"all bees buzz",
       "all apes are bold",
       "an angry ape ate a big bug"}

We make the following observations about this example:

1.  The prefix class of "a" in L_1, written "[a]L_1",
    or simply "[a]" if the context is understood, is
    the set of all words in L_1 that begin with "a".

2.  The prefix class of "bu" in L_1, written "[bu]L_1",
    or simply "[bu]" if the context is understood, is
    the set of all words in L_1 that begin with "bu",
    namely, the set {"bug", "buzz"}.

3.  The prefix class of "all" in L_2, written "[all]L_2",
    or simply "[all]" when the context is understood, is
    the set of all phrases in L_2 that begin with "all",
    namely, the set {"all bees buzz", "all apes are bold"}.

In general terms, a prefix, whether it belongs to the language or not,
can be used to "stand for", that is, as a proxy, a representative, or
a symbol for, the associated prefix class, which constitutes a subset
of the language in question.

In graphical terms, the path up to a point in a lexical or literal
cactus can be used, under the proper alternative interpretation,
to stand for the whole class of paths in the cactus that run
from the root, through that point, to a syntactic terminus,
in so many ways extending the initial path in question.

The situation that we have just now been looking at
is only a very special case of a much more general
phenomenon, falling under a principle that I will
describe this way:

Information is form before matter.

That is not a definition -- it is only an emphasis.
I am often tempted to express the idea by saying that
information is form, not matter, but the more I reflect
on it the less certain I become that form and matter are
not all one in the end, so the best that I can do for now
is to emphasize what seems fair to stress in the meantime.

One of the consequences of this principle is that all codes are
abstract, formal, and symbolic to some degree, which means that
no code has the power to determine its interpretation perfectly.

I will not attempt to prove this principle -- not knowing how long the
line between form and matter will hold, it would probably be pointless
to try -- I will merely call attention to examples of it as they arise.

The most pressing pertinent example
arises in the case of logical cacti,
so let us now turn to consider that.

Jon Awbrey

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