SUO: Re: Examples! Examples! Examples!
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EEE. Note 5
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Generalities: Informing the Media
There are a few dimensions or distinctions
of a practical nature that we need to keep
an eye out for as we work through this set
of examples. I will discuss these issues
here in no particular order.
1. Casual/Formal
There is the distinction between the "casual" and the "formal",
in other words, the line that we draw somewhere on a spectrum
that stretches from the haunts of the informal, the partially
formalized, and the possibly even never wholly formalizable,
into the regions of the completely formalized formal object
of all-around reflection and enough-in-depth contemplation.
Of course, we should not expect to formalize this boundary --
it's a slippery slope in both directions -- but it's still
a good idea to maintain a rough sense of orientation with
respect to it.
A useful guideline is provided by keeping track of the
number of elements in a given space. Spaces that have
infinite or very large cardinalities are things that
we may hope to denote or to mention in formal terms,
even if many times we only wave our hands at them,
but we do not come close to touching them the way
that we can manage and manipulate finitary signs.
For example, the numbers of positions and propositions in the
TLC example are 2^25 = 33,554,432 and 2^(2^25) = 2^(33,554,432),
respectively. A person might just consider making a truth table
with 33 million rows of binary vectors, but is not likely to finish
filling it in with all of the possible columns of boolean functions.
3. Mention/Use
Let me adopt the familiar distinction between mention and use
and use it in a way that has pragmatic computational imports.
In the sense that I have in mind, we use the finitary signs
that are within our grasp to mention the more intangible
objects that are within our reach only in a mediate or
a virtual sense.
4. Object/Sign
That brings up the distinction between objects and signs, which is
another one of those overlapping waves on the shoreline types of
distinctions, since some things can be signs one moment and then
objects the next, for instance, as we reflect on them and even
formalize them to some degree of adequacy, while other things,
though mentionable, are hardly a bit more than that, leaving
us in the waving hands sort of usage with regard to them.
5. Generator/Relation
In algebra, generally speaking, things that increase the number
of elements in a space are called "generators" while things that
decrease the number of elements in a space are called "relations",
with a specific technical sense that is usually understood in situ,
but if it isn't, then the word "relator" may be used in its place.
A specification of an algebraic structure in terms of generators
and relations is called a "presentation", and is customarily
presented as a divided set {g_1, ..., g_m | r_1, ..., r_n},
with two finite sets, of generators g_i and relators r_j.
For example, the TLC alphabet !A! = {a_1, ..., a_25} is
a set of generators that can be thought of as generating
the space of positions A = <|!A!|> = <|a_1, ..., a_25|>,
using brackets of the form "<|...|>" as "generator bars",
and inducing the generation of the space of propositions
A^ = (A -> B) = {f : A -> B}, while the TLC axiom !a! in
Table 1 can be thought of as a relator that shrinks both
of these spaces more effectively than a $120/hr analyst.
However, in the kinds of examples that we are concerned with,
there are actually two sorts of spaces to worry about, which
brings to the forefront the next dimension or distinction.
6. Model/Theory
This is really just a special case of the object/sign distinction,
but it's a good idea to emphasize it once more in this particular
context of use.
In the TLC example, the "model space" is the universe of discourse
A% = [!A!] = [a_1, ..., a_25] that combines the positions of A with
the propositions of A^ = (A -> B).
The "syntactic space" is the formal language L = L(!A!) c (!A! |_| !M!)*
of well-formed strings in the Cactus Language, or pick your own favorite
language for the task, that we may variously describe as "expressions",
"formulas", "sentences", "terms", "wffs", or whatever fits the moment.
A "zeroth order theory" (ZOT) is an arbitrary subset of L.
So we have, just for starters, at least two lattices:
1. The model lattice is (Pow(A), c), ordered by set-theoretic inclusion
or the "contained as a subset" relation that is here denoted by "c".
This is isomorphic to the proposition lattice (A^, =>) that consists
of the propositions f in (A -> B), ordered by the logical implication
relation that is here denoted by "=>".
2. The theory lattice is (Pow(L), c), ordered by inclusion as usual.
But wait, there's more ...
Jon Awbrey
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