ONT Re: Apposite Purposes Of Logical Languages Objectified (APOLLO)
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With these logical tools now hanging from our belts
let us take a more experienced look at one of those
factorization diagrams that are of such fundamental
utility in supporting the structure of this subject.
Everyone will recognize the similarity that exists between
the diagram that I drew before to illustrate the operation
of hypostatic abstraction and the diagram that I draw here
to indicate the so-called "factorization" of a proposition
about some world or another.
| F
| X o---------->o B
| \ ^
| \ /
| c \ / f
| \ /
| v /
| o
| B^k
|
| c(x) = <x_1(x), ..., x_k(x)>
In reading this diagram, one thinks of X as being
a "real world" or an "unknown reality" (UR) and F
as forming a logical proposition about this world,
whose logical values, ranging through B = {0, 1},
may vary from point to point across the domain X.
The diagram illustrates the "factoring" of the proposition
F : X -> B through the "medium" or the intermediary of the
k-dimensional binary code space B^k. Through this medium,
F is factored into the functional composition f(c(x)),
where c : X -> B^k is the "coding" of each x in X as
a k-bit vector in B^k, and where f is the mapping
of these codes into a range that one interprets
as true-or-false values in the boolean space
B = {0, 1} = {false, true}.
One way to harmonize the pictures of functional factorization
and hypostatic abstraction might be to reconfigure the latter
as follows:
| Object Formal
| Domain F Values
| X o---------->o B
| \ ^
| \ /
| c \ / f
| \ /
| v /
| o
| B^k
|
| Hypostatic
| Intermediary
There are numerous ways of thinking about their relationship
but I think that this one is worth contemplating for a while.
Jon Awbrey
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