[Inquiry] Re: Examples! Examples! Examples!

[Jon Awbrey]

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EEE.  Note 19

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Aside from looking out my window every now and then,
I am still staring at this stereoscopic latticework:

o-------------------------------------o-------------------------------------o
|                                     |                                     |
|                 TLC                 |             1 : TLC -> B            |
|                  o                  |                  o                  |
|                 / \                 |                 / \                 |
|                /   \                |                /   \                |
|               /     \               |               /     \               |
|              /       \              |              /       \              |
|             /  2^TLC  \             |             /  TLC^   \             |
|            /           \            |            /           \            |
|           /  Pow (TLC)  \           |           / (TLC -> B)  \           |
|         ...             ...         |         ...             ...         |
|           \   2^(2^25)  /           |           \   2^(2^25)  /           |
|            \  subsets  /            |            \ functions /            |
|             \         /             |             \         /             |
|              \       /              |              \       /              |
|               \     /               |               \     /               |
|                \   /                |                \   /                |
|                 \ /                 |                 \ /                 |
|                  o                  |                  o                  |
|                 { }                 |             0 : TLC -> B            |
|                                     |                                     |
o-------------------------------------o-------------------------------------o
Figure 4.  Subset Lattice Pow(TLC) and Proposition Lattice TLC^ = (TLC -> B)

I see that I need to clear up some points of terminology
that arise through my loose use of elliptical constructs.

For example, when I speak of TLC = <|a_1, ..., A_25|> as
a "model space" and 2^TLC as a "lattice of models", what
I really mean is that the elements of TLC are the models
of each and every proposition in the language L(!TLC!)
that employs the vocabulary !TLC! = {a_1, ..., a_25}.
Of course, I do not mean to say that these elements
are the formal models, much less the natural models,
of the axiom !a!_1 for TLC, displayed in Table 1,
that we are currently contemplating.  In short,
it would be more proper to refer to TLC and
Pow(TLC) as a "space of interpretations"
and a "lattice of interpretations",
respectively.

One issue that needs to be mentioned, and constantly kept in mind,
but not fussed over too much at this early stage of investigation,
is what has generally been called the "theory laden" character of
observation, in this Example, the obvious fact that data as coded
is nothing that merits being called "Nature's Own Language" (NOL),
and if we chance to call it "raw data", no one should dream for a
moment that it's a bit un-pre-processed or unrefined for all that.
The upshot of this reflection is that an indivisual coding scheme,
like !TLC!, is always an abductive construct, and always deployed
subordinate to the total hypothesis that needs be implicated with
any brand of experimental approach to Nature.  As such, languages
and their associated conceptual frameworks are eminently fallible.

Jon Awbrey

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