[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 mean is that the elements of
TLC collectively comprise the models of the various propositions in the
language L(!TLC!) that employs the vocabulary !TLC! = {a_1, ..., a_25}.
Of course, I do not mean to imply these elements are the formal models,
much less the natural models, of the axiom !a!_1 for TLC that we are
currently contemplating, as it was displayed in Table 1.  In short,
it would probably be more precise 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
are nothing that merit 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 individual 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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