[Inquiry] Re: Simple Meanings In Limnal Expressions -- Commentary

[Jon Awbrey]

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SMILE.  Commentary Note 6

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Re: SMILE 4.  http://stderr.org/pipermail/inquiry/2005-November/003166.html
In: SMILE.    http://stderr.org/pipermail/inquiry/2005-November/thread.html#3166

We continue to examine the breed of algebraic inquiry,
vaguely suggestive of an inverse problem, that Peirce
described in the following terms:

| Again, given certain data concerning 'x',
| we may ask, what else needs to be known in
| order to compel 'x' to be !v! or to be !f!.
|
| C.S. Peirce, 'Collected Papers', CP 4.250

Filling in the missing pieces of the analogy brings us to this problem:
Required, to re-interpret 'x' as a function of the form 'x' : M -> !B!,
where M is a functional domain that will support the interpretation of
phrases like "certain data concerning 'x'" and "what else needs to be
known about 'x'" in a more objective and systematic way, that is, in
a less impressionistic and meta-linguistic way than thus expressed.

Last time we contemplated a couple of standard approaches to the problem,
where 'x' is reconstrued as a function on a universe or a set of thereof.

With a little imagination, it's possible conceive of a suitable space of
possible "states of information" (SOI's) that we might find ourselves in.
In this setting, "suitable" means that we must needs bound the SOI-space
to the SOI's that we actually need to think about for a given discussion,
to avoid being overmastered by the usual welter of boundless possibility.

Let !SOI! be that suitable space of information states.
Then we may view 'x' under either one of the following
informationally and logically equivalent descriptions:

   1.  'x' is a relation of the form 'x' c !SOI! x !B!.

   2.  'x' is a function of the form 'x' : !SOI! -> Pow(!B!).

Here, Pow(!B!) is the "power set" of !B!, that is, the set of subsets of !B!,
specifically, Pow(!B!) has these four elements, {}, {!f!}, {!v!}, {!f!, !v!},
or, the empty set, a singleton of !f!, a singleton of !v!, and the total set.

In this reading of the matter, we think of the "data or knowledge" (DOK) that
we possess with regard to 'x' in a particular context and moment of discussion
as being embodied or stored in a particular state of information m in M = !SOI!.

In the first view of the situation, we have the relation 'x' c M x !B!,
and then we are given a particular state of information m in M, all of
which conspires to determine the subset of !B! that is compatible with
the relation 'x' and the DOK m.

In the second view of the situation, which amounts to a mere phrase transition,
we have the function 'x' : M -> Pow(!B!), and then we are given the infomation
m in M, all of which combines to determine the element of Pow(!B!) that is the
value of the function 'x' on the argument m.

To Be Continued ...

Jon Awbrey

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