[Inquiry] Re: Sign Relations -- Commentary

[Jon Awbrey]

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SR.  Commentary Note 33

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Peirce List,

Correcting a few typos in the last installment, and continuing.

Let's look at one of Peirce's most endearing impressions of
determination, the example of a raindrop falling on a stone.
If reluctantly, it's our duty now to analyze its impression,
and to articulate its inherent nuances in unrecondite terms.

| To determine means to make a circumstance different from what
| it might have been otherwise.  For example, a drop of rain
| falling on a stone determines it to be wet, provided the
| stone may have been dry before.  But if the fact of
| a whole shower half an hour previous is given,
| then one drop does not determine the stone to
| be wet;  for it would be wet, at any rate.
|
| C.S. Peirce, 'Chronological Edition', CE 1, 245-246
|
| Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865),
|'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866',
| Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.
| Cf: DET 13.  http://stderr.org/pipermail/inquiry/2004-December/002209.html
| In: DET.     http://stderr.org/pipermail/inquiry/2004-December/thread.html#2197

In saying that a Rain Drop determines a Dry Stone to become a Wet Stone,
we're not speaking of a single sequence of occurrences merely, but of a
system, the Stone, that has distinct states, Dry and Wet, and that will
undergo definite changes of state under the action or influence of some
some kind of agency, in this case a Rain Drop, more accurately observed,
a mere sample from a more general system of activity, a Rain Storm, say,
and which in fact determines a "transformation" of the "phase space" of
the Stone into itself, that is, a function from the domain of states to
itself, having the form Rain : {Dry, Wet} -> {Dry, Wet}, as shown below.

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` Dry Stone ` O ` ` ` O ` Wet Stone ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` `\` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` \ ` ` | ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` `\` ` | ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` Rain`Drop ` ` ` v ` v ` Rain Drop ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `\` | ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` \ | ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `\| ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` Dry Stone ` ` ` ` ` O ` Wet Stone ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

Thus Peirce's Parable Of The Rain Drop quite well illustrates
the minimal features implied in talking about "determination".

This much illustration will do as a first rough sketch of the
Weather-Stone system, but closer examination reveals a number
of additional details that require articulation before we can
definitively say what's essential to determination in general.

But before we do that let's consider a rather different brand
of determination, as comparison of maximally diverse examples
can frequently help us to "triangulate" the sought for target.

Consider that nugget from the "Old Elements":  Two points determine a line.
What this means, taken within the implied context of a particular geometry,
is that every choice of two distinct points in the implied geometric space
determines a line in the same space.  Obviously, or as would be evident to
our geometric imagination, many different pairs of distinct points will be
found to determine the same line.  So we get a many-to-one mapping like so:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` `D_1` `D_2` `D_3` `D_4` `D_5` `D_6` `D_7` `D_8` `D_9` ` `
` ` ` o ` ` o ` ` o ` ` o ` ` o ` ` o ` ` o ` ` o ` ` o ` ` `
` ` ` `\` ` | ` `/` ` ` `\` ` | ` `/` ` ` `\` ` | ` `/` ` ` `
` ` ` ` \ ` | ` / ` ` ` ` \ ` | ` / ` ` ` ` \ ` | ` / ` ` ` `
` ` ` ` `\` | `/` ` ` ` ` `\` | `/` ` ` ` ` `\` | `/` ` ` ` `
` ` ` ` ` \ | / ` ` ` ` ` ` \ | / ` ` ` ` ` ` \ | / ` ` ` ` `
` ` ` ` ` `\|/` ` ` ` ` ` ` `\|/` ` ` ` ` ` ` `\|/` ` ` ` ` `
` ` ` ` ` ` o ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` o ` ` ` ` ` `
` ` ` ` ` `L_1` ` ` ` ` ` ` `L_2` ` ` ` ` ` ` `L_3` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

Adopting the term "dyad" for a pair of distinct points, the Figure
shows the "incidence relation" between dyads D_i and lines L_j for
a hypothetical case where the dyad D_1 determines the line L_1 and
so on down the line, er, series, where the dyad D_9 determines the
line L_3.  As would be the usual thing in any geometry, then, each
dyad determines just one line, but each line is determined by many
different dyads.  This can be expressed in a very succinct fashion
by saying that there is a many-to-one function J : !D! -> !L! with
!D! being the domain of dyads and !L! being the co-domain of lines.
It should be obvious that this is a formal, informational, logical,
or mathematical brand of determination, not a causal determination.

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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