ONT Re: Inquiry Into Symbolization
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This is one of those puzzles that I have been puzzling away at for almost as long
as I can remember. I have gotten fairly well acquainted with the various pieces
of the puzzle, but haven't quite figured out yet how they ought to fit together.
It all seems to have something to do with an intricate relationship
among concepts, kinds (of the natural kind, naturally), and symbols.
I know that I am always reminded of it when I read what Peirce says
on the issues of "symbolization" and "symbolizability". And I have
the impression that there is a vast order of generalization in the
works here, taking the topics of "observation" and "observables",
along with "computation" and "computables", and even "conception"
and "conceivables" under its wing with plenty of room left over.
Still, the best that I seem able to do at this juncture in time
is just to keep assembling the pieces together and just to keep
staring at them till the right sorts of connections occur to me.
The pieces of the puzzle are these:
1. Remember that for Peirce "concepts are a species of symbols", and so
to talk about "symbolization" and "symbolizability" is tantamount to
invoking a generalization of "conceptualization" and "conceivability".
So the whole scene in question is taking place on the stage set by Kant,
whose depiction of the creation, development, and elimination operators
that work on conceptions Peirce has already intoned in his prologue to
the entire drama:
| The essential of a thing -- the character of it --
| is the unity of the manifold therein contained.
| 'Id est', the logical principle, from which as
| major premiss the facts thereof can be deduced.
|
| CSP, CE 1, page 6.
|
| Charles Sanders Peirce,
|"Private Thoughts, Principally On The Conduct Of Life" (Number 37, August 1860),
|'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866',
| Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.
2. The idea about the function of conceptions that Peirce obtained from Kant:
| This paper is based upon the theory already established, that the function of
| conceptions is to reduce the manifold of sensuous impressions to unity, and that
| the validity of a conception consists in the impossibility of reducing the content
| of consciousness to unity without the introduction of it. (CSP, CP 1.545, CE 2.49).
|
| http://www.door.net/arisbe/menu/library/bycsp/newlist/nl-frame.htm
3. Peirce's "Note On A Limited Universe Of Marks" (NOALUOM).
| http://suo.ieee.org/ontology/msg03204.html
|
| CSP, SIL, pages 182-186. (Cf. CE 4, pages 450-453, CP 2.517-531).
|
| Charles Sanders Peirce, "Note A. On A Limited Universe Of Marks" (1883),
| CSP (ed.), 'Studies in Logic, by Members of the Johns Hopkins University',
| Reprinted with an Introduction by Max H. Fisch & a Preface by Achim Eschbach,
| in 'Foundations of Semiotics, Volume 1', John Benjamins, Amsterdam, NL, 1983.
|
|'Writings of Charles S. Peirce: A Chronological Edition, Volume 4, 1879-1884',
| Peirce Edition Project, Indiana University Press, Bloomington, IN, 1986.
4. An image that I have about the relationship between artificial kinds and natural kinds
in terms of a mapping, morphism, restriction, or quotient relation between two lattices.
I worked this out once in application to the "apterous biped" definition of the human,
but cannot quite recall the punchline.
Let me begin again with this last bit and see if I can get a little bit further this time.
Consider the joke definition of a Human Being as a Featherless Two-legged Critter.
By way of a schematic formalization, I set out the matter in the following manner:
| A B
| o o
| |\. ./|
| | \ . . / |
| | \ . . / |
| | \ . . / |
| | \ / |
| | . \ / . |
| | G |
| | . / \ . |
| | / \ |
| | . / \ . |
| | / \ |
| |./ \.|
| |/ \|
| o o
| H P
|
| Figure 1. On Being Human
|
| A = Apterous = featherless animal
| B = Bipedal = two-legged being
| C = Critter = creature, creation
| G = glb(A, B) = A |^| B
| H = Human Being
| P = Plucked Chicken
Figure 1 outlines the subject matter, to wit, the category "human being" (H)
here defined as falling under the head of an "apterous biped" (G = A |^| B),
hence bound by the set-theoretic intersection G of the respective extensions
of the two concepts, "apterous" (= featherless) and "bipedal" (= two-legged).
Now the wise-cracking sort of person, one who ignores the naturally implicit
constraints of the discussion to what are often described as "natural kinds",
will naturally be compelled to pipe up, "But what of the plucked chicken? --
a two-legged critter without feathers? -- is that your idea of human being?"
Now, we know that the response to this witlesscism must invoke the distinction
between what one calls an "artificial kind" and a "natural kind", respectively,
even though it is difficult to say just how this difference makes a difference.
Here is one possible way to view the situation:
| SET NAT NAT
|
| A B A B A B
| o o o o o o
| |\. ./| | / \ /
| | \ . . / | | / \ /
| | \ . . / | | / \ /
| | \ . . / | | / \ /
| | \ / | | / \ /
| | . \ / . | | / \ /
| | G | | / G
| | . / \ . | | / =
| | / \ | | / =
| | . / \ . | | / =
| | / \ | | / =
| |./ \.| | / =
| |/ \| |/ =
| o o o o
| H P H H
|
| Figure 2. On Being Human, All Too Human
Think of the initial set-up as being cast in a lattice of arbitrary sets.
Within that setting, the "greatest lower bound" (glb) of the extensions
of A and B is their set-theoretic intersection, G = glb(A, B) = A |^| B.
This G covers the desired class H but also admits the risible category P.
Now, suppose that we are clued into the fact that not all sets in SET
are admissible, allowable, natural, pertinent, relevant, or whatever,
to the aims of the discussion in view, and that only some mysterious
'je ne sais quoi' subset of "natural kinds", NAT c SET, is at stake,
a limitation that, whatever else it does, excludes the set P and all
of that ilk from beneath glb(A, B). Though we cannot quite say how
we apply this information, we know it by its effects to give us the
lattice structure in the next frame, where H = glb(A, B), and thus
in this more natural setting the proposed definition works okay.
An alternative way to look at the transformation of our views
from the arbitrary lattice SET to the natural lattice NAT,
is illustrated in the last frame, where the equal signs
indicate that the nodes for G and H are identified.
In this picture, the measure of the interval that
once existed between G and H, now shrunk to nil,
gives a rough indication of the quantity of
information that went into forming the
natural end result.
Or something like that ...
Jon Awbrey
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