ONT Re: Extension x Comprehension = Information
- To: Arisbe <arisbe@stderr.org>, Gdsemiocom <gdsemiocom@univ-perp.fr>, Ontology <ontology@ieee.org>
- Subject: ONT Re: Extension x Comprehension = Information
- From: Jon Awbrey <jawbrey@oakland.edu>
- Date: Tue, 05 Mar 2002 12:34:01 -0500
- CC: Jean-Luc Delatre <jld@club-internet.fr>
- References: <3C67D0B9.8976D5C1@oakland.edu> <3C692DF1.543B27EB@oakland.edu> <3C6A7966.F1633CEC@oakland.edu> <3C6BC310.289EA552@oakland.edu> <3C6E9809.981AB8F3@oakland.edu> <3C6FB3B0.B4C53F32@oakland.edu> <3C7190D2.F8C1DC47@oakland.edu> <3C731C71.1AC1A4BC@oakland.edu> <3C7345B2.694E2D40@oakland.edu> <3C747ED4.F5498C81@oakland.edu> <3C764A36.22C2BB14@oakland.edu> <3C7721FF.68BC8077@oakland.edu> <3C796095.A39A83D6@oakland.edu> <3C7A6D93.5A8A1A7C@oakland.edu> <3C7BC4D2.DAC53BA@oakland.edu> <3C7D4C92.FCA190AB@oakland.edu> <3C7D7F8D.F0642808@club-internet.fr> <3C7DA340.B32E9C73@oakland.edu> <3C7DD7F9.32A94B74@club-internet.fr> <3C7E3ADD.83084FBC@oakland.edu> <3C7F49D8.F73ACE12@club-internet.fr> <3C7F8A57.28C437F2@oakland.edu> <3C825324.72DE1C4A@club-internet.fr> <3C842799.61A9E74F@oakland.edu> <3C8448ED.8F9A929B@oakland.edu>
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CH = Chris Hillman
CP = Charles Peirce
JA = Jon Awbrey
JD = Jean-Luc Delatre
Jean-Luc,
I will make an effort to get through your comments this time,
keeping to short replies for now, and then I will go back to
the drawing board and try to rethink things more concertedly.
Added later. I can see that I am going
to have to break this job into pieces.
My browser balks at editing files of
more than 36 kb!
CH: http://www.math.washington.edu/~hillman/papers.html
JA: i have only skimmed the "newconcept" paper and the "talk" abstract
at this point -- thanks again for the ghostscript tip, as my plugin
finder did not know about it -- since i was piqued by the idea of
"galois information theory", but i did not yet find the equivalent
of peirce's notion of information there, so clue me in if you think
it's there somewhere.
JD: Hey! This is *your* job!!!
yes, sorry to be so lazy. i will eventually work my way through the paper.
but i read my first paper on "galois theory of models" almost 20 years ago,
and so far, in spite of many tantalizing titles since then, i have not run
across anybody who rediscovered the equal of what i consider to be peirce's
crucial insights about information in this connection. i think that there
are probably systemic reasons for this, having to do with the cognitive
styles that are engrained into most mathematical thinkers in our times.
JD: I will kindly try to translate for you from Peirce incredibly awkward and
convoluted (yet, enlightened for the time) explanations, taken from your
messages, into straight lattice theoretic statements matched to Hillman's
so you will perhaps see that it's almost the same, just neater.
I am also commenting Peirce statements & yours.
CP: | Yet there are combinations of words and combinations of conceptions
| which are not strictly speaking symbols. These are of two kinds
| of which I will give you instances. We have first cases like:
|
| 'man and horse and kangaroo and whale',
|
| and secondly, cases like:
|
| 'spherical bright fragrant juicy tropical fruit'.
JD: First thing first, here Peirce uses two different
kinds of words in the two sentences.
JD: 1) 'man', 'horse', etc., ... stand for what is called 'intents' in
Hillman's terminology and thus is a set of 'predicates' shared by all
objects satisfying the concept ('man' -> 'Apterous', 'Biped', etc., ...)
Translated to Hillman's 'man' is the name of the subset of his Y set
which emcompass 'Apterous', 'Biped', etc... as *elements* of the Y set.
These could be called essential attributes of 'man' and this is what
Peirce call 'Content'.
JD: 2) 'spherical', 'bright, as well as 'Apterous', 'Biped', etc., ...
are attributes, that is what Hillman calls 'predicates' (Page 2).
Although each one have an extension as well as 'man' and 'horse'
have, the extension of a 'predicate' is just the counter-image
of the singleton set: <bright objects> = <| {'bright'}, while
<man objects> is the intersection of all the counter-images of
the "essential attributes" of 'man' (Peirce's 'Content', my
wording on "essential") giving Hillman's 'extent', that is
Peirce's 'Sphere'.
JA: first off, peirce is using just one kind of words in the two cases.
they are all called "terms". you may of course opt to use two kinds
of words to interpret him, if you wish, but then you will be in danger
of introducing distortions in the process of forming this interpretation.
JA: indeed, one of the reasons that i prefer peirce's account is that he does not
begin by lading these ontological notions into his logical cart at the outset.
so his treatment is somewhat less theory-laden to start. this is a good idea,
especially if we desire to use our logic to reason about the possibility of a
distinction between two kinds of terms, without prejudicing the matter in the
very form of the logic we use.
I might add that my suspicion of such distinctions
goes back to my childhood in physics, when I found
out about particle-wave duality, and promised that
I would never let myself be so easily fooled again.
Application? The Subject-Predicate distinction.
These are roles that terms play, not essences.
JD: As you can see, the X set of Hillman is the set (probably better named only
a collection ...) of all 'things' in the universe. While the Y set is the set
of 'attributes', some subsets of which are distinguished as 'Content' (Peirce)
or 'intents' (Hillman) and given a name which stands for the 'words' (Peirce)
or 'concepts' (Hillman), although both are happily confusing the intentional
and extensional meaning and you never know, unless you check carefully, if
they are talking about subsets within X or Y, THAT's duality!!!
JA: no, i think that peirce is quite clear about the distinction
between intensions (= attributes, properties, or qualities)
and extensions (= things in the universe of discourse).
moreover, he is admirably clear about the distinction
between the role of a sign and the role of an object,
which distinction almost everybody else confounds.
When things get too confusing I need to draw a picture.
Peirce is implicitly working within a relational context
that will eventually come to be called a "sign relation".
This is a 3-adic relation whose role-places or role-players
are conventionally appellated as the "object", the "sign",
and the "interpretant sign" (or "interpretant" for short).
In extension, then, a sign relation L would be a subset
of a cartesian product, L c !O!x!S!x!I!.
Here are a couple of small examples of sign relations, formalizing
a fragment of noun and pronoun usage for two agents, A and B, and
a syntactic domain (the union of the sign and interpretant domains)
that are given as follows:
| Object domain: !O! = {A, B}.
| Sign domain: !S! = {"A", "B", "I", "You"}.
| Interp domain: !I! = {"A", "B", "I", "You"}.
|
| Syntactic set: !S! |_| !I! = {"A", "B", "I", "You"}, in this special case.
For compactness, though, I will code "I" and "You" as "i" and "u", respectively.
| Object domain: !O! = {A, B}.
| Sign domain: !S! = {"A", "B", "i", "u"}.
| Interp domain: !I! = {"A", "B", "i", "u"}.
|
| Syntactic set: !S! |_| !I! = {"A", "B", "i", "u"}, in this special case.
Table 1. Sign Relation L(A)
o---------------o---------------o---------------o
| Object | Sign | Interpretant |
o---------------o---------------o---------------o
| A | "A" | "A" |
| A | "A" | "i" |
| A | "i" | "A" |
| A | "i" | "i" |
o---------------o---------------o---------------o
| B | "B" | "B" |
| B | "B" | "u" |
| B | "u" | "B" |
| B | "u" | "u" |
o---------------o---------------o---------------o
Table 2. Sign Relation L(B)
o---------------o---------------o---------------o
| Object | Sign | Interpretant |
o---------------o---------------o---------------o
| A | "A" | "A" |
| A | "A" | "u" |
| A | "u" | "A" |
| A | "u" | "u" |
o---------------o---------------o---------------o
| B | "B" | "B" |
| B | "B" | "i" |
| B | "i" | "B" |
| B | "i" | "i" |
o---------------o---------------o---------------o
These are just an interrelated couple of well-bounded and concrete examples,
but they are still complex enough to illustrate many significant properties
of sign relations in general.
One way to read these 3-adic sign relations is to recognize that
the 2-adic relations formed by the sign and interpretant columns
constitute equivalence relations, and this property allows us to
graph the pertinent facts as shown in the next couple of Figures:
o-----------------------------o-----------------------------o
| Objective Framework | Interpretive Framework |
o-----------------------------o-----------------------------o
| |
| o "A" |
| · ^ |
| · | |
| · | |
| · | |
| · v |
| A o · · · · · o "i" |
| |
| |
| |
| |
| B o · · · · · o "u" |
| · ^ |
| · | |
| · | |
| · | |
| · v |
| o "B" |
| |
| Figure 3. Sign Relation L(A) |
o-----------------------------------------------------------o
o-----------------------------o-----------------------------o
| Objective Framework | Interpretive Framework |
o-----------------------------o-----------------------------o
| |
| o "A" |
| · ^ |
| · | |
| · | |
| · | |
| · v |
| A o · · · · · o "u" |
| |
| |
| |
| |
| B o · · · · · o "i" |
| · ^ |
| · | |
| · | |
| · | |
| · v |
| o "B" |
| |
| Figure 4. Sign Relation L(B) |
o-----------------------------------------------------------o
These are the kinds of pictures that we like to see when it comes to
the relationships between objects and signs. A double arrow between
a pair of signs indicates that they are equivalent to each other in
the associated equivalence relation. Since an equivalence relation
on a set partitions it into disjoint and exhaustive parts, another
way of describing the situation in each of these Figures is to say
that L(A) and L(B), or their projections on the subspace !S! x !I!,
each partitions the set of signs into subsets that correlate with
the objects in !O!. Except for the annoying circumstance that each
sign relation carries out this partition in a different way, leaving
us with two partitions instead of just one, this would be the perfect
picture of how we naturally tend to think that signs and objects ought
to behave in regard to each other.
To be continuous ...
Jon Awbrey
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