ONT Re: Inquiry Into Inquiry
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Subj: Theory of Inquiry, Types of Inference, Types of Signs
Date: Sun, 20 May 2001 12:00:06 -0400
From: Jon Awbrey <jawbrey@oakland.edu>
To: Arisbe <arisbe@stderr.org>,
SemioCom <semiocom@listbot.com>,
Standardize Unto Others <standard-upper-ontology@ieee.org>
CC: Cathy Legg <cathy@coombs.anu.edu.au>,
David Low <low@cdi.com.au>,
Arien Malec <arien_malec@yahoo.com>,
Jean-Marc Orliaguet <jmo@medialab.chalmers.se>,
John F Sowa <sowa@bestweb.net>,
David Whitten <whitten@NETCOM.COM>
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| "And hast thou slain the Jabberwock?
| Come to my arms, my beamish boy!
| O frabjous day! Callooh! Callay!"
| He chortled in his joy.
|
| Lewis Carroll (Charles Lutwidge Dodgson) "Jabberwocky" <<<---<<<
| http://www76.pair.com/keithlim/jabberwocky/poem/jabberwocky.html
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Inquiry SIG,
This completes, at least for a while, my retrospective account
of "What I Did Last Summer". When it is said and done, I will
turn to elaborating its logical implications and its practical
consequences for the application that we have before us, given
e-special aptness and recursive reference to the criterion of
a "Body Electric Of Maintainable Ontology Software" (BEOMOS).
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Note 5
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Subj: Re: Arguments and the Categories
Date: Thu, 11 May 2000 14:04:08 -0400
From: Jon Awbrey <jawbrey@oakland.edu>
To: Peirce Discussion Forum <peirce-l@lyris.acs.ttu.edu>
For ease of reference, and to keep the analyses
of Abduction and Induction on the same page for
the purposes of comparison, I will repeat here
the fundamentals of what happened last time,
making just a couple of additional remarks,
and then moving on to the new material.
Abduction to a Case:
| M is, for instance, P^1, P^2, P^3, and P^4;
| S is P^1, P^2, P^3, and P^4:
| Therefore S is M.
|
| Here the first premiss amounts to this,
| that "P^1, P^2, P^3, and P^4" is a likeness of M,
| and thus the premisses are or represent a likeness
| of the conclusion.
Peirce's analysis of this pattern of abductive argument
can be understood according to the following paraphrase:
Fact: S => P^1, S => P^2, S => P^3, S => P^4.
Rule: M => P^1, M => P^2, M => P^3, M => P^4.
---------------------------------------------------
Case: S => M.
When X => each of A, B, C, D, ...
then X => the Greatest Lower Bound (GLB) of A, B, C, D, ..., which is to say,
then X => the Logical Conjunction, A & B & C & D, to give it a nickneme, "N",
then X => N.
Most succinctly, the argument can be summarized as follows:
Where N = P^1 & P^2 & P^3 & P^4:
Fact: S => N.
Rule: M => N.
----------------
Case: S => M.
In this piece of Abduction, it is the GLB or the Conjunction
of the ostensible Predicates that is the operative Predicate
of the Argument, to wit, the Predicate that is Common
to both the Fact and the Rule of the Inference.
Finally, the reason why one can say that N is an iconic sign
of the object M is that N can be taken to denote M by virtue
of the qualities that they share, namely, P^1, P^2, P^3, P^4.
Notice that the iconic denotation is symmetric, at least in principle,
that is, icons are icons of each other as objects, at least potentially,
whether or not a particular interpretive agent is making use of their
full iconicity during a particular phase of semeiosis.
The situation is diagrammed in Figure 3.
| P^1 P^2 P^3 P^4
| o o o o
| \* \ / */|
| \ * \ / * / |
| \ * \ / * / |
| \ * \ / * / |
| \ *\ /* / |
| o N o |
| | | * | |
| | | * | |
| | | | |
| | | | * |
| | | | * |
| o | o M
| \ | / *
| \ | / *
| \ | / *
| \ | / *
| \|/*
| o
| S
|
| Figure 3. Abduction to the Case S => M
In a diagram like this, even if one does not bother to
show all of the implicational or the subject-predicate
relationships by means of explicit lines, then one may
still assume the "transitive closure" of the relations
that are actually shown, along with any that are noted
in the text that accompanies it.
We pick up the story with the following episode of Induction.
Induction to a Rule:
| S^1, S^2, S^3, and S^4 are taken as samples of the collection M;
| S^1, S^2, S^3, and S^4 are P:
| Therefore All M is P.
|
| Hence the first premiss amounts to saying
| that "S^1, S^2, S^3, and S^4" is an index of M.
| Hence the premisses are an index of the conclusion.
As it happens, I think that I can explain what I just called a "peculiarity"
of Peirce's language here, simply by referring to our ordinary informal usage,
and this will save me the trouble of looking through his early writings, where
I recall seeing this usage before. Remember, our common logical, mathematical,
and set-theoretic language of "unions" and "intersections" was not fully worked
out at this time (1860's), at least, not in all the glories or mirages, depending
on your point of view, of its current axiomatic treatment. Of course, these old
folks had the concepts, more or less, but if I remember correctly from my first
encounter with Peirce's work -- and my memory is always a doubtful proposition
when I'm talking about three days, much less thirty years! -- Peirce was still
at this time, or soon to be, writing about "aggregations" and "compositions",
and these of two kinds, "absolute" and "relative", but the intuitive meanings
that were attached to the "absolute" or the "non-relative" variety of terms,
and bounded by their corresponding concepts and their rudimentary definitions,
were analogous to but not exactly identical to our modern notions of "unions"
and "intersections", that is, the set-theoretic operations that are associated
with the logical usage of "OR" and "AND", respectively.
So what you have to understand -- what all this preambling is leading up to --
is the following multiplicity of meaning in Peirce's usage at this point,
namely, that the "AND" in his account of Abduction and the "AND" in his
account of Induction are two different words, or tokens of the same
polymorphous sign, if you will, but with a diversity of meanings,
the first corresponding to conjunction and intersection, and the
second corresponding to disjunction and union. And this is just
done in accord with a perfectly natural natural language idiom.
Now, if you will just try to remember the way that we often speak
in informal circumstances and in ordinary language -- I know, it
gets harder to remember all the time! -- but it is true that we
often use the word "AND", especially when referring to samples
of "dry goods", like handfuls of beans and bags of wool, to
speak of their more aggregarious union, and not so much,
since it barely makes sense in this setting, of their
contentious intersection.
Peirce's analysis of this pattern of inductive argument
can be understood according to the following paraphrase:
Case: S^1 => M, S^2 => M, S^3 => M, S^4 => M.
Fact: S^1 => P, S^2 => P, S^3 => P, S^4 => P.
---------------------------------------------------
Rule: M => P.
When X <= each of A, B, C, D, ...
then X <= the Least Upper Bound (LUB) of A, B, C, D, ..., which is to say,
then X <= the Logical Disjunction, A v B v C v D, to give it a nomen, "L",
then X <= L.
More succinctly, the argument can be summarized as follows:
Where L = S^1 v S^2 v S^3 v S^4:
Case: L => M.
Fact: L => P.
----------------
Rule: M => P.
In this bit of Induction, it is the LUB or the Disjunction
of the ostensible Subjects that is the operative Subject
of the Argument, to wit, the Subject that is Common
to both the Case and the Fact of the Inference.
Finally, the reason why one can say that L is an indexical sign
of the object M is that L can be taken to denote M by virtue of
the instances that they share, namely, S^1, S^2, S^3, S^4.
Notice that the indexical denotation is symmetric, at least in principle,
that is, indices are indices of each other as objects, at least potentially,
whether or not a particular interpretive agent is making use of their full
indiciality during a particular phase of semeiosis.
The situation is diagrammed in Figure 4.
P
o
/|\*
/ | \ *
/ | \ *
/ | \ *
/ | \ *
o | o M
| | | * |
| | | * |
| | | |
| | * | |
| | * | |
o L o |
/ */ \* \ |
/ * / \ * \ |
/ * / \ * \ |
/ * / \ * \ |
/* / \ *\|
o o o o
S^1 S^2 S^3 S^4
Figure 4. Induction to the Rule M => P
Now, depending on the architectronic principles and the basic parameters
of one's personal favorite framework for ontology, whether it is one of
the standard ones, for instance: (1) yet another version of set theory,
(2) what the mathematicians, by waylay of an admitted theft of Kant and
even of Carnap, though as of yet an unconfessed theft of Peirce, are wont
to call "Category Theory", or (3) the exceptions that prove the rule, to wit,
any old or any new "Ontological Hierarchy" (OH), "Objective Framework" (OF),
or whatever "Form of Synthesis-Analysis" (FOSA) for pragmatic objects that
Ontogenies Recapitulating Phylogenies can rustle up from the grubstake of
even the slightest, all too human imagination, then the fearful symmetry
and the vertical revertigo of Abduction and Induction may or may not be
broken and stop right here. For it does not hold in general, that is
to say, across every sort of ontology that might be conceived, that
the Anode (the way up) and the Cathode (the way down) lend their
lines of conduction and succession to be switched at will --
I am sure that you can imagine the dire consequences that
might ensue if one's brain and one's body were not wired
for that option! But the matter of this concern is
a charge that I must defer to another time.
Jon Awbrey
P.S. I feel like I ought to roll the credits about here,
but the task of tracing back through this thread of
inquiry, in order to enumerate all of the characters
that are contributary to it, would certainly delay its
release beyond the "must see" summer season! So I will
have to leave that scholarly duty to a moment of leisure
and a calmer time, or else to some indefinitely ultimate,
e-future, e-ventual, e-masochistic, e-telexial e-historian!
J.A.
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