SUO: Re: Building Ontologies Through Signs And Inquiries (BOTSAI)
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Jean-Marc,
Let me see if I have built up enough of my stock material
to begin to address one or two or three of your questions,
say, this one:
JM: Now the fact that one looks upon abduction from a different perspective,
i.e. considered as the relatively genuine genus (thirdness) of a previous
division instead of being the most degenerate (firstness), would in principle
give in the next generation three subclasses of abductions. In that case, since
there is a perspective from which you see it that way, can you make explicit the
three distinct ways in which abduction can "mediate" between deduction and induction?
No, not yet.
First of all, let me be honest and say that I do not really understand all this
classification mania, whether it's pragmatists or ontologists that fall into it.
I know, you will say that whatever it is, one thing for sure is that Peirce had
it bad, but even there I do not read him quite the same way. Maybe it's due to
the historical accident that I happened on his work by chance, without the boon
of any professorial guidance, and only delved into the logical and mathematical
source materials, soaking them up solely for the lights that they threw on my
early questions about the grounds of reasoning. So now whenever I run into
one of Peirce's more "classical" stretches, it does not seem out of place,
because I think back to concrete material that it organizes, and I never
get the impression, as I do with many of our contemporary writers on
both pragmatism and ontology, that he was doing it for its own sake.
I am more incited by how inquiry works,
and how we can learn to work it better,
than I am by the mere nomination of it.
So let me regress, in the service of the inquiry,
to a more primitive station in inquiry's journey
of self-discovery, say here:
| I shall now show how the three conceptions of reference to a ground,
| reference to an object, and reference to an interpretant are the
| fundamental ones of at least one universal science, that of logic.
|
| CSP, OANLOC, CP 1.559
|
| Charles Sanders Peirce, "On a New List of Categories" (1867),
| Cf. 'Collected Papers', CP 1.545-567.
| Cf. 'Chronological Edition', CE 2, pages 49-59.
| http://members.door.net/arisbe/menu/library/bycsp/newlist/nl-frame.htm
I will return to this extremely important paragraph anon,
but for the moment let us hurry on down to the end of it:
| In an argument, the premisses form a representation of the conclusion,
| because they indicate the interpretant of the argument, or representation
| representing it to represent its object. The premisses may afford a likeness,
| index, or symbol of the conclusion. In deductive argument, the conclusion is
| represented by the premisses as by a general sign under which it is contained.
| In hypotheses, something 'like' the conclusion is proved, that is, the premisses
| form a likeness of the conclusion. Take, for example, the following argument:
|
| [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. That it is different with induction another example
| will show:
|
| [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.
Peirce's analysis of the patterns 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
If X => each of A, B, C, D, ..., then we have the following equivalents:
1. X => the greatest lower bound (glb) of A, B, C, D, ...
2. X => the logical conjunction A & B & C & D & ...
3. X => Q = A & B & C & D & ...
More succinctly, letting Q = P^1 & P^2 & P^3 & P^4,
the argument is summarized by the following scheme:
| Case Abduction
|
| Fact: S => Q
| Rule: M => Q
| ---------------.
| 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 Q is an iconic sign
of the object M is that Q 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 1.
| P^1 P^2 P^3 P^4
| o o o o
| \* \ / */|
| \ * \ / * / |
| \ * \ / * / |
| \ * \ / * / |
| \ *\ /* / |
| o Q o |
| | | * | |
| | | * | |
| | | | |
| | | | * |
| | | | * |
| o | o M
| \ | / *
| \ | / *
| \ | / * Case
| \ | / * S=>M
| \|/*
| o
| S
|
| Figure 1. 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.
Peirce's analysis of the patterns 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
If X <= each of A, B, C, D, ..., then we have the following equivalents:
1. X <= least upper bound (lub) of A, B, C, D, ...
2. X <= the logical disjunction A v B v C v D v ...
3. X <= L = A v B v C v D v ...
More succinctly, letting L = P^1 v P^2 v P^3 v P^4,
the argument is summarized by the following scheme:
| Rule Induction
|
| 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 2.
| P
| o
| /|\* Rule
| / | \ * M=>P
| / | \ *
| / | \ *
| / | \ *
| o | o M
| | | | * |
| | | | * |
| | | | |
| | | * | |
| | | * | |
| o L o |
| / */ \* \ |
| / * / \ * \ |
| / * / \ * \ |
| / * / \ * \ |
| /* / \ *\|
| o o o o
| S^1 S^2 S^3 S^4
|
| Figure 2. Induction to the Rule M => P
As it happens, I think that I can explain what I once 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.
Jon Awbrey
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