SUO: Re: Conformance, 2-Faceted Models, Reformation
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John Velman wrote:
>
> One nice feature of "model as Janus" is that it
> provides a bridge for explaining what appear to be
> two different uses of the word 'model', the logician's
> use and the engineer's use.
>
> John V.
Indeed, this dual model of models has been with us
since the time that Aristotle dusted off the word
"paradigm" and put it into technical parlance with
his analysis of reasoning by "analogy" or "example"
as a complex form syllogism combining the "figures"
or the patterns of inductive and deductive reasoning.
Peirce gave even more complex articulations of this
process, involving all three, ab-, de-, & in-duction,
the pictures of which I will sketch for you sometime.
Here is my most recent summary mention of this fact:
http://suo.ieee.org/email/msg06788.html
Here is a new edition, straightening out a few digressions:
I keep coming back to the following two pictures or configurations:
1. There is that A-frame construction in Aristotle's discussion
of reasoning by way of Analogy, Example, or Paradigm, which
he articulates as a combination of Induction and Deduction.
| Atrocious Adversity
| A
| o
| /*\
| / * \
| / * \
| / * \
| / * \
| / * \
| / R u l e \
| / * \
| / * \
| / Bellicose \
| / Battles \
| F a c t B F a c t
| / Between \
| / Bordermates \
| / * * \
| / * * \
| / C a s e C a s e \
| / * * \
| / * * \
| / * * \
| / * * \
| / * * \
| o o
| C D
| Contest: Debacle:
| Athens versus Thebes Thebes versus Phocis
|
| Figure 1. Aristotle's "War Against Neighbors" Example
|
| A = Atrocious, Adverse to All, Avoid This!
| B = Belligerent Battle Between Brethren.
| C = Contest of Athens against Thebes.
| D = Debacle of Thebes against Phocis.
The cardinal- or hinge-point to note about Aristotle's example
of reasoning by example is that the middle term B serves as an
explanation of 'why' the major term A should be considered as
applicable to the contemplated instances of conflict, C and D,
instance C a future contingent whose advisability of rendering
actual was presently, at that time in Athens, being disputed,
instance D already a part of the discussants' previous history,
from which they might reasonably be expected to have learned.
2. The other picture is John Sowa's Model-Theory-World Triptych:
http://www.bestweb.net/~sowa/ontology/mthworld.gif
Aristotle's A-Frame structure of analogy equips us with an archetype
for understanding the relation between abstraction and analogy, plus
the relationship of models and morphisms. Let us trace it like this:
| Comprehension Theory
| C T
| o----------------------<----------------------o
| /|\ Denotation .
| / | \ .
| / | \ .
| / | \ .
| / | \ .
| / | \ .
| / R u l e \ .
| / | \ .
| / | \ .
| / | \ .
| / | \ .
| F a c t A F a c t .
| / Abstraction \ .
| / * * \ .
| / * * \ .
| / * * \ .
| / C a s e C a s e \ .
| / * * \ .
| / * * \ .
| / * * \ .
| / * * \ .
| / * Arrow * \ .
| o---------------------->----------------------o
| X <------------------<----->------------------> E
| Unknown Analogy Effective
| Reality Facsimile
|
| Figure 2. Abstraction Over An Appropriate Arrow Of Analogy
I have labeled the top node "Comprehension" because I believe that this
is the classical word for the conjunction of all of the intensions that
a collection of subjects have in common, but my impression is hazy here,
and so I will just have to use it this way provisionally for the moment.
The middle term A is an explanatory, pertinent, or relevant Abstraction,
the property that accounts for all of the rest of the properties in the
Comprehension.
I am still not sure about the use of the word "comprehension" here,
but let me use it provisionally for the complex of properties that
are held in common by a set of instances, objects, or situations,
and that are intended to be captured by a fitting theory of them.
Then we can say that all of these objects, whether in the object
domain or in the analogue model domain, are logical models that
satisfy the theory. In sign-theoretic terms, the theory T is
a sign in a suitable sign domain S, while everything else in
Figure 2 is an object in the hierarchical object domain O.
I remembered what was bothering me about my earlier use
of the word "comprehension" to label the summit point
in Aristotle's paradigm of analogy. As it happens,
I had already written out an amendment this summer,
but that was before vacation ...
In my anticipation of a future development whose time is not yet prepared,
I created a potential for a certain confusion in the picture that I drew
last time, attempting to show the connections among abstract hypostases,
reasoning by analogy, the invocation of formal models, and the combined
intensions, or "comprehension", that is common to an object and its icon,
that is, its formal likeness. So let me try to make amends this time around,
and maybe even foreshadow where I believe this discussion is eventually going.
If you take what I sketched above quite literally -- and I gave you
no reason to do otherwise -- then it just fails to make good sense.
For if the comprehension C is the conjunction of "all" intensions,
then whatever intension evolves as the accountable abstraction A
is already among them, and so we have C => A. But the picture
appears to suggest that A => C. Of course, this could happen,
but it does not represent the most generic state of affairs.
By way of excusing myself, let me explain what caused me to say this.
First, I am jumping ahead in my thoughts to a text of Peirce's where
he overlays a type of abductive reasoning on this picture of analogy.
Second, the order of development that I am trying to diagram here is
not a static hierarchy of implications but a dynamic evolution, thus
involving at least two distinct moments in time.
I think that I can fix up the rest of this discussion by redrawing and
relabeling slightly the adaptation that I made of Aristotle's A-Frame.
| Initial
| Comprehension
| A = IC
| o
| /|\
| / | \
| / | \
| / | \
| / | \
| / | \
| / R u l e \
| / | \
| / | \
| / | \
| / | \
| F a c t B = EC F a c t
| / Evolved \
| / Comprehension \
| / * * \
| / * * \
| / C a s e C a s e \
| / * * \
| / * * \
| / * * \
| / * * \
| / * Arrow * \
| o---------------------->----------------------o
| X <------------------<----->------------------> E
| Unknown Analogy Effective
| Reality Facsimile
|
| Figure 3. Dynamic Comprehension, or Conceptual Evolution Over Time
Let us now consider the two chief moments in the development of the argument
as they work out in Aristotle's paradigmatic example of analogical reasoning.
The apical node is labeled "Initial Comprehension" (IC) to indicate
the conjunction of the intensions that are implicit in an agent's
initial understanding of what the analogue subjects have in common.
The middle node is labeled "Evolved Comprehension (EC) to indicate
the conjunction of the intensions that are evolved in an agents's
reflective understanding of the situation or the state of affairs.
In Aristotle's original example the argument began with the conjunction
of many qualities that were called to mind in regard to several well-known
battles, whose effects could be summed up by saying that that their results
turned out to be "Adverse" (A) for all sides. After a moment of reflection,
the reasoning agent becomes aware of a new intension, whose significance has
gone previously unnoticed, to wit, the common case that all of these debacles
were instances of "Battles Between Bordermates" (B), and this serves as the
pivot point for an extension of the argument by analogy to an anticipated
application, one that is actively being contemplated but still avoidable,
amounting to a prediction of adversity arising from a future contingent
war against neighbors, anticipated but not yet engaged.
Jon Awbrey
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