SUO: Re: Inquiry As Theory Change
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JA = Jon Awbrey
JS = John Sowa
MW = Matthew West
JA: For some time now I have been concerned with the dynamics of inquiry processes,
that is, with the trajectories of changing theories as they evolve through time,
and the forces acting on initial theories that push them toward their next state.
JA: I have started out with "zeroth order theories" (ZOT's), working on the
heuristic principle of "zeroth things first", and it happens that many
of the most important phenomena can already be observed at this level.
As long as we are working within simple concept hierarchies, the sorts
of orders that can be embedded in, or factored out of boolean lattices,
and that is much more often than we often suspect, then the examination
of theory change in ZOT's can be quite enlightening about many aspects
of what is involved in more complicated settings. Furthermore, in the
case of "zeroth order logic" (ZOL), it is possible to generate in an
exhaustively complete fashion a form of logical representation that
is information-equivalent to having a survey of all the models of
a propositional formula. The computational complexity of doing
this is another matter, but for the sake of gaining conceptual
clarity about model theory in the most simple of cases, it can
be very helpful to be able to do this.
JA: An important thing to realize about this variety of "formal model theory"
is that it's all done with mirrors, that is, we never leave the stage of
signs and images in manifesting these models. Nor is it even the act of
some Mandrake the Magician or David Copperfield who gestures mysteriously
over a "theory of the elephant" and thereby produces an elephant on stage.
All that is really happening is that one sort of language about elephants
is being transformed, rather prosaically and utterly syntactically, into
another sort of langauge about elephants. The illusion of dealing in
the thing itself is managed only because the target language is one
whose relationship to things we hardly think to question, largely
by dint of our longer familiarity with its uses and its ways.
MW: I am beginning to think that how terms in FOL relate
to the domain (life, the universe, and everything)
is about the model theory, from what I have been
able to grasp of Chris M's recommended reading
on model theory recently. However, it is no
clearer to me how the link is made.
Thinking of theory as "rational universals, beloved beliefs, elegant rules" (RUBBER)
and content domain as "real or actual data" (ROAD) can make it tough to see exactly
how they meet. But the fact is that these two realms are not quite incommensurable,
and each is empowered to present a proxy representation of its informational impact
within a common sensorium of affective impressions and cognitive expressions, where
they can be compared. This compatibility between general predicates and particular
subjects is another one of the key insights in Peirce's logic -- actually it was in
Aristotle's logic, though seldom noticed there -- and it is the very reason that he
was so critical of the illusion of the individual, the imaginary atomic entity that
it would take an infinite amount of information to pin down with logical exactitude.
It is critical to any account of inquiry, in other words, to any explanation
of how theories change, that it be possible to compare and contrast theories
and data on a common ground, where they will impinge on one another's claims
and be obliged to hammer out one brand of compromise or another between them.
Jon Awbrey
JS: That is a point I have been trying to explain
by means of the following diagram:
http://www.jfsowa.com/figs/mthworld.gif
JS: Following are two paragraphs from Section 7 of my article
"Signs, Processes, and Language Games" that describe the
diagram mentioned above. If you want to read them in
context, see the full paper:
http://www.jfsowa.com/pubs/signproc.htm
JS: Or see Ch. 6 of my KR book, which goes into more detail.
JS: "Theories, Models, and the World" [excerpt]
JS: The problems of knowledge soup result from the difficulty of matching
abstract theories to the physical world. The techniques of fuzziness,
probability, defaults, revisions, and relevance are different ways
of measuring, evaluating, or accommodating the inevitable mismatch.
Each technique is a metalevel approach to the task of finding or
constructing a theory and determining how well it approximates
reality.
JA: One of the key insights in Peirce's logic is that certain negative operations,
like the "amphecks" (Nand, Nnor), are more generative and more primitive than
the more popularly attended positive operations. When it comes to the issue
of a "logic of inquiry", there is a not unrelated point to be seen, I think,
in the circumstance that the rawest relationship that a real datum can have
to a finely rationalized theory is the brute negativity of puncturing that
theory, and this is the point at which inquiry begins, with the mismatch,
the failure of some previously taken for granted theory that we had been
automatically acting on right up to the point where it let us down in
the face of actual occurrences.
JS: To bridge the gap between theories and the world, Figure 5 shows models
as Janus-like structures, with an engineering side facing the world and
an abstract side facing the theories.
JS: On the left is a picture of the physical world, which contains more detail and
complexity than any humanly conceivable model or theory can represent. In the
middle is a mathematical model that represents a domain of individuals D and
a set of relations R over D. If the world had a unique decomposition into
discrete objects and relations, the world itself would be a universal model,
of which all accurate models would be subsets. But as the examples in this
book have shown, the selection of a domain and its decomposition into objects
depends on the intentions of some agent and the limitations of the agent's
measuring instruments. Even the best models are approximations to a limited
aspect of the world for a specific purpose. Engineers express that point in
a pithy slogan: All models are wrong, but some are useful.
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