ONT Re: Inquiry Driven Learning Environments (IDLE's)
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| Document History
|
| Subject: Extensions Of Mind
| Subhead: Essays And Reports On Intelligent Systems
| Contact: Jon Awbrey <jawbrey@oakland.edu>
| Version: Draft 3.04
| Created: 10 Sep 1993
| Revised: 11 Feb 2002
| Advisor: C.C. Wagner
| Setting: Oakland University, Rochester, Michigan, USA
| Excerpt: Division 4 (An OAR For Odysseus: Offline Analytic Resource)
| Excerpt: Subdivision 4.3 (Multi-Sage Deconstruction: Post*Modern Elephant)
| Excerpt: Section 4.3.2 (Prospective Needs Analysis: Talking About Tables)
| Excerpt: Subsection 4.3.2.3 (Mosaic Paradigm: Twin Tables for Every Law)
4.3.2.3 Mosaic Paradigm: Twin Tables for Every Law (cont.)
One-place relations are too simple to illustrate all of the important issues.
It may be good to keep in mind the following complications on the same theme.
Consider the complementary pair of 2-column relational tables
T = ACO = "Area Code Of" and (T) = (ACO) = "Not Area Code Of".
Without getting into latitudes and longitudes, specialize the
assignment of area codes as it applies to cities and treat ACO
as a relation between cities and area codes. Then ACO and (ACO)
are complementary subsets of the cartesian product Cities x Codes.
Under the counterfactual assumption that no city is divided across
several area codes, ACO amounts to a function from Cities to Codes.
o-------o------------o------o o-------o------------o------o
| ACO | City | Code | | (ACO) | City | Code |
o-------o------------o------o o-------o------------o------o
| | Amarillo | 806 | | | Ann Arbor | 806 |
| | Ironwood | 906 | | | Boston | 001 |
| | Washington | 202 | | | Chicago | 202 |
| | Detroit | 313 | | | Detroit | 011 |
| | Galveston | 409 | | | Escanaba | 409 |
| | Lansing | 517 | | | Flint | 101 |
| | Cheboygan | 616 | | | Galveston | 616 |
| | Houston | 713 | | | Houston | 111 |
| | ... | ... | | | ... | ... |
o-------o------------o------o o-------o------------o------o
Next consider a pair of 3-column tables that are derived from those above.
The Info Table provides Area Code Information by means of a Yes/No value in
its last column that tells whether the preceding <City, Code> pair is a valid
match or not. The complementary MisInfo Table records the opposite of what is
true about every match or misfit pair. The Info relation, of course, resembles
the answer key to a True/False exam. Other variations along these lines might
arise in characterizing the compatibility of <Hardware, Software> combinations,
the suitability of means and ends, or the empirical results of benchmark trials
for <Tool, Task> applications. Refining the binary values into numeric grades,
this may finally be the occasion for using something like certainty factors or
confidence ratings, so long as we make it clear what the "meaning in practice"
of these numbers might be.
o--------o------------o------o-----o o--------o------------o------o-----o
| Info | City | Code | Y/N | | (Info) | City | Code | Y/N |
o--------o------------o------o-----o o--------o------------o------o-----o
| | Amarillo | 806 | Y | | | Amarillo | 806 | N |
| | Ironwood | 906 | Y | | | Ironwood | 906 | N |
| | Washington | 202 | Y | | | Washington | 202 | N |
| | Detroit | 313 | Y | | | Detroit | 313 | N |
| | Galveston | 409 | Y | | | Galveston | 409 | N |
| | Lansing | 517 | Y | | | Lansing | 517 | N |
| | Cheboygan | 616 | Y | | | Cheboygan | 616 | N |
| | Houston | 713 | Y | | | Houston | 713 | N |
| | Ann Arbor | 806 | N | | | Ann Arbor | 806 | Y |
| | Boston | 001 | N | | | Boston | 001 | Y |
| | Chicago | 202 | N | | | Chicago | 202 | Y |
| | Detroit | 011 | N | | | Detroit | 011 | Y |
| | Escanaba | 409 | N | | | Escanaba | 409 | Y |
| | Flint | 101 | N | | | Flint | 101 | Y |
| | Galveston | 616 | N | | | Galveston | 616 | Y |
| | Houston | 111 | N | | | Houston | 111 | Y |
| | ... | ... | ... | | | ... | ... | ... |
o--------o------------o------o-----o o--------o------------o------o-----o
The discussion that we had above about three aspects of relational tables --
their conceived intensions, their complete extensions, and their current
actualizations -- may now be visualized against the backdrop of the twin
table representation. The extension of a table can now be understood in
both of the ordinary senses of the word. We imagine that we keep adding
examples to the table T, and keep adding counterexamples to the table (T),
approaching as a limit the complete extension of the parallel arrangement.
The actual or the ideal limit of this extension, exhausting the enumeration
of instances, would be information-equivalent to a realized intension of the
concept, and this comprehensive intension, or "comprehension", may be treated
as determining the whole empirical development of the entire conceptual scheme.
This is where it begins to get tricky. I introduce one of the axioms
for classical propositional calculus, provide it with an interpretation
in the form of a semantic integrity constraint that applies to the terms
for tables, and demonstrate how this principle governs the developmental
life-cycle of a typical relational database.
Given the syntactic conventions for propositional calculus laid down
so far, the principle of non-contradiction can be written as follows:
| T (T) = ().
|
| For example: People (People) = ().
Interpreted as a principle of logical or semantic integrity for the
twin table model of a database, this axiom means the following things.
Every expression in the propositional calculus must now be understood as
referring to a possible table. The expression "()" refers to a table that
should not "normally" have any entries in it. That is, it connotes a concept
that has no examples, only counterexamples. Its counterexamples belong to the
complementary table denoted by "(())". As a matter of fact, every possible data
item constitutes an example of this latter concept -- subject most likely to some
type restrictions on the universe of discourse -- its table has no counterexamples.
Based on these descriptions, it is easy to identify (()) as being nothing other
than the whole universe of discourse X. It is a table so under-constrained in
intension as to be all-inclusive in extension. On the other hand, () falls
under the heading of a "normally empty table", recognizable as signaling
an "exception" or "objection" condition, and perhaps implemented as an
Error Table ET. It is a table so over-constrained in intension as to
be devoid of any extension whatsoever under all normal circumstances.
A situation that forces the appearance of items in this table, in
effect, items defined as falling outside the "normal universe",
marks the end of the current "normal phase" of development in
the life-cycle of the database, and tells us that a phase of
problem-solving, remediation, and possible restructuring of
the whole knowledge base is now in order.
| Scholium. Thomas Kuhn's ideas about alternating phases of
| normal science and revolutionary science come to mind here.
| We might be looking at a very primitive archetype of a far
| broader dialectical process in the evolution of knowledge.
| Whether it turns out as a case of "revolutionary science"
| or "civil war science", however, seems to depend on which
| way the tables turn at the upshot of the restive interval.
With this preamble, I can now give a full interpretation of the
axiom "T ·(T) = ()". It says that, unless something very serious
has gone wrong, the tables T and (T) cannot have any common items.
In the phase-based implementation of this twin table paradigm, any
common items that do turn up, in spite of everything, must be placed
in the Error Table, ET = (). In operational terms, this inadvertent
state must generate a signal indicating the downfall of normal routine.
The abnormal or exceptional phase continues until the table of errors
can be reduced to nil. Returning to a state of intellectual peace
requires an intervening process of explanation seeking. In order
to be successful, the explanation that is ultimately found must
point out what specific features and what specific constraints
in the knowledge base have caused the unexpected clashes of
class indices to fall out in the exact way that they have.
Jon Awbrey
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