ONT Re: Inquiry Driven Systems
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| Document History
|
| Subject: Inquiry Driven Systems: An Inquiry Into Inquiry
| Contact: Jon Awbrey <jawbrey@oakland.edu>
| Version: Draft 8.70
| Created: 23 Jun 1996
| Revised: 06 Jan 2002
| Advisor: M.A. Zohdy
| Setting: Oakland University, Rochester, Michigan, USA
| Excerpt: Section 1.3.4 (Discussion of Formalization: Concrete Examples)
| Excerpt: Subsection 1.3.4.13 (Formalization of OF: Objective Levels)
|
| http://members.door.net/arisbe/menu/library/aboutcsp/awbrey/inquiry.htm
1.3.4.13 Formalization of OF: Objective Levels (cont.)
Another way to formalize the defining structure of an OG
can be posed in terms of a "relative membership relation"
or a notion of "relative elementhood". The constitutional
structure of a particular OG can be set up in a flexible
manner by taking it in two stages, starting from a level
of finer detail and working up to the bigger picture:
1. Each OM is constituted by what it means to be an object within it.
What constitutes an object in a given OM can be fixed as follows:
a. In absolute terms, by specifying the domain of objects that
fall under its purview. For the present, I assume that each
OM inherits the same object domain X from its governing OG.
b. In relative terms, by specifying a converse pair of 2-adic
relations that (redundantly) determine two sets of facts:
i. What is an instance, example, member, or element of what,
relative to the OM in question.
ii. What is a property, quality, class, or set of what,
relative to the OM in question.
2. The various OM's of a particular OG can be unified under
its aegis by means of a single triadic relation, one that
names an OM and a pair of objects, and that holds when one
object belongs to the other object in the sense identified
by the relevant OM. If it becomes absolutely essential to
emphasize the relativity of elements, one may resort, a bit
humorously, to calling them "relements", in this way jostling
the mind to ask: "Relement to what?"
The last and most likely the best way that one can choose to
follow by way of forming an objective genre G is to present
it as a triadic relation, in one of these fashions:
1. G = {<j, p, q>} c J x P x Q.
2. G = {<j, x, y>} c J x X x X.
For some reason the ultimately obvious method seldom presents itself
exactly in this wise without diligent work on the part of the inquirer,
or one who would arrogate the roles of both its former and its follower.
Perhaps this has to do with the problematic role of "synthetic a priori"
truths in constructive mathematics. Perhaps the mystery lies encrypted,
no doubt buried in some obscure dead letter office, due to the obliterate
indicia on the letters "P", "Q", and "X" inscribed above. No matter --
at the moment there are far more pressing rounds to make.
Given a genre G whose OM's are indexed by a set J and whose objects
comprise a set X, there is a 3-adic relation among an OM and a pair
of objects that exists when the first object belongs to the second
object according to that OM. This is called the "standing relation"
of the OG, and it can be taken as one way of defining and establishing
the genre G. In the way that 3-adic relations ordinarily give rise to
2-adic operations, the associated "standing operation" of the OG can be
regarded as a brand of assignment operation that makes one object belong
to another in a certain sense, namely, in the sense that is indicated by
the designated OM.
There is a "partial converse" of the standing relation that transposes
the order in which the two object domains are mentioned. This is called
the "propping relation" of the OG, and it can be taken as an alternate way
of defining the genre G, proceeding by way of its converse G^ in one of the
following manners:
1. G^ = {<j, q, p> in J x Q x P : <j, p, q> in G}.
2. G^ = {<j, y, x> in J x X x X : <j, x, y> in G}.
The following conventions are useful for discussing the set-theoretic
extensions of the staging relations and staging operations of an OG:
1. The standing relation of an OG is denoted by the symbol ":<-",
pronounced "set-in". Thus :<- c J x P x Q, or :<- c J x X x X.
2. The propping relation of an OG is denoted by the symbol ":>-",
pronounced "set-on". Thus :>- c J x Q x P, or :>- c J x X x X.
Often one's level of interest in a genre is "purely generic".
When the relevant genre is regarded as an indexed family of
2-adic relations, G = {G_j}, then this generic interest is
tantamount to having one's concern rest with the union of
all of the 2-adic relations in the genre.
|_|^J G = |_|^j G_j = {<x, y> in X x X : <x, y> in G_j for some j in J}.
When the relevant genre is contemplated as a triadic relation, G c J x X x X,
then one is dealing with the projection of G on the object domain dyad X x X.
G_XX = Proj_XX (G) = {<x, y> in X x X : <j, x, y> in G for some j in J}.
On these occasions, the assertion that <x, y> is in |_|^J G = G_XX
can be indicated by any one of the following equivalent expressions:
G : x -<- y, x -<G<- y, x -<- y : G,
G : y ->- x, y ->G>- x, y ->- x : G.
At other times explicit mention needs to be made of the interpretive
perspective or individual dyadic relation (IDR) that links two objects.
To indicate that a triple consisting of an OM j and two objects x and y
belongs to the standing relation of the OG, <j, x, y> in :<-, or equally,
to indicate that a triple consisting of an OM j and two objects y and x
belongs to the propping relation of the OG, <j, y, x> in :>-, all of the
following notations are equivalent:
j : x -<- y, x -<j<- y, x -<- y : j,
j : y ->- x, y ->j>- x, y ->- x : j.
Assertions of these relations can be read in various ways, for example:
j sets x in y.
j sets y on x.
j makes x an instance of y.
j makes y a property of x.
j thinks x an instance of y.
j thinks y a property of x.
j attests x an instance of y.
j attests y a property of x.
j appoints x an instance of y.
j appoints y a property of x.
j witnesses x an instance of y.
j witnesses y a property of x.
j interprets x an instance of y.
interprets y a property of x.
j contributes x to y.
j attributes y to x.
j determines x an example of y.
j determines y a quality of x.
j evaluates x an example of y.
j evaluates y a quality of x.
j proposes x an example of y.
j proposes y a quality of x.
j musters x under y.
j marshals y over x.
j indites x among y.
j ascribes y about x.
j imputes x among y.
j imputes y about x.
j judges x beneath y.
j judges y beyond x.
j finds x preceding y.
j finds y succeeding x.
j poses x before y.
j poses y after x.
j forms x below y.
j forms y above x.
In making these free interpretations of genres and motifs, one needs to
read them in a "logical" rather than a "cognitive" sense. A statement
like "j thinks x an instance of y" should be understood as saying that
"j is a thought with the logical import that x is an instance of y",
and a statement like "j proposes y a property of x" should be taken to
mean that "j is a proposition to the effect that y is a property of x".
These cautions are necessary to forestall the problems of intentional
attitudes and contexts, something I intend to clarify later on in this
project. At present, I regard the well-known opacities of this subject
as arising from the circumstance that cognitive glosses tend to impute
an unspecified order of extra reflection to each construal of the basic
predicates. The way I plan to approach this issue is through a detailed
analysis of the cognitive capacity for reflective thought, to be developed
to the extent possible in formal terms by constructing sign relational models.
By way of anticipating the nature of the problem, consider the following
examples to illustrate the contrast between logical and cognitive senses:
1. In a cognitive context, if j is a considered opinion that p is true,
and j is a considered opinion that q is true, then it does not have
to automatically follow that j is a considered opinion that p and q
are true, since an extra measure of consideration might conceivably
be involved in cognizing the conjunction of p and q.
2. In a logical context, if j is a piece of evidence that p is true,
and j is a piece of evidence that q is true, then it necessarily
follows by these very facts alone that j is a piece of evidence
that p and q are true. This is analogous to the situation where,
if a person j draws a set of three lines AB, BC, and AC, then j
has drawn a triangle ABC, whether j recognizes the fact at first,
on reflection, with extended contemplation, or never does at all.
Some readings of the staging relations are tantamount to statements
of (a possibly higher order) model theory. For example, the predicate
P : J -> %B%, defined by P(j) <=> "j proposes x an instance of y", is a
proposition that applies to a domain of propositions, at least, elements
with the evidentiary import of propositions, and its models are therefore
conceived to be certain propositional entities in J. And yet, all of these
expressions are just elaborate ways of stating the underlying assertion that
says that there exists a triple <j, x, y> in the relevant genre G.
Jon Awbrey
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