ONT Re: Apposite Purposes Of Logical Languages Objectified (APOLLO)
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Yet another topic that is far too elusive for us to take up now (YATTIFTEFUTTUN),
would be to worry too much about the ontological status of the abstract objects
and hypostatic abstractions that we are calling "lattices" and "partial orders".
When our concerns with the so-called "real world" are not too pressing, we will
tend to treat them as terminal objects, and they will serve as the main objects
of "discussion and thought" (DAT) for many an hour at a time, but it is equally
true that we are ultimately interested in these objects because they capture by
way of analogy and example properties of real world situations in which we find
ourselves also caught. Taken in this role, the abstract objects are serving as
signs, specifically, as icons, of the actual instances and the objective states
of affairs that constitute our primary, our real, and our long-term pragmata.
In order to accommodate as many different styles of thinking as possible,
I have already developed a range of different ways of rationalizing how
to place these hypostatic objects within an ontological hierarchy, and
I can bring that into play if this problem ever amounts to an obstacle.
Let me go back to the "hypostatic abstraction" diagram and revise
it in a way that will exhibit both the category theoretic and the
sign theoretic relationships that are involved in the situation.
Here is more or less how the diagram looked before,
using only slightly different labels, like Peirce's
optional term "subjectival" instead of "hypostatic".
| Subjectival
| Abstraction
| o
| ^ \
| / \
| / \
| / \
| / \
| / v
| Real World o------------>o Formal Calculus
Let us now reconfigure this diagram in a way that comports
a little better with Aristotle's picture of analogy. Here,
the light lines represent relations of logical implication,
the wavy line represents a sign theoretic denotation, and
the heavy lines represent category theoretic arrows. Thus,
the Comprehension is the logical conjunction of properties
that are shared by the real world application domain and
the abstract model domain. A given member of the family
of formal calculi that we are considering may be taken
whole cloth as a sign that denotes this Comprehension.
Moreover, given a certain latitude of interpretation,
we may take this whole formal calculus as denoting
also either the Abstraction or the Application.
These relationships can be made a lot clearer
in a category theoretic setting, but this is
enough to illustrate the general idea.
|
| Comprehension o<~~~~~~~~~~~~o Formal Calculus
| ^ ^ > ^
| / \ % %
| / \ % %
| / % \ %
| / % \ %
| / % \ %
| Application o% % % % % % >o Abstraction
That should be enough to justify, or at least to make excuses for,
the level of abstraction and generality that I'll be needing here.
Next time I return to the level of concrete and specific examples.
Jon Awbrey
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