SUO: Re: Examples! Examples! Examples!
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EEE. Note 2
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I am going to start with zeroth order theories,
because those are the kind that I know how to
analyze rather completely and also to express
in computable if not always tractable form.
It should be rather trivial for anybody working with
first order theories to re-write all of these examples
in their favorite language. If they do this, then it
would provide us with some concrete material to explore
the issues of logical transformation, including the
subtopics of (1) mappings between ontology theories,
(2) theory formation, (3) theory transformation,
(4) representation invariant ontologies (RIO's).
Example 1. John Sowa's "Top Level Categories"
The first example that comes to mind is John Sowa's
"Top Level Categories", in the version that is here:
http://www.bestweb.net/~sowa/ontology/toplevel.htm
There are several different way to express this content in my
favorite zeroth order language -- called the "Cactus Language"
because its strings are parsed into the species of graphs that
graph-theorists call "cacti". I will outline the first version
that I worked out and then translate it into ordinary syntax.
The "alphabet" !A! (also called the "basis", "lexicon", "vocabulary", ...),
is a finite set of strings, that can be regarded either as "sentences" or
as "terms" depending on the level of application. In the present example,
they appear to be the names of what some people call "distinctive features"
of things that exist, or things that are covered by the ontology. Here, !A!
is a set of 25 terms, !A! = {a_1, ..., a_25} = {"Abstract", ..., "Structure"}.
After this point, one tends to drop the quotation marks around strings, using
them only when necessary to avoid a clear and present danger of confusion.
| !A!
|
| a_1 = "Abstract"
| a_2 = "Actuality"
| a_3 = "Continuant"
| a_4 = "Description"
| a_5 = "Form"
| a_6 = "History"
| a_7 = "Independent"
| a_8 = "Intention"
| a_9 = "Juncture"
| a_10 = "Mediating"
| a_11 = "Nexus"
| a_12 = "Object"
| a_13 = "Occurrent"
| a_14 = "Participation"
| a_15 = "Physical"
| a_16 = "Prehension"
| a_17 = "Process"
| a_18 = "Proposition"
| a_19 = "Purpose"
| a_20 = "Reason"
| a_21 = "Relative"
| a_22 = "Schema"
| a_23 = "Script"
| a_24 = "Situation"
| a_25 = "Structure"
The "(finite) axiomset" $A$ for this example is just a (finite) set
of zeroth order axioms or propositional constraints. Inasmuch as we
can consider all of these propositions to be conjoined (conjuncted?)
into a single proposition, we are at liberty to view the whole thing
as a "one-axiom theory", if it makes us feel any better to do so.
What the axiom does is this. It removes from immediate consideration
particular regions of the universe of discourse that might otherwise
be thought to have content.
I use a percent sign suffix and square brackets around the alphabet to
denote the universe of discourse that is generated by a given alphabet,
writing A% = [!A!] = [a_1, ..., a_n]. The universe of discourse is
a two-layer object, consisting of the set of "positions" or "cells"
in the universe, written A = <|!A!|> = <|a_1, ..., a_n|>, and also
the set of "propositions", that is, the mappings from points in A
to the boolean domain B = {0, 1}, often pictured as various ways
of shading a venn diagram, written A^ = (A -> B) = {f : A -> B}.
Looked at this way, the universe of discourse for the TLC example
potentially has 2^25 positions and 2^(2^25) propositions, but the
axiomset $A$, which we may think of as singling out a single one
of these propositions, eliminates all but a certain number of
cells from consideration. This informational constraint can
be thought of as a type of quotient operation, whose result
we may indicate as A%/$A$, commonly read as "A% mod $A$".
To be continued ...
Jon Awbrey
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