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Re: [ontolog-forum] Foundation ontology, CYC, and Mapping

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "John F. Sowa" <sowa@xxxxxxxxxxx>
Date: Sat, 13 Feb 2010 12:30:32 -0500
Message-id: <4B76E1B8.1070804@xxxxxxxxxxx>
Cory,    (01)

I'd like to give some examples that may clarify that issue:    (02)

CC> ... foundational concepts are very similar to the "minimally
> axiomatized micro theories" I remember John-S describing as
> a workable foundation, yet John does not see primitives
> as workable - why the difference?    (03)

My objection to using "primitives" as a foundation is that the
meaning of a primitive changes with each theory in which it occurs.
For example, the term 'point' is a "primitive" in Euclidean geometry
and various non-Euclidean geometries.  But the meaning of the term
'point' is specified by axioms that are different in each of those
theories.    (04)

Note that there are two kinds specifications:    (05)

  1. Some terms are defined by a *closed form* definition, such as    (06)

          '3' is defined as '2+1'.    (07)

     In a closed-form definition, any occurrence of the term on the left
     can be replaced by the expression on the right.    (08)

  2. But every formal theory has terms that cannot be defined by a
     closed-form definition.    (09)

For example, both Euclidean and non-Euclidean geometries use the
term 'point' without giving a closed-form definition.  But calling
it undefined is misleading because its "meaning" is determined by
the pattern of relationships in the axioms in which the term occurs.    (010)

The axioms specify the "meaning".  But the axioms change from one
theory to another.  Therefore, the same term may have different
meanings in theories with different axioms.    (011)

For example, Euclidean and non-Euclidean geometries share the
same "primitives".  The following web site summarizes Euclid's
five "postulates" (AKA axioms):    (012)

    http://www.cut-the-knot.org/triangle/pythpar/Fifth.shtml    (013)

The first four are true in Euclidean and most non-Euclidean
geometries.  By deleting the fifth postulate, you would get
a theory of geometry that had exactly the same "primitives",
but with fewer axioms.  That theory would be a generalization
of the following three:    (014)

  1. Euclid specified a geometry in which the sum of the three
     angles of a triangle always sum to exactly 180 degrees.    (015)

  2. By changing the fifth postulate, Riemann defined a geometry
     in which the sum is < 180 degrees.    (016)

  3. By a different change to the fifth postulate, Lobachevsky
     defined a geometry in which the sum is > 180 degrees.    (017)

This gives us a generalization hierarchy of theories.  The theories
are generalized by adding axioms, specialized by deleting axioms,
and revised by changing axioms (or by deleting some and replacing
them with others).    (018)

I have no objection to using collections of vague words, such as
WordNet or Longman's, as *guidelines*. But the meanings of those
words are ultimately determined by the axioms, not by the choice
of primitives.    (019)

Note to RF:  Yes, the patterns of words in NL text impose strong
constraints on the meanings of the words.  That is important for
NLP, but more explicit spec's are important for computer software.    (020)

John    (021)

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