Hi John, (01)
Thanks for your useful comments on the orthogonality of ontologies! I
have been working for quite a while in the area of discovering
various types of relationships for mapping axioms across different
ontologies, and have seen many interesting types of relationships. I
am doing exactly what you have mentioned in 4 below - that is,
clustering various ontologies by formulating a semantic distance
between the axioms. (02)
However, I often find that the relationships do not fall into the
five neat categories you have enumerated. They are showing more fuzzy
relationships - such as similar contexts that have been used for
defining two classes (by way of similar restrictions, etc.) or
prototypical patterns of defining certain classes of axioms. The
axioms are related, but in looser or more subtle senses. These
relationships are useful in disambiguating the contextual aspects for
the concepts across ontologies or formulating templates for extending
an ontology or reuse of axioms across ontologies. (03)
Is there a way to fit these types of observations as formal
relationships into your lattice theory? I wonder if you have had such
experiences with your graph matching algorithms. (04)
Thanks for your insights!
Mala (05)
Mala (06)
At 02:06 AM 3/11/2008, John F. Sowa wrote:
>Folks,
>
>I agree with Bill Andersen's point that ontologies are not vectors
>and therefore the notion of orthogonality does not apply.
>But I'd like to add some additional comments:
>
> 1. The term 'orthogonal' has been widely used in a metaphorical
> sense, and I'll admit that it is tempting to continue using it
> in the sense of some sort of logical independence.
>
> 2. However, it is extremely dangerous to use technical terms in
> a metaphorical sense in technical fields in which they are not
> naturally defined -- especially because it may become useful to
> use them in a new technical sense in which they are applicable.
>
> 3. For example, since the early 1960s, vector methods have been
> used in information retrieval to define a *semantic distance*
> measure between documents. In that case, two documents that
> have nothing in common (as defined by that measure) could truly
> be called 'orthogonal' because the cosine of the angle between
> their two vectors would be 0 -- indicating a right angle.
>
> 4. It's conceivable that similar measures could be used to compute
> the semantic distance between the axioms of two different
> ontologies. If so, the term 'orthogonal' should be reserved
> for such techniques, and it should be avoided as a metaphorical
> term.
>
> 5. However, there also are logic-based methods for defining
> semantic relationships between different ontologies that are
> much more relevant to most of the work with ontologies.
>
>These method are closely related to what I've been discussing in
>terms of lattices of theories, but the methods are useful even
>without bothering to organize the theories in a lattice.
>
>The basic operator is material implication, which I'll represent
>by an arrow: ->. Given two sets of axioms A1 and A2 stated in
>some logic L, following are the basic relationships:
>
> 1. Equivalence: A1 -> A2 and A2 -> A1. The two sets of axioms
> define exactly the same theory.
>
> 2. Generalization: A1 is more general than A2 iff A2 -> A1
> and not A1 -> A2. Every application (or model) of A2 is
> also a model of A1, but A1 may have applications (or models)
> that are not applications of A2.
>
> 3. Specialization: the inverse of generalization. A1 is more
> specialized than A2 iff A1 -> A2 and not A2 -> A1.
>
> 4. Least common generalization: The set of axioms A3 define
> a theory T3 that is the least common generalization of the
> theories defined by A1 and A2 iff A1 -> A3, A2 -> A3, and
> there is no set of axioms A4 that is more specialized than
> A3 for which A1 -> A4 and A2 -> A4.
>
> 5. Greatest common specialization: the inverse of the least
> common generalization.
>
>Two special cases are important:
>
> 6. The universal theory U, which has no axioms whatever and is
> true of every possible model.
>
> 7. The absurd theory ~U, which is inconsistent and therefore
> contains all possible statements in the given logic L.
>
>In terms of these concepts, we can define the following notions:
>
> 8. Logical independence: Two theories A1 and A2 are said to be
> logically independent iff their least common generalization is
> the universal theory U. This property implies that the truth
> or falsity of one set of axioms is completely unrelated to
> the truth or falsity of the other. In other words, for any
> model M, the truth or falsity of A1 on M has no implications
> for the truth or falsity of A2 on M.
>
> 9. Contradiction: Two theories A1 and A2 are contradictory if
> their greatest common specialization is the absurd theory ~U.
> In other words, there is no model M in which both A1 and A2
> are true.
>
>When talking about sets of axioms, I suggest that we use these
>nine terms instead of vague notions of orthogonality: equivalence,
>generalization, specialization, least common generalization,
>greatest common specialization, universal theory U, absurd theory ~U,
>logical independence, and contradiction.
>
>John Sowa
>
>
>
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Mala Mehrotra
Pragati Synergetic Research Inc. MS 19-46Q, NASA Research Park,
Moffett Field, CA 94035
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