Folks, (01)
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: (02)
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. (03)
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. (04)
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. (05)
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. (06)
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. (07)
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. (08)
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: (09)
1. Equivalence: A1 -> A2 and A2 -> A1. The two sets of axioms
define exactly the same theory. (010)
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. (011)
3. Specialization: the inverse of generalization. A1 is more
specialized than A2 iff A1 -> A2 and not A2 -> A1. (012)
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. (013)
5. Greatest common specialization: the inverse of the least
common generalization. (014)
Two special cases are important: (015)
6. The universal theory U, which has no axioms whatever and is
true of every possible model. (016)
7. The absurd theory ~U, which is inconsistent and therefore
contains all possible statements in the given logic L. (017)
In terms of these concepts, we can define the following notions: (018)
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. (019)
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. (020)
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. (021)
John Sowa (022)
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