I received an offline note with some objections to the lattice
of theories as a useful framework for the Foundation Ontology.
Following is a slightly edited version of my reply. (01)
John Sowa (02)
-------- Original Message -------- (03)
The lattice is a purely theoretical structure that embodies
all possible generalization and specialization relations among
theories. Every implementation of any special case is an
implementation of that theory. (04)
> I am not convinced that the lattice of all possible theories
> is the most efficient solution to the problem. (05)
That is like saying that you don't like integers because there
are infinitely many of them and some functions over the integers
are difficult to compute. (06)
The fact that the theory of lattices or the theory of integers
embodies a very wide range of useful relationships is good.
The fact that some relations may be hard to compute is not an
argument against the theory. If you don't need them, you don't
have to compute them. If you do need them, the lattice is not
a hindrance, and it can be a help. (07)
> I am concerned, for example, about the relation of the Cyc and
> SUMO and BFO and DOLCE. I don't think that any one of the
> relations applies to any two of those, as whole theories. (08)
Of course it applies. It says that they are cousins, not supertypes
or subtypes of one another. But the lattice also shows how to find
common supertypes: (09)
1. If you can find any subset of axioms that is common to all three
of them, it is automatically the axiomatization of a theory that
is a common supertype (or "core") of all three. (010)
2. If you can find axioms common to two out of the three, it defines
a common supertype of those two. (011)
3. If you can't find any common axioms (or can't find all you'd
like to find), you might find some set of simpler axioms that
imply different axioms in each of the three. That set of
simpler axioms is also a common supertype. (012)
4. Any core that you can propose is guaranteed to be a common
supertype of any theory derived by adding axioms to that core. (013)
This is an illustration of how the theory shows you how to think
about the problem. Any core you propose is going to belong to
cases #1, #2, #3, #4 above or some variation of them. Some of
the common axioms may be easier to find than others, but the
methods of testing them to see whether they are indeed common
are all based on the relationships embodied in the lattice. (014)
All of those techniques plus many others are implementations
of that theory. Some of them may be easier to implement than
others. But the lattice displays all the possible relationships
among theories. Any implementation of any subset of those
relationships counts as an implementation of the lattice. (015)
If you ignore the lattice, that is like playing with integers
without any theory about how they are related to one another. (016)
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