Re: SUO: OpenCyc Motion Open for Discussions
Jon,
The models and the theories are distict, but dual
mathematical structures. I was primarily talking
about the theories, but it is important to
acknowledge the relationship to the theories.
JA> John, please explain to people that you are really
> talking about a lattice of models and not strictly
> speaking a lattice of theories.
The theories are the intentions and the models are
the extensions -- they are two separate structures.
There is a duality between intensions and extensions,
which has been recognized in various forms since
the time of Aristotle.
JA> If I add another poorly chosen axiom, say, one
> that contradicts the three for a group and the one
> more for abelian, then I get an empty set of models,
> hitting bottom in model space, but in theory space,
> every available sentence in now a theorem.
Yes, that's the duality. The most general theory,
which has the exmpty set of axioms, has the largest
possible collection of models -- every model.
The most specialized theory at the bottom, the
absurd theory, which includes all possible
propostions, has the empty set of models.
I was talking about theories because they are
the things that represent ontologies. The models
are the things that the ontologies are about.
Both are important.
John