Re: SUO: OpenCyc Motion Open for Discussions
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
John Sowa wrote:
> As soon as add one axiom to the empty set,
> you have specialized your theory to another node
> farther down in the lattice. Then when you type
> a second axiom, you have specialized it to another
> theory that is a specialization of the previous one.
John, please explain to people that you are really
talking about a lattice of models and not strictly
speaking a lattice of theories.
For example, groups and abelian groups, the latter having exactly one additional axiom.
The lattice in which this axiom adding move takes you down is a lattice of models.
Every abelian group is a group. In other words, the set of abelian groups is
a subset of the set of groups, and this is what makes you say "down",
not the size of the corresponding theories.
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.
Jon Awbrey
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o