Re: Fwd: SUO Quo Vadis
Avril,
Some comments on your comments:
> That holds only for set theories that have the empty set...
Yes, of course. In my lattice of theories, I put a theory
named Collections above Sets and Mereology.
For many applications, I would personally prefer a theory of
collections without a null element, but I would not use the
term "set theory" for a theory of collections without an
empty or null element.
> Also graphs of a generic type, graphs that have only
> one node TOP=BOT, are both trees and lattices.
Yes, but that graph, which is often called a _seed_, is also
a chain. Therefore, my previous statement is still true.
> Still, it is easier to think nominalistically that TOP
> describes all objects except itself
Three points:
1. A lattice of types would correspond to a lattice of
monadic predicates.
2. The lattice of theories is a lattice of closures of
sets (or collections) of axioms. It's related to #1,
but it's not identical to #1.
3. If you do define a lattice of types, why would you insist
that the predicate that is true of the lattice of types not
be in that lattice? Notice my example (Figure 2) of the
sample generalization hierarchy of theories, in which the
theory of theories is rather far down the hierarchy.
> And what do you lose by using a tree instead of lattice
> [in the FCA lattices]?
If you remove the bottom node, you would *not* have a tree
because it would have multiple paths from many nodes to
the top. It would, in fact, be an irregular graph that
would *almost* be a lattice. I have no idea why the bottom
node bothers you. It's an obvious addition that is needed
to preserve the duality of the infimum and supremum operators.
But if you don't like the bottom node, throw it out. I intend
to keep it in my lattices, but you can do anything you please.
If you have a complaint about FCA, discuss it with Wille, Ganter,
and the FCA gang.
> If every category describes something that exists as...
You definitely do *not* want to require every category to be
limited to just things that exist. For example, you might
want to design a new type of airplane that is a subtype of
a few existing types. But before you build your first exemplar,
nothing of that type exists. That is a very normal case in
any kind of design and development project.
> By the way, every a priori truth is a tautology, or at least
> quite close to a tautology. Your pragmatic TOP might contain
> the whole a priori ontology after all!
So what? It has no content and says nothing about anything.
> But the theory revision operators do not answer the question:
> "why do we need a BOT that is true of nothing?"
For symmetry. For the same kinds of reasons why it is useful
to have the number 0, which counts nothing.
> I'm interested in hearing what you think about the relations
> of tautologies and a priori truths.
I'm not a big fan of a priori truths. So if the only a priori
truths are the tautologies, then I'm happy.
> Reaching an priori ontology and a pragmatic ontology might
> walk hand in hand.
I doubt it, but if you want to search for such things,
be my guest.
John