Re: Fwd: SUO Quo Vadis
John, I hope we can reach a consensus here.
> 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.
You could classify collection systems in many ways:
-axiomatic or non-axiomatic (natural)
-existence of the least element
-existence of atoms (one or many types)
-existence of axiom schemas
-existence of classes
-existence of memberOf operator
Cantor's idea was just to have the hierarchy of infinities,
but I'm sure he didn't wish for incoherence. Ok, empty set
is practical in that task, but there can be hierarchies of
infinities without the least element too. Empty set, as it
is in ZF, causes misunderstandings and is conceptually
incoherent.
I'd put mereology under the category of Boolean algebras,
and all collection formalisms that have the memberOf
operator under the category Set theories. But this is not
a very important issue.
> Three points:
>
> 1. A lattice of types would correspond to a lattice of
> monadic predicates.
That is clear.
> 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.
I don't quite understand what you mean by a closure of
axioms. Closure of a collection X is X with all collection
formalisms that do not have the memberOf operator, like
in mereology. However, with KPU and CST the transitive
closure of set {a,{b}} is {a,b,{b}}.
> 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.
But even that Theory cannot explain how it explains something.
That sort of a belief is similar to the belief that a man
washes himself. A quote from D.M.Armstrong's Theory of
Universals 1978 19.VI:
One hand washes another; both wash the rest of the body.
Perhaps the trickiest sort of case is that where a person
loves or hates himself. But even here genuine self-relation
seems avoidable. If a man loves himself, then it is not
that self-loving state which he loves, but other aspect
of himself. It is possible that he should love the self-
loving state, but this seems to demand a new, second-
order, loving state which is distinct from the original
one.
Our own headmaster Ilkka Niiniluoto just reminded me last
friday that Armstrong's credit is that he took the discussion
to a higher level.
> > 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.
Ok, but I could manage with that irregular graph as well.
Frame-based ontologies can also be irregular graphs, if they
have a multiple inheritance system of categories (classes).
Frame-based ontologies can also be lattices, but I believe
that this is very unusual.
> 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.
It bothers me because of the incoherence. "T > Circle, Square"
is a tree. In lattice "T> Circle, Square > BOT", BOT describes
a contradiction, a round square.
> > 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.
Of course, but in that case the category would describe something
that exists inside the brains of a human being. And if that
person dies or forgets what the category describes, the category
would be like a circle in sand in some deserted island.
> > 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.
It can say a lot about e.g. the principles of classification,
that also a pragmatic ontology must follow. I'll get back to
you with this issue.
> > 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 think that lattice drives the ontologist to find what all
things have in common, which is great. But that analogy only
proves that BOT is very problematic: you cannot divide by 0.
Forcing a lattice causes BOT to have three possible meanings:
1) it describes proper parts of Substance similarly to all
other categories.
2) it describes contradiction (round square): 1/0
3) emptiness: 0/1.
> > 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.
I'll get back to this issue after some evaluation.
Avril