Re: Fwd: SUO Quo Vadis
Before going back to the foundations, a few questions and
comments about lattices.
Quoting "John F. Sowa" <sowa@bestweb.net>:
> > Lattice has always the minimal node. If some categories,
> > or nodes of the lattice are not comparable, the minimal
> > node describes absurdity or contradiction.
>
> There are two kinds of lattices to consider: a lattice
> of *theories*, which I discuss in the theories.htm paper.
> Another kind is a lattice of concept *types*. Neither
> kind has the slightest difficulty in handling those two
> points. Furthermore, these are two distinct lattices,
> and I would recommend *both*.
>
> For a lattice of types, see the section on lattices
> in my tutorial on math and logic:
>
> http://www.jfsowa.com/logic/math.htm#Lattice
JSLattice:
"All finite lattices are bounded, and so are
many infinite ones. In a lattice of subsets, the universal
set U itself is top, and the empty set {} is bottom"
That holds only for set theories that have the empty set,
but not for mereology and not for combined set theory that
has no empty set. The empty set is the 'ontological
embarrassment' of set theories. {} is a subset of every set
because it has the same 'empty member' than every other set.
That is conceptually incoheret. In combined set theory and
in mereology, A has to have a real actual common member with
B in order to be a proper subset of B.
JSLattice:
"In fact, the only graphs that are both trees and lattices
are the simple chains (which are linearly ordered)."
Also graphs of a generic type, graphs that have only
one node TOP=BOT, are both trees and lattices.
> For the lattice of theories, the top node consists of
> all propositions that are provable from the empty set of
> axioms and are true of everything. That node contains
> all tautologies and every other proposition such as
> "Every unicorn is a unicorn" that can be proved from
> zero axioms and zero assumptions about existence.
At first it seemed very unnatural to have nothing on the
TOP and more on the BOT, but the duality of intension and
extension is clear e.g. with RDFS ontologies; the TOP
usually does not have (many) properties, and the deeper a
category is, the more properties it has.
Still, it is easier to think nominalistically that TOP
describes all objects except itself TOP={redBall, redCar,
blueCar} and the successors of TOP describe less objects;
TOP>A,B>BOT where A={redBall,redCar} and B={redCar,blueCar}.
Visually:
TOP
/ \
A B
\ /
BOT
The intersection of A and B is BOT={redCar}. There we have
a lattice, but if TOP={a,b} where a and b are atoms, then
only a tree can be formed: {a,b}>{a},{b}. Visually:
{a,b}
/ \
{a} {b}
It is not intentional to force the empty set as the BOT.
Or can you imagine a good reason for it?
Good examples of the absurdity of BOT are e.g. the lattices
of "Figure 6: Lattice constructed by the method of formal
concept analysis" and "Figure 7: Revised lattice with new
attributes". I cannot understand the meaning of BOT in those
lattices. There is no use in forcing BOT in those cases.
Tree would be a coherent choice. And what do you loose by
using a tree instead of lattice? You loose only the difficulty
of having a category that describes absurdity or emptiness:
something that is both alcoholic and non-alcoholic.
> If you really insist on saying something about existence,
> I would be perfectly happy to include just one axiom in
> the top note: "There exists something." Such an axiom
> is not necessary in general, but there are some proof
> procedures that require an assumption of that form. But
> I would *not* give that something any name whatever or
> characterize it in any way.
That something exists is a perfect premiss for a priori
ontology, but as a premiss of a pragmatic ontology, it is
hard to say. If every category describes something that
exists as a premiss, then it is useless to state it again.
But yes, if TOP is only the place for tautologies, then
"something exists" belongs to TOP, because it is a tautology
as long as something exists. 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!
> At the bottom of the lattice is the absurd theory, which
> consists of all propositions that are provable from a
> contradiction. It contains *all* axioms and is true of
> nothing.
>
> > The third option is the only useful one: all categories
> > are comparable...
>
> To compare any two theories, you can navigate the lattice
> by using the four theory-revision operators. All of this
> is covered in the reference. Please read it. It's only
> 13 pages long:
>
> http://www.jfsowa.com/logic/theories.htm
I read it with pleasure, thank you. But the theory revision
operators do not answer the question: "why do we need a
BOT that is true of nothing?" What is a revison, expansion,
contraction or analogy of absurdity or something that does
not exist? Something that does not exist is not analogous
with anything, and what is analogous with something that
cannot be defined? Ockham's razor shaves absurdity and
emptiness.
> Regarding subjectivity: Please note that I did *not* use
> that word, and it is totally irrelevant to this discussion.
> I used the words "goal", "purpose", and "intention" --
> all of which are implemented quite nicely in chess-playing
> programs, which do not have any resemblance to human thinking.
>
> > Human cognition in given too much value here. I was
> > suggesting those kinds of a priori premises of ontology
> > that hold disregarding subjective humans.
>
> Without those concepts, we are discussing pure physics.
> If you're interested in that, I would highly recommend
> _The Road to Reality_ by Roger Penrose, which is an
> excellent summary of the math and physics underlying
> modern physics. It's an 1100-page tome, but it is not
> necessary to read it all at once. It is written in such
> a way that every mathematical concept is discussed with
> intuitively understandable examples, so it can be read
> by people who had studied math & physics many years ago
> and have forgotten everything they learned.
I'll see if it can be found from the library, after I've
finished Marx's evaluation of atomistic theories of
Demokritos and Epikuros. There is a great difference in
the atoms that the physicists talk about and the atoms
only as primitive indivisible elements.
> On the other hand, if you're interested in any kind of
> ontology that can be applied to anything having to do with
> society, government, engineering, business, medicine, law,
> or any other human activity, *YOU CANNOT IGNORE HUMANS*.
>
> My preference would be to use Peirce's terms Firstness,
> Secondness, and Thirdness. Those are purely objective terms
> that can be used to define and describe purposes, goals,
> and intentions. But if I can't get people to read a simple
> paper about lattices, I despair of getting them to read
> anything by Peirce.
So, the ideal pragmatic ontology is the kind that suits for
every human? I get the idea that it has the basic elements
and their relations, and these just have to be pragmatically
applied like Peirce's 123 is applied in various fields. Just
see the punctum, and connect it to the studium, connect
denotation to connotation, icon to index, a sign to
signification, 1 to 3 through 2. I have nothing against that,
but I'm not a great fan of the continental flavour in it.
The a priori ontolgy should be the kind that all 'intelligent'
beings reach the same one, not only humans. I'm interested in
hearing what you think about the relations of tautologies and
a priori thruths. Reaching an priori ontology and a pragmatic
ontology might walk hand in hand.
Avril