Re: Fwd: SUO Quo Vadis
John, let's finish this before Christmas. Please
address the issues.
Quoting "John F. Sowa" <sowa@bestweb.net>:
> Reflexivity or even self reference does not, by itself,
> create any paradox or infinite regress. There's nothing
> wrong with writing an autobiography to describe yourself.
Yes, but then you would be writing about yourself, not about
_how_ you write about yourself. And if you write about how
you write about yourself, you don't write about how you
write about how you write about yourself. It is plain that
self reference does create an infinite regress. You must
accept this.
But a regress is not the same as a paradox. A round
square is a paradox, a regress is a regress. A paradox
might lead to a regress.
> > That category cannot describe all theories, because
> > then it would describe itself, which causes a vicious
> > infinite regress.
>
> Cantor's paradox (AKA Russell's paradox) concerns the
> set of all sets that are not members of themselves.
>
> What causes the paradox is *not* self reference or
> reflexivity. The paradox is created by the following
> axiom, which is used in one form or another in most
> versions of set theory:
>
> For any monadic predicate P, there exists a set S,
> which consists of all x for which P(x) is true.
But there is an obvious self reference.
S={x|x~x}
S is the set of all sets that are not members
of _themselves_.
The axioms were created for set theories in order to avoid
paradoxes in the first place. Please create a paradoxical
set by using the axioms of e.g. ZF, BNG, KPU, CST.
> Mereology does not suffer from the Cantor-Russell
> paradox for the simple reason that it doesn't include
> this axiom.
All that can be done with mereology can be done with Boolean
algebras. That is why I'm suspicious about your antipathy
towards set theory and sympathy towards Lesniewski's mereology.
A good page about the history of the field: http://www.ltn.lv/~podnieks/gt2.html
Boole lived 1815-1864. Lesniewski was born in 1886, and created
mereology that is isomorphic with Boolean algebra, that was
created by Boole before Lesniewski was even born!
This is what I read form the above link:
----
The first paradox in Cantor's set theory was discovered already in 1895
by Cantor himself. Cantor did not publish about the problem, he only
communicated about it with David Hilbert. Hilbert proposed his own simpler
version of a paradox, and, after this, around 1900, Hilbert's Goettingen
collaborator Ernst Zermelo made one more step - he simplified Hilbert's
paradox, thus, in fact, inventing Russell's paradox before Russell! Still,
this discovery remained "Goettingen folklore" until Russell's publication
in 1903. The creation of the de facto set theory, ZF, is mainly credited
to Zermelo and Fraenkel (1921) in addition to Cantor.
----
Mereology is simply a set of atoms, without the brackets
if you wish.
In mereology a,b is_a_subset_of a,b,c
In ZF/KPU/CST/ {a,b} is_a_subset_of {a,b,c}
> The Liar paradox is caused by the use of a very specific
> kind of self reference:
>
> This sentence is false.
>
> But consider the following example:
>
> This sentence is true.
>
> You can safely assume that sentence is true without
> causing any paradox.
Yes, of course you can assert tautologies about everything, like
"everything that exists, exists", and "every possible thing is
divisible or indivisible" but a theory must explain something.
You have a hierarchy of theories. All those theories are members
of the set of the theories. Theories={t1,t2,t3,...,theory,...}.
Now, you also have a theory called 'theory' that describes,
suggests, or explains something. I ask, what does that theory
stand for, what does it explain or describe? It cannot be a
tautology, since only TOP contains tautologies.
> Gödel's famous paper on undecidability is based on
> a sentence of the following form:
>
> This sentence is true, but unprovable.
>
> Gödel's ingenious construction was to demonstrate that
> a sentence of this form about ordinary arithmetic could
> be true.
May peace be upon Gödel.
> > the actual definition of an absurd BOT is impossible.
>
> The definition is trivial: the absurd theory is the
> deductive closure of (p and not p) where p is any
> proposition whatever. If you are using first-order
> logic, that closure consists of every well-formed
> sentence in FOL.
>
> > When we force a lattice within an actual domain
> > ontology, then we have to separate the three meanings
> > of BOT: absurdity, emptiness, and a normal category.
>
> The problem exists in your own mind. Nobody said that you
> have to do any such thing. So just mediate on the matter
> until the desire goes away.
Ok, let's forget this issue, I'll meditate that swamp of
contradictions away.
> > What benefit does it give to have this sort of BOT?
> > Why do you want to keep it?
>
> Because it simplifies the structure, it preserves the duality
> of intension and extension, it means that the infimum and
> supremum operators apply to any pair of theories, and
> most of all, it does not cause any problems whatever.
>
> > ... slightly amusing, but mostly painful.
>
> A dog might bite you, but a mathematical structure won't.
> The fact that the theories may be organized in a lattice
> allows anybody who wants to use lattice operators to do so.
> But if you don't like lattices or don't want to use them,
> just ignore them, and the pain will go away.
I *do* especially like lattices of categories, but I don't like
the absurd type. I'll use lattices, but I'll have only those
kinds of categories that describe something that exists. I
consider two theories to be contradictory by _not imposing
a common subcategory for them_. You impose BOT as a symbol
of contradiction, I use BOT as a normal category.
> > What about this formulation: when a being realizes a
> > category, the category describes something that exists.
>
> No. You're trying to assume the same kind of axiom that
> causes problems in set theory.
>
> There are very good reasons for *not* assuming that axiom:
> It's often unknown or unprovable whether something of type
> X exists, and in order to resolve that issue, you must be
> able to talk about X.
I agree, you must be able to talk about it with a language.
The meaning of category is the hardest issue here.
If there are things outside of the scope of the present
languages, these things must be discluded from the ontology
because it is impossible to describe them. According to
Armstrong, there are no negative properties."X is something
that is outside of the capabilities of human understanding"
is not a genuine property. Another example of a negative
property is "X's length is not 28.01 meters".
A person realizes a category, e.g. a commercial about a
perfect Christmas. At the same moment, the category describes
something for that person: a perfect Christmas. If the perfect
Christmas exists somewhere e.g. in the form of a good Christmas
atmosphere inside a house in Russia, then it exists that way,
and it exists also as an idea in the person's mind. If the
perfect Christmas does not exist in the form of a good
Christmas atmosphere inside a house, then it exists at least
in the person's mind. If nobody ever percepts the category,
then the category potentially describes a variety of things
to some agents.
Avril