Re: Fwd: SUO Quo Vadis
Avril,
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.
> 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.
Mereology does not suffer from the Cantor-Russell
paradox for the simple reason that it doesn't include
this axiom.
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.
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.
> 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.
> 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.
> 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.
John