|
Adrian,
I believe the short answer is no
- it is a result of one of Godel's theorems. I guess the starting
point here is "Some Metamathematical Results on Completeness and Consistency",
1930:
"If to the Peano axioms we add the logic of
Principia Mathematica (with the natural numbers as the individuals) together
with the axiom of choice (for all types) we obtain a formal system S for which
the following theorem's hold:
I : The system S is not complete: that is, it
contains propositions A for which neither A or NOT A is
provable..."
I don't know the area well enough to say at which
point non-completenss comes in, whether it is the axioms of Principia
Mathematica or the axiom of choice, or even just Peano's
axiomatisation.
Regards,
Sean Barker,
Bristol
----- Original Message -----
Sent: Thursday, December 10, 2009 9:46
PM
Subject: Re: [ontolog-forum] Form and
content
Hi
Sean --
You wrote
...of course, you wouldn't be able to
axiomatise even the integers in many description logics, since they are
designed to prevent you describing problems that are
undecidable.
If that is
indeed so, it would seem to be shortcoming of those particular DLs.
There are simple recursive definitions of the
integers, and the question of whether an arbitrary string satisfies such a
definition is decidable.
Cheers, -- Adrian
Internet
Business Logic
A Wiki and SOA Endpoint for Executable Open
Vocabulary English over SQL and RDF
Online at www.reengineeringllc.com
Shared use is free
Adrian
Walker
Reengineering
On Thu, Dec 10, 2009 at 3:49 PM, sean barker <sean.barker@xxxxxxxxxxxxx>
wrote:
Adrian,
The great thing about numbers is there are so many different
sorts to choose from: Naturals, integers, rationals, reals, complex,
quarternons and so
on, not to mention the modulo groups, the transfinite numbers and
the floating point numbers (which, not being a metric space, make a mess of
most theorems about numbers). I suspect if you get into the hard maths, then
that will raise questions about the axiom of choice and whether you should
use it. And, of course, you wouldn't be able to axiomatise even the integers
in many description logics, since they are designed to prevent you
describing problems that are undecidable.
Sean Barker
Bristol, UK
*** WARNING ***
This message has originated outside your organisation,
either from an external partner or the Global Internet.
Keep this in mind if you answer this message.
Hi John, You wrote ...anybody who needs a
different ontology for numbers can use that instead.Somehow I'm
reminded of the saying "the nice thing about standards is that there are so
many of them to choose from" Seriously, if we can't agree on a
standard for something as basic as a number, what are the chances for
interoperability without the need for expensive and continuing human
intervention? To get a measure of how far there is still to go on
this journey, here's a real world problem description that serves as a nice
example of an interoperability requirement: http://www.economist.com/businessfinance/displaystory.cfm?story_id=15016132
Cheers, -- Adrian Internet Business Logic A Wiki and SOA
Endpoint for Executable Open Vocabulary English over SQL and RDF Online
at www.reengineeringllc.com Shared use is
free Adrian Walker Reengineering
Most people on this list understand the distinction
between form
and content, but some discussions tend to blur the
distinction.
I received an offline question related to that point, and
I thought
it might be useful to forward the answer to the
list.
John
Sowa
___________________________________________________________________
>
May I know what is the difference between semantic network
>
and ontology?
A semantic network is a graphical form for knowledge
representation.
It can be used to express content of any
kind.
An ontology is a formal definition of content, which could
be
represented in many different KR languages and notations.
The
distinction between form and content is critical, but some
KR languages
incorporate some ontology into the basic notation.
Therefore, they
would combine some content with the form.
For example, a temporal
logic is likely to have at least a minimal
ontology for time built into
the notation and rules of inference.
A KR language that includes
arithmetic will have an ontology for
numbers and operations on numbers
built into the basic language.
Common Logic is a version of logic
that is as neutral as possible
about ontology. For example, CL
includes syntax for numerals, but
it does not assume any axioms about
the integers represented by
those numerals. One application might
represent integers of
arbitrary length, but another might have an upper
limit of 2^63.
Some people have complained about the lack of an
ontology for
numbers in CL. But there are many different
ontologies that can
be added to CL as needed. One example is the
Mathematical toolkit
in the ISO standard for Z notation. It's a
fine ontology, it's
defined by an ISO standard, and anyone who wants it
can use it
in conjunction with CL. But anybody who needs a
different
ontology for numbers can use that instead.
John
Sowa
|