[Top] [All Lists]

Re: [ontolog-forum] Foundation ontology, CYC, and Mapping

To: ontolog-forum@xxxxxxxxxxxxxxxx
From: "doug foxvog" <doug@xxxxxxxxxx>
Date: Thu, 4 Feb 2010 01:47:49 -0000 (GMT)
Message-id: <4905.>
> On Wed, 2010-02-03 at 13:36 -0500, Patrick Cassidy wrote:
>> Rob,
>> [RF] > > Examples of incompatible theories: the axiomatic set theories
>> > > of mathematics.
>> >
>> The math theories you mention may or may not be all describable in terms
>> of the same primitives - but I can't visualize the incompatibilities you
>> are
>> referring to (and decide whether they are not expressible by common
>> primitives) unless I can see the logical axioms that demonstrate a
>> logical
>> contradiction.  My first attempt to find them came up null.    (01)

> It is very easy to find examples: ZF set theory is incompatible with
> AFA, the theory that results from removing the axiom of foundation from
> ZF and replacing it with an anti-foundation axiom that asserts the
> existence of, e.g., self-membered sets.  Again, ZF + the axiom of choice
> is inconsistent with ZF + the axiom of determinacy.    (02)

> There is really only one axiom common to all set theories, viz.,
> extensionality: sets with the same members are identical and perhaps
> that is the only primitive principle you need.    (03)

The definition should start out that a set has members.  Restrictions
on the permissible types of members can identify different types of set,
or in given contexts may limit what is considered to be a generic set.
Sets are timeless; they cannot change members.  If an element which is not
in a set is added to a set, the result is a different set.    (04)

> That said, virtually all
> set theories in broad use seem to be based in a common set of intuitions
> and share a pretty solid axiomatic core -- Kripke-Platek set theory
> would probably be a good shot at that core: KP is basically ZF without
> the axioms of infinity and power set and with weaker versions of the
> axiom schemas of separation and replacement.  (See Barwise, _Admissible
> Sets and Structures_ to see KP in action (more exactly KPU, KP + "there
> are things other than sets")).    (05)

These axioms apply to different types of set.  If the FO is to include
the different set theories, then it would distinguish different subclasses
of fo:Set, e.g., fo:KPSet, fo:ZFSet, fo:KPUSet, fo:NFSet, etc.  The
different axioms would apply to the appropriate subclasses of set.  Then
mappings would be established between sets as defined in external ontologies
(e.g., sumo:Set) and the appropriate subclass of fo:Set.    (06)

> There are much sharper incompatibilities between ZF-style theories and
> theories based on Quine's NF (notably, in these theories, there is a set
> of all sets) but NF-style theories are not in wide use and are mostly of
> theoretical interest (see, e.g., Thomas Forster's excellent book _Set
> Theory with a Univerals Set).    (07)

Once it is recognized that the various types of set are different, but
related, concepts, they each can be defined using a common vocabulary
with constant meaning.  Axioms that apply to one type of set need not
apply to a different class.    (08)

It does not matter if different ontologies define their concept of Set
(or Line or Point) differently.  They just need to map their concept
to different concepts in the FO, or to specify rules that differentiate
the concept in their ontologies with those in the FO.    (09)

--  doug foxvog    (010)

> Chris Menzel    (011)

doug foxvog    doug@xxxxxxxxxx   http://ProgressiveAustin.org    (012)

"I speak as an American to the leaders of my own nation. The great
initiative in this war is ours. The initiative to stop it must be ours."
    - Dr. Martin Luther King Jr.
=============================================================    (013)

Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/  
Config Subscr: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/  
Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx
Shared Files: http://ontolog.cim3.net/file/
Community Wiki: http://ontolog.cim3.net/wiki/ 
To join: http://ontolog.cim3.net/cgi-bin/wiki.pl?WikiHomePage#nid1J
To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx    (014)

<Prev in Thread] Current Thread [Next in Thread>