Hi Pat,
Not sure what happened to the old subject line – have added
one that is reasonably informative.
A few small comments:
PH>You took my rhetorical question differently than I
intended. I was making a point only about which discipline should guide our
choice of technical vocabulary. If a word (like "individual") has one
meaning in logic and knowledge representation, and a different meaning in
metaphysics and statistics, then I would have thought it was generally agreed
here that the former would be the meaning that an unguarded use of the word was
intending to convey, when talking to this forum.
I have a slightly different view. I think the ontology
enterprise is heterogeneous (multi-disciplinary) and to claim that it is the solely
the business of logic and knowledge representation (which you may not have
been) is likely to hamper progress. In general, I would try to be inclusive.
Where we have trouble is where there is a good body of relevant knowledge in
one discipline and we need to appraise other disciplines of it – this can
be difficult. As you know, if we were going to argue priority, then I would
vote for metaphysics over logic.
CP>One of the things that continues to surprise me it that
the current ZF contain only sets. This seems to be the outcome of
mathematicians desire to avoid any contact with the real world at the beginning
of the last century.
PH>No, it arose from the desire of mathematicians to
produce a "pure", minimal, set theory as a foundation for
mathematics, in a 50-year attempt to rescue something from the ashes of
Hilbert's programme. Which, by the way, while a fascinating and deep topic, IMO
has very little to do with our ontological business in this forum.
I do not see why you say ‘no’ – aren’t
we saying the same thing? Wouldn’t a ‘"pure",
minimal, set theory’ ‘avoid any contact with the real [material]
world’? Isn’t the empty set a good example of something with no
contact to the material world?
PH>On the other hand, if you are asking me to speak as an
ontologist, of course I want to distinguish ur-elements from sets.
"Ur-element" here just means something that isn't a set, and of course
I want to be able to talk about sets of anything.
Doesn’t one need to adjust ZF if your universe includes
set of ur-elements? (The article mentions this - Axiomatizations
of set theory that do invoke urelements include Kripke-Platek set theory with
urelements, and the variant of Von
Neumann–Bernays–Gödel set theory described in Mendelson (1997:
297-304). In type theory, an object of type 0 can be called an
urelement; hence the name "atom.").
So if one wants ur-elements one would need to go for these
adjusted theories.
PH> But "ur-element" is a very odd term for
on ontologist to use.
Which is perhaps why someone plumped for an equally odd term ‘Individual’.
Chris
From: Pat Hayes [mailto:phayes@xxxxxxx]
Sent: 13 February 2009 17:37
To: Chris Partridge
Cc: '[ontolog-forum] '
Subject: Re: [ontolog-forum] (no subject)
On Feb 12, 2009, at 4:35 AM, Chris Partridge wrote:
Many thanks for taking the trouble to make such a full answer.
Like and agree with much of what you say, but a few points
Can I start with your last comment.
PH> Priority??
And aren't we, in this forum, talking about logics (in a broad sense, ie
formalisms for description) and KR, rather than statistics or metaphysics?
I appreciate that this is your (and others) view.
However, there is another view (and another view of logic) which
I think John was espousing in an earlier set of emails (in relation to
Aristotelian syllogisms), which is that logic is a formalism for describing the
way the world is – or more grandly, what exists. And that in some way the
form of the logic reflects the structure/nature of the world.
A colleague pointed out to me something you may be familiar
with, “ARISTOTLE'S LOGIC: A COMPARISON OF LUKASIEWICZ'S AND
CORCORAN-SMILEY'S RECONSTRUCTIONS”
Though this is not exactly the point we are discussing, it
illustrates the kinds of tensions that can arise between the ‘formalisms for description’ and
‘formalism for describing the way the world is’.
However, I expect we will just have to agree to disagree.
You took my rhetorical question differently than I intended.
I was making a point only about which discipline should guide our choice of
technical vocabulary. If a word (like "individual") has one meaning
in logic and knowledge representation, and a different meaning in metaphysics
and statistics, then I would have thought it was generally agreed here that the
former would be the meaning that an unguarded use of the word was intending to
convey, when talking to this forum.
With respect to the intended meaning of individual (a point you
raised), we were trying to find out what the OWL sense was. It is explicitly
mentioned several times in the specification. See some extracts below.
Yes, of course, I should have checked the text before
replying. This is all in the introduction rather than the normative content,
and I am virtually certain that every usage of "individual" here uses
it in the sense I outlined, i.e. to mean "an element of the universe of
discourse". Note that this is a rather ticklish point for the OWL
docs as OWL-Full and OWL-DL contemplate quite different universes of discourse,
the latter being severely restricted compared to the former. The URI
owl:sameIndividualAs, in particular, was introduced explicitly for OWL-DL use,
since OWL-DL requires there to be a class/property/Individual trichotomy.
We definitely were not using it in its metaphysical sense.
1.
Introduction (Informative)
…
This
document contains two formal semantics for OWL. One of these semantics, defined
in Section
3, is a direct, standard model-theoretic semantics for OWL ontologies
written in the abstract syntax. The other, defined in Section
5, is a vocabulary extension of the RDF semantics [RDF
Semantics] that provides semantics for OWL ontologies in the
form of RDF graphs. Two versions of this second semantics are provided, one
that corresponds more closely to the direct semantics (and is thus a semantics
for OWL DL) and one that can be used in cases where classes need to be treated
as individuals or other situations that cannot be
handled in the abstract syntax (and is thus a semantics for OWL Full). These
two versions are actually very close, only differing in how they divide up the
domain of discourse.
Appendix
A contains a proof that the
direct and RDFS-compatible semantics have the same consequences on OWL
ontologies that correspond to abstract OWL ontologies that separate OWL individuals,
OWL classes, OWL properties, and the RDF, RDFS, and OWL structural vocabulary. Appendix
A also contains the sketch of
a proof that the entailments in the RDFS-compatible semantics for OWL Full
include all the entailments in the RDFS-compatible semantics for OWL DL.
Finally a few examples of the various concepts defined in the document are
presented inAppendix
B.
- [26 June 2003] Per a
decision of the Web Ontology working group on 26 June 2003 to replaceowl:sameIndividualAs with owl:sameAs,
recorded in http://lists.w3.org/Archives/Public/www-webont-wg/2003Jun/0364.html,
made changes to Section
2.2, Section
3.3, Section
4.1, Section
4.2, Section
5.2, andAppendix
A.1.
- [30 June 2003] Fixed a
bug in the semantic conditions for owl:hasValue noticed
by Jeremy Carroll, changing the conditions for the value from a property
to an individual or a data value in Section
5.2.
- [23 July 2003] In
response to a substantive post-last-call change to the RDF semantics,
changing the if-and-only-if conditions for rdfs:subClassOf and rdfs:subPropertyOf to only-if conditions, added
if-and-only-if conditions forrdfs:subClassOf,
over OWL classes, and rdfs:subPropertyOf, over OWL individual-valued
properties and over OWL datatype properties, to Section
5.2.
- [22 July 2003] In
response to http://lists.w3.org/Archives/Public/public-webont-comments/2003Jul/0011.html andhttp://lists.w3.org/Archives/Public/public-webont-comments/2003Jul/0041.html,
changed several uses of ``object'' to ``individual''
or ``individual-valued'' in Section
2 and Section
5.2 and made other
editorial changes to Section
5.2.
I agree that a logician may not like the term individual –
I prefer element – but in it what is intended in the ordinary language
sense. You may prefer ur-element.
I don't believe that there is a single ordinary language
sense of "individual".
PH> It is not a
metaphysical classification: it does not separate the ontic universe into two
kinds of thing, one kind more 'individuated' than the other. (Speaking
personally, now, I have never understood what such a distinction could possibly
mean.)
I think, in logic, it may be the distinction between ur-elements
and sets.
No. Not in logic, at any rate, which does not even recognize
this as a logical distinction. After all, consider: ZF set theory is
axiomatized in logic.
One of the things that continues to surprise me it that the
current ZF contain only sets. This seems to be the outcome of mathematicians
desire to avoid any contact with the real world at the beginning og the last
century.
No, it arose from the desire of mathematicians to produce a
"pure", minimal, set theory as a foundation for mathematics, in a
50-year attempt to rescue something from the ashes of Hilbert's programme.
Which, by the way, while a fascinating and deep topic, IMO has very little to
do with our ontological business in this forum.
But “The Zermelo set theory of 1908 included urelements.
It was soon realized that in the context of this and closely related axiomatic
set theories, the urelements were not needed because they can easily be modeled
in a set theory without urelements. Thus standard expositions of the canonical
axiomatic set theories ZF and ZFC do not mention urelements.” http://en.wikipedia.org/wiki/Urelement
Quite. This clearly reflects the use of set theory to
provide a consistent foundation for mathematics: Hilbert's programme.
Have you come across NFU?
Hah! It was the first set theory I ever read, in fact:
my introduction to set theory itself, as an undergraduate. Talk about a baptism
of fire. In comparison, ZFC was like drinking a milkshake.
But NFU is appallingly complicated, subtle and delicate to
use, compared to ZFC. Im not even sure its relative consistency has been fully
investigated. I think of it as an Edsel among set theories now.
Could you live with an urelement / set distinction?
Im not sure what you are asking. Speaking as a judge of
rival set theories, I see no reason at all to not follow the mathematical herd
and stick to ZFC (but allowing ur-elements, of course.) If I need a rival
one, I'll use Axelrod's nonwellfounded set theory which has been shown to be
relatively consistent with ZFC. But I see no real purpose in re-hashing these
old debates about sets. Sets are now a thoroughly explored topic, and the
overwhelming consensus among mathematicians is that ZFC provides the best -
most secure, most useful, most thoroughly investigated, the nearest anyone is
ever going to get to a "standard" - account of sets. For virtually
all ontology purposes, we can get by (as 99% of working mathematicians do) with
a naive set theory plus the very occasional appeal to the axiom of choice. Set
theory is simply not an interesting topic for us to be discussing on this
forum. We have better things to do.
On the other hand, if you are asking me to speak as an
ontologist, of course I want to distinguish ur-elements from sets.
"Ur-element" here just means something that isn't a set, and of course
I want to be able to talk about sets of anything. Just as Russell and Zermelo
and Quine all the other pioneers did. Note that ZFC does not prohibit
ur-elements: it just ignores them. But "ur-element" is a very odd
term for on ontologist to use. It reminds me of my favorite example of a
high-level classification, found on a Burger King package, which divides up the
universe into "Double cheeseburger" and "Other"
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