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Re: [ontolog-forum] (no subject)

To: Chris Partridge <mail@xxxxxxxxxxxxxxxxxx>
Cc: "'[ontolog-forum] '" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Pat Hayes <phayes@xxxxxxx>
Date: Fri, 13 Feb 2009 11:36:31 -0600
Message-id: <49098ADF-CD35-4E2A-9B73-54E79B74A626@xxxxxxx>

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.

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. 

http://en.wikipedia.org/wiki/New_Foundations  http://plato.stanford.edu/entries/quine-nf/ Intriguingly, NFU has ur-elements and a universal set – which, from what you say, you approve of.

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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