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Re: owl:Thing (was Re: Managing Co-reference)

From: Peter Ansell <ansell.peter@gmail.com>
Date: Thu, 22 May 2008 09:04:23 +1000
Message-ID: <a1be7e0e0805211604m3b0ac5b0h759a9b5fa03f2679@mail.gmail.com>
To: "Bijan Parsia" <bparsia@cs.man.ac.uk>
Cc: "Semantic Web" <semantic-web@w3.org>

2008/5/22 Bijan Parsia <bparsia@cs.man.ac.uk>:
> On May 21, 2008, at 10:17 PM, Peter Ansell wrote:
>
>> 2008/5/21 Bijan Parsia <bparsia@cs.man.ac.uk>:
>
> [snip]
>>>
>>> This isn't how they are defined. They are defined in terms of the model
>>> theory, to wit, in any interpretation (thus any model) owl:Thing contains
>>> all the elements of the interpretation (intuitively, all the individuals)
>>> and owl:Nothing is the empty set. Obviously, the universal set (or a
>>> given
>>> domain) and the empty set are complements, hence the tautologies/theorems
>>> you list above. You, of course, don't need such distinguished symbols,
>>> since
>>> you do have negation in OWL, you can always introduce them by a
>>> definition.
>>> Pick an arbitrary class name, C, then owl:Thing == (C or ~C) and
>>> owl:Nothing
>>> == (C and ~C). Obviously, once you have one, you could always define the
>>> other in the manner you list above.
>>
>> The problem I see might be more clear as follows:
>>
>> owl:Thing : {owl:Nothing ^ ~owl:Nothing}
>
> If "^" means "and" that's wrong. It should be "or"
>
>> owl:Nothing : {~owl:Thing}
>
> That's fine. I'm not sure what the braces mean.
>
>> substituting owl:Thing into owl:Nothing you get,
>>
>> owl:Nothing : {~{owl:Nothing ^ ~owl:Nothing}}
>
> Uhm. I think you mean owl:Nothing = ~owl:Nothing & owl:Nothing.
>
> That's fine.
>
>> I just don't see how that can be a consistent definition for an empty
>> set.
>
> Well, remember that in propositional logic you can substitute pretty freely.
>
> If C then C&~C.
>
> Is perfectly grammatical and meaningful. It can even be true (e.g., if C is
> false). It can't be true if C is true.
>
>> How can you define something as the complement of a set which
>> includes itself in it?
>
> It doesn't include itself. Equivalence isn't a membership relationship. it
> says that the two sides *have all the same members*.
>
> So, Nothing = ~owl:Thing should be read as:
>
> The class named "Nothing" contains, in every interpretation, exactly the
> individuals that are not instances of the class named "Thing". Since Thing,
> by definition, contains every individual, Nothing must be empty.
>
> Similarly, Nothing = Nothing and ~ Nothing should be read as:
>
> The class named "Nothing" contains, in every interpretation, exactly the
> individuals that are instances that are members of the class Nothing (well,
> we can stop here since there are no instances).
>
> Nothing = C and ~C
>
> The class named "Nothing" contains, in every interpretation, exactly the
> individuals that are instances of the class C and *not* instances of the
> class see. obviously there are no such instances, so Nothing is the empty
> set.
>
>> The definition may be in the model but it
>> appears inconsistent in this interpretation.
>
> Nope. See the simple model theoretic given above.
>
>> I am not big on playing
>> games with "empty sets" or "zero" to make things appear consistent as
>> they usually mask higher conceptual problems that (would/may) manifest
>> themselves when you actually get to utilising them.
>
> I don't know what you mean here. Nothing and Thing are very bog standard.
> And they are definable in the logic without reference to those symbols,
> e.g.,
>
> myThing = C or ~C (for some arbitrary class name "C")
> myNothing = C and ~C (for some arbitrary class name "C").
>
> Thus, strictly speaking, they are syntactic sugar. So they are not add ons
> or weird things, they are deeply part of the logic which is, after all, a
> fragment of first order logic.
>

I have a feeling I am mixed up with the idea of classes and instances
and their logic representations. Some of what you say doesn't make
sense to me, but you are very sure about it so I probably should do
some more reading. I get hung up on the idea that you even need to
define a special class for something which can never have any
instances.

Am I wrong in saying that you start off with nothing, and them
immediately use that to define everything, and then have subsets of
everything, except for in a case where the subset would be nothing and
it would then be sectioned off into its own world? It just doesn't
seem meaningful, even though it may be proved consistent once I
understand classes and individuals and instances etc.

Peter
Received on Wednesday, 21 May 2008 23:04:58 GMT

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