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

From: Peter Ansell <ansell.peter@gmail.com>
Date: Thu, 22 May 2008 07:17:38 +1000
Message-ID: <a1be7e0e0805211417l5366b160u6b2d3dd46341e09b@mail.gmail.com>
To: "Bijan Parsia" <bparsia@cs.man.ac.uk>
Cc: "Semantic Web" <semantic-web@w3.org>

2008/5/21 Bijan Parsia <bparsia@cs.man.ac.uk>:
>
> Hi Peter,
>
> On 21 May 2008, at 12:58, Peter Ansell wrote:
>
> [snipped what I couldn't follow]
>
>> [http://www.w3.org/2002/07/owl]
>>
>> <Class rdf:ID="Thing">
>>  <rdfs:label>Thing</rdfs:label>
>>  <unionOf rdf:parseType="Collection">
>>    <Class rdf:about="#Nothing"/>
>>    <Class>
>>      <complementOf rdf:resource="#Nothing"/>
>>    </Class>
>>  </unionOf>
>> </Class>
>>
>> <Class rdf:ID="Nothing">
>>  <rdfs:label>Nothing</rdfs:label>
>>  <complementOf rdf:resource="#Thing"/>
>> </Class>
>
> 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}
owl:Nothing : {~owl:Thing}

substituting owl:Thing into owl:Nothing you get,

owl:Nothing : {~{owl:Nothing ^ ~owl:Nothing}}

I just don't see how that can be a consistent definition for an empty
set. How can you define something as the complement of a set which
includes itself in it? The definition may be in the model but it
appears inconsistent in this interpretation. 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.

>> Sorry if this sounds like uninformed rambling and I am missing an
>> important point why these two Classes need to obliquely reference each
>> other using complement in this way as the basis for a consistent
>> ontological system.
>
> I hope I've clarified!
>
> In many logics, the distinguished symbols are known as Top and Bottom, or
> \top and \bot
>
> (see: <http://www.artofproblemsolving.com/LaTeX/AoPS_L_GuideSym.php#others>)
Received on Wednesday, 21 May 2008 21:18:19 GMT

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