RE: ISSUE-119: What can be done against the Russell paradox?

Peter F. Patel-Schneider wrote:

>> When I want to construct an OWL 2 Full, which is at least as
>> expressive as OWL 2 DL, then I should only ask for all those results
>> which OWL 2 DL produces. I should not ask OWL Full for a cow, when I
>> ask DL only for a glass of milk. If DL provides me from some given
>> premise that Alice belongs to the /set/ of all self-lovers, then I
>> cannot expect more than exactly this from OWL Full, too.
>
>But this is precisely what Theorem 2 does.  In DL to ask whether Alice
>belongs to the set of all self-lovers one asks whether the KB entails
>Alice in SelfRestrition(loves).  In an RDF-based language, one asks the
>same question couched in RDF-speak, i.e., does an RDF graph OWL
>Full-entail an RDF graph stating that Alice is in the set of
>self-lovers.

This is not the same. While DL really just says that Alice is an instance of 
the set of all self-lovers, Full does not only make this set related 
assumption, but /additionally/ demands that there exists some individual in 
the universe, which represents that set.

Class axioms in DL directly refer to subsets of the universe. They are 
indifferent about how many individuals the universe contains, or what these 
individuals represent. In Full, however, every assertion is about individuals 
in the universe. In particular, when one talks about sets, one needs to have 
an individual in the universe which acts as a "proxy" for the respective set. 
In other words, making assertions about subsets of the universe has a side 
effect on the universe itself. This is a distinct semantic difference between 
DL and Full, and it is visible.

Having only a few of these "artifact" individuals wouldn't perhaps be a big 
deal. But the comprehension principles fill up the universe with infinitely 
many such proxy individuals, representing all kinds of specific sets. So with 
the comprehension principles, the "little semantic difference" between DL and 
Full becomes very big.

[...]

>> Bottom line: The "latent" collapse of OWL Full is not something which
>> is inherent to an RDFS-based language which tries to compete with
>> OWL DL w.r.t. semantic expressivity. Instead, this problem seems to
>> result solely from the strong and (technically) unfair demands
>> which Theorem 2 puts on OWL Full in comparison to OWL DL.
>
>I'm open to other ways of doing this, but I don't see any.

For the case of our self-lovers: Following Jeremy's idea to explicitly state 
the necessary comprehension principles as additional premises of an entailment 
seems save: If we have an OWL 2 Full with all comprehension principles, then

  {
	:alice :loves :alice
  } |= {
	:alice rdf:type _:s
	_:s rdf:type owl:SelfRestriction
      _:s owl:onProperty :loves
  }

holds IF AND ONLY IF

  {
	_:s rdf:type owl:SelfRestriction
      _:s owl:onProperty :loves
	:alice :loves :alice
  } |= {
	:alice rdf:type _:s
	_:s rdf:type owl:SelfRestriction
      _:s owl:onProperty :loves
  }

holds. So dropping the comprehension principles and adopting Jeremy's idea 
would *in this case* lead to the same comparison result as Theorem 2.

Intuitively, I regard this approach to "balance" the semantic side effect 
which additionally introduces a proxy individual into the universe. What 
remains is the information that Alice is an instance of the /set/ of 
self-lovers (the "essential" information which we are interested in). In fact, 
the comprehension principles also do such a balancing, but Jeremy's method 
just adds what is actually needed, and when it's needed. As a result, the 
universe doesn't get polluted so much with proxy individuals anymore.

I will have to check whether the observation above holds generally, and how to 
formulate a comparison criterion as an alternative to Theorem 2.

...after a few days of vacation. :)

Cheers,
Michael