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

Peter F. Patel-Schneider answered to me:

>> What I would like to know is the following: What was the motivation to
>compare
>> OWL DL (a description logic) with OWL Full (an RDFS-based logic) on
>the basis
>> of common entailments?
>
>This comes from the RDF semantics, which concentrates on entailment.
>Because of this, it made the most sense to compare using entailments.

In general, I think, one cannot compare two arbitrary RDF-based languages
in this way.

I can build from RDFS a new language RDFS* by consistently swapping all
occurrences of the URIs 'rdfs:subClassOf' and 'rdfs:subPropertyOf'
in all semantic conditions of RDFS. The new language RDFS* will be
RDF-based, and will have the same vocabulary as RDFS. And since only
vocabulary names have been consistently substituted, the semantic
expressivity should not have changed. But since the syntax is now
interpreted differently for the two languages, they do not produce
the same entailments. For example:

   * { w type C . C subClassOf D } RDFS-entails { w type D },
     but this is /not/ an RDFS* entailment

   * { s p o . p subClassOf q } RDFS*-entails { s q o },
     but this is /not/ an RDFS entailment

So for our case of OWL Full vs. OWL DL: If OWL Full would /not/ fulfill
Theorem 2, this would /not/ necessarily mean that OWL Full cannot
compete with OWL DL w.r.t. semantic expressivity. Additional properties
of OWL DL and OWL Full have to be taken into account in order to be
allowed to draw such a conclusion.

>Note that description logics generally talk about particular inference
>services, like subsumption and classification, not general entailment.
>
>> I.e. why is there a criterion (Theorem 2) of the style:
>> "If the RDF graph O1 DL-entails the RDF graph O2, then O1 is also
>required to
>> Full-entail O2."? I would expect that such a criterion only makes
>sense, if
>> the semantic meaning of the syntactic expression O1 is the same both
>under OWL
>> DL and OWL Full semantics, and ditto for O2.
>
>The whole idea of Theorem 2 was to try to determine how close the
>semantic meanings were between OWL 1 DL and OWL 1 Full.

I know that this was the idea, and I strongly agree that we need a
criterion for this purpose. But I do not easily accept that the
current formulation of Theorem 2 is an adequate approach to compare
the semantic expressivity of OWL Full and OWL DL.

I restate my example of my previous mails here:

     G_L := {
         :alice rdf:type owl:Thing .
         :loves rdf:type owl:ObjectProperty .
         :alice :loves :alice .
     }

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

The graph G_L has a similar meaning in both OWL 2 DL and OWL 2 Full, which
can
be stated intuitively (I don't want to be too formalistic here) by:

    (M-G_L)
    "The individual Alice loves herself."

The meaning of the graph G_R is in OWL 2 DL given by:

    (M-G_R-DL)
    "Alice is an instance of the /set/ of all self-lovers."

But under OWL 2 Full semantics, we have the following meaning of G_R:

    (M-G_R-Full)
    "There exists some /individual/ in the universe,
    which is a class resource
    having the set of all self-lovers as its class extension,
    and Alice is an instance of this set."

These two meanings differ significantly: The former meaning
follows from the latter, but the converse is not true.

Further, the entailment "G_L|=G_R" means under OWL-DL semantics:

    (M-E-DL)
    "Whenever Alice loves herself,
    then Alice is an instance of the set of all self-lovers."

It's easy to believe that this is true, and this is definitely a desirable
conclusion, both for OWL DL and OWL Full. But under OWL Full semantics, the
same entailment (or better: the same expression "G_L|=G_R") means:

    (M-E-Full)
    "Whenever Alice loves herself, 
    then there has to exist some class resource,
    which has the set of all self-lovers as its class extension,
    with Alice being an instance of this set."

This is a much stronger demand than for OWL DL. It is a bit like whenever we
ask DL to provide us a glass of milk, then we expect Full to provide us a
cow.
While this is actually /sufficient/ for OWL Full to compete with OWL DL (we
can get the glass of milk from the cow, of course), it sets the bar
unnecessarily high for OWL Full.

Further, what Theorem 2 demands here, i.e. that the graph G_R must be
entailed
by G_L in OWL Full, is a very *undesirable* result. This result *is*
desirable
in OWL DL, where it only states the set assertion (S-G_R-DL). But in OWL
Full
the meaning of G_R is given by (S-G_R-Full). This is *not* something I would
ever expect to hold. Yes, Alice has to be an instance of the /set/ of all
self-lovers. But why should there be such an /individual/ in the universe,
which has this set as its class extension? I would certainly expect that
it is /satisfiable/ to have such an individual. But I would also expect that
it is satisfiable to /not/ have it. Having such an individual in
/every/ model is a very strong and counter-intuitive assumption, IMO.

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.

The asymmetric demands, which Theorem 2 puts on OWL DL and OWL Full
force OWL Full to include some very strong additional semantic conditions,
namely the comprehension principles. A comprehension principle for
self-restrictions will provide for every existing property p a class
/resource/, which has the set {x|p(x,x)} as its class extension.
Or more figuratively, comprehension principles provide me
/precautionary/ a cow for milk, a fountain for water, and a brewery
for beer, just for the case that I should ever become thirsty some
day. :) This doesn't only lead to undesirable results (as I stated
above), but also brings OWL Full near to its collapse, as one can see
from the problem with the Russell paradox.

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.

>> Btw: What do you mean by "the N3 way"?
>
>No syntax for OWL Full, no semantics for OWL Full.

I guess that this would make my job considerably easier. But I didn't
expect this job to become a walk in the park, anyway... :-)

Cheers,
Michael