Re: Universal Quantification - common misconception

On 22 Mar 2011, at 11:18, Niall Murphy wrote:

> Hello,
> 
> I am confused by the statement in the OWL 2 Primer that "any individual that is not a “starting point” of the property hasChild is class member of any class defined by universal quantification over hasChild.",

That's clumsily put.

Rephrase: In order for *ALL* your children to be happy, it suffices that you have NO children at all.

This is sometimes known as "vacuously satisfying the restriction".

> referring to this statement.
> 
> EquivalentClasses(
>     :HappyPerson
>     ObjectAllValuesFrom( :hasChild :HappyPerson )
> )

The way to see this is to consider the following class:

EquivalentClasses(
	:ChildlessHappyPerson 
	ObjectAllValuesFrom(:hasChild owl:Nothing)
)

It follows that ChildlessHappyPerson SubClassOf: HappyPerson.

> My understanding (or attempt thereof) is that the any individual that is not not a "starting point" of the property hasChild can be expressed as follows:
> 
> EquivalentClasses(
>     :NotStartingPointOfHasChild
>     ObjectComplementOf(
>         :ObjectSomeValuesFrom(:hasChild :Person)
>      )
> )

Nope. At least, not by itself. You could still have children that are not persons. (Assuming the right additional axioms are in play, that might suffice).

> And the class of individuals defined by universal quantification over hasChild as follows:

There are potentially many such classes.

> EquivalentClasses(
>     :UniversalQualificationOverHasChild
>     ObjectAllValuesFrom(:hasChild QualoverHasChild)
> )

Bit garbled there. I think my example is clearer.

> So because NotStartingPointOfHasChild has no individuals who are members of :UniversalQualificationOverHasChild,

Er...if things came out as in my example, it SHOULD have members who are UniversalQualificationOverHasChild. That's the point.

> or to put it another way, all none of them are related to UniversalQualificationOverHasChild by hasChild,

This is off the rails. We're not talking about the successors.

> they all meet the restriction and are therefore members.
> 
> If the above is true, does the solution of including an ObjectSomeValuesFrom(:hasChild :HappyPerson) restriction in the example class expression prevent entailment of a paradox, or have I just been spending too long looking at OWL semantics?

Nothing at all to do with paradox! The main reason for this somewhat curious condition is to ensure certain closure conditions for negation. I.e., that negated existentials always have a corresponding universal. 

If you define (P only C) as not(P some not C) (in the standard way, it's clear that you can fit the second expression either by having ALL your Ps be Cs or by having no Ps at all.

Hope this helps.

Cheers,
Bijan.

Received on Wednesday, 23 March 2011 09:56:36 UTC