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Re: Universal Quantification

From: Pat Hayes <phayes@ihmc.us>
Date: Wed, 5 May 2004 15:23:31 -0500
Message-Id: <p06001f0cbcbef9f85f2c@[10.0.100.76]>
To: "Stephen Rhoads" <stephen.rhoads@db.com>
Cc: ewallace@cme.nist.gov, www-rdf-logic@w3.org

>stephen.rhoads@db.com wrote:
>>Can someone please explain the meaning and consequences of the following
>>in layman's terms:
>>
>>"Also notice that the correspondence of owl:allValuesFrom with the
>>universal quantifier means that an owl:allValuesFrom constraint for a
>>property P is trivially satisfied for an individual that has no value for
>>property P at all. To see why this is so, observe that the
>>owl:allValuesFrom constraint demands that all values of P should be of
>>type T, and if no such values exist, the constraint is trivially true."
>
>It is an easy mistake for a non-logician to assume that
>owl:allValuesFrom implies owl:minCardinality 1 (or owl:someValuesFrom). 
>After all, intuitively shouldn't allValuesFrom just be a more restricted
>version of someValuesFrom? The text quoted above makes clear that this is
>not the case.  If you want to say that any instance of some specified class
>must have some property values of a certain type AND only values of that
>type, then you must explicitly make both restrictions.
>
>An example:
>
>:Quote1
>     a owl:Class ;
>     rdfs:subClassOf
>	[ a owl:Restriction ;
>	  owl:onProperty :effectivePeriod ;
>	  owl:allValuesFrom :Duration
>	] .
>
>is not equivelant to
>
>:Quote2
>     a owl:Class ;
>     rdfs:subClassOf
>	[ a owl:Restriction ;
>	  owl:onProperty :effectivePeriod ;
>	  owl:allValuesFrom :Duration
>	] ;
>     rdfs:subClassOf
>	[ a owl:Restriction ;
>	  owl:onProperty :effectivePeriod ;
>	  owl:someValuesFrom :Duration
>	] .
>
>In a Quote1 instance, the presence of Duration value is optional while it is
>mandatory in a Quote2 instance.

Right. Another way to think about it is to ask what would it would 
take to make the claim false. To falsify  an allValuesFrom you would 
need to exhibit a Value that was not From the class. But if there 
were no values at all, then you couldn't possibly do that, so you 
couldn't possibly falsify it, so (following, er, a logician's logic) 
it has to be considered to be true. In other words, asserting an 
allValuesFrom is consistent with there being no values at all.

The phrase 'trivially satisfied' is a logician's codeword for cases 
like this, where the normal intuitions are kind of run off the tracks 
by taking something to a (logically possible) extreme, such as in 
this case there being nothing there at all. If its any consolation, 
in just about every such case, people have tried to make logics which 
capture intuition better, and they always break in some worse way, 
and one can get used to the apparently unintuitive cases fairly 
quickly. In this case, the great advantage of having things be this 
way is that it makes complementOf work properly, which may not be 
obvious until you try to use it.
(If you have no kids, are all your children graduates of Yale? In 
English you can say that is a dumb question, but logic has to say 
either yes or no, and yes makes more sense than no does, because if 
you say no then it would follow that one of your kids wasn't a Yale 
graduate, but you have no kids...)

If that didn't help, forget it.

Pat Hayes

>-Evan
>
>Evan K. Wallace
>Manufacturing Systems Integration Division
>NIST
>ewallace@nist.gov


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Received on Wednesday, 5 May 2004 16:24:52 UTC

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