Re: Rich semantics and expressiveness

Bernard Vatant wrote:
> See http://www.w3.org/TR/2004/REC-owl-guide-20040210/#ComplexClasses
> In section 5.1.3
> "Therefore, a typical usage pattern for /complementOf/ is in combination 
> with other set operators"
> Which means (as the NonFrenchWine class shows) the typical usage is 
> relative complement, as in your examples below. Relative complements are 
> intersections, so you can't define relative complement if you have not 
> the notion of absolute complement. :-)

Isn't this backwards?  Assuming you have intersection it's clear that 
every system that has intersection and absolute complement must have 
relative complement.  As a counterexample to the other direction 
consider ordinary mathematics, which has relative complement by virtue 
of the Comprehension axiom (that those elements of a set satisfying a 
given predicate themselves form a set, i.e. sets and 
classes-defined-by-predicates intersect to form sets) but not absolute 
complement because then you'd leave yourself open to Russell's paradox.

One could amplify the remark "Usually this refers to a very large set of 
individuals" in 5.1.3 with the further remark that "The universal class, 
consisting of all individuals, can be defined as the complement of the 
empty class."

Judging from the examples one could replace absolute complement by 
relative complement in OWL-DL without inconveniencing anyone.  This 
would then bring OWL-DL into line with ordinary mathematical practice as 
well as making it more amenable to theoretical analysis and more 
tractable from both implementation and verification standpoints.

Perhaps more importantly it would make the existence of the universal 
class an option for an ontology.  This is because any ontology with 
relative complement has the universal class if and only if it has 
absolute complement.  (For the "if" direction, U = ~(C-C) where ~ is 
absolute complement, C is any class, and C-C is its complement relative 
to itself.)  Making absolute complement mandatory removes that option.

Vaughan Pratt

Received on Thursday, 22 February 2007 20:21:21 UTC