[protege-discussion] Complement of classes

[Thomas Russ]

On Aug 26, 2010, at 11:44 AM, Marcelino Sente wrote:

> Which is the difference?
> Forall hastopping ~MeatTopping
> ~(Exists hastopping MeatTopping)

These are equivalent.

This can be proven by appropriate application of de Morgan's laws for  
disjuction and conjuction, their extension to existential and  
universal quantifiers, the definition of implication and the meaning  
of the restrictions.

Briefly, these items can be rewritten in a predicate calculus form as  
the following:

    ForAll x: hasTopping(a,x) -> ~MeatTopping(x)
   ~[Exists x: hasTopping(a,x) ^ MeatTopping(x)]

If you start from the first one, you can transform it into the second:

    ForAll x: hasTopping(a,x) -> ~MeatTopping(x)
    ForAll x: ~hasTopping(a,x) v ~MeatTopping(x)    ; Def'n of ->
    ForAll x: ~[hasTopping(a,x) ^ MeatTopping(x)]   ; de Morgan
    ~[Exists x: hasTopping(a,x) ^ MeatTopping(x)]   ; de Morgan