Equivalence between two domain expressions in OWL 2

Question

The text, Foundations of Semantic Web Technologies by Pascal Hitzler, Markus Krtzsch, and Sebastian Rudolph says on page 162.

And this answer says,

You can write p has range D as

⊤ ⊑ ∀pD

which says that ⊤ (or owl:Thing , ie, everything) is such that every one of its values for p must be a D . By using inverse properties, you can get domain axioms as well. p has domain C is equivalent to

⊤ ⊑ ∀p ^-1 .C

As we can see, the two expressions for range from the book and the answer match. I can not find how the expressions for domain are equivalent.

In the expression from the mentioned answer,

⊤ ⊑ ∀p ^-1 .C

if we apply ∀p to both the sides, we get,

∀p.⊤ ⊑ C

The above differs from the one given in the book.

What is it I could be missing here?

Answer 1

In Description Logics, as well as in most logic (if not all), it is possible to express the same thing in different ways. In particular, DL general concept inclusion axioms X ⊑ Y can be equivalently expressed in different ways, such as X ⊓ ¬ Y ⊑ ⊥ or ⊤ ⊑ ¬ X ⊔ Y . If X or Y use ∀ or ∃, then it is usually possible to convert the axiom to an equivalent one that use the other quantifier. One such example is:

∃p.⊤ ⊑ C

which is equivalent to:

⊤ ⊑ ∀p ⁻ .C

My understanding of your confusion is the following: you correctly remarked that if X ⊑ Y holds then ∀ r . X ⊑ ∀ r . Y holds too for any r and any (atomic or complex) concepts X and Y . So you start from ⊤ ⊑ ∀p ^-1 .C to derive ∀p.⊤ ⊑ ∀p.∀p ^-1 .C, which is a correct entailment. But you seem to assume that ∀p.∀p ^-1 .C is somehow equivalent to C, from which you conclude that ∀p.⊤ ⊑ C. This is not correct.

(As a side note, the classic notation for the inverse property of p is p ⁻ ).

You must understand that if an element e of the universe does not have a relationship r with anything, then it is a member of the class ∀ r . X (for instance, ∀hasChild.Human includes all those who do not have any children, so it includes at least every human beings). So, the class ∀p.∀p ⁻ .C contains all things that do not have a relation p with anything, which have no reason to be in C.QED.

I should add a proof that ∃p.⊤ ⊑ C is equivalent to ⊤ ⊑ ∀p ⁻ .C but I will leave this as an exercise to the reader;-)