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

enter image description here

And this answer says,

You can write p has range D as

⊤ ⊑ ∀p.D

which says that ⊤ (or owl:Thing, i.e., 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?

Masroor
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    Henriette Harmse has a post explaining the issues introduced by universal restrictions and equivalences. You don't have a stated equivalence here but top on the left side of an inclusion has the same effect. https://henrietteharmse.com/2018/05/10/understanding-owl-universal-property-restrictions/ – Ignazio Sep 01 '22 at 11:57
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    What do you think "apply ∀p to both the sides" means? Regardless, ⊤ ⊑ ∀p-1.C does not imply ∀p.⊤ ⊑ C. In fact, it precisely implies (and is in fact equivalent to) ∃p.⊤ ⊑ C, so both _Foundations of Semantic Web Technologies_ and [the stackoverflow answer](https://stackoverflow.com/questions/21843521/owl-dl-property-restrictions-and-or-domain-range/21858504#21858504) are correct. – Antoine Zimmermann Sep 01 '22 at 19:55

1 Answers1

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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 ;-)

Antoine Zimmermann
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