Universal quantification of OWL

Question

I am half reading OWL2 primer and I have a problem understanding universal quantification

Given example

EquivalentClasses( :HappyPerson ObjectAllValuesFrom( :hasChild :HappyPerson ) ) 

It says that someone is a happy person, for sure, if all their children are happy people. But what if John Doe has no children, he can be an example of HappyPerson? What about his parent?

I also find this part very confusing, she says:

Therefore, according to our statement, every childless person will be qualified as happy.

but won't break the constructor of ObjectAllValuesFrom ()?

Answer 1

I think primer really explains this pretty well, especially the following:

Natural language indicators to use universal quantitative assessment - the words "only", "exclusively" or "nothing but".

To simplify this, consider the expression you provided:

HappyPerson ≡ ∀ hasChild . HappyPerson

This suggests that HappyPerson is the one who only has children, who are also HappyPerson (also happy). Logically, this actually says nothing about the existence of cases of happy children. It simply serves as a universal restriction for any children that may exist (note that this includes any instances of HappyPerson that have no children).

Compare this to the existential quantifier , there is (∃):

HappyPerson ≡ ∃ hasChild . HappyPerson

This suggests that HappyPerson is the one who has at least one child, who is also HappyPerson . According to (∀), this expression actually implies the existence of a happy child for each instance of HappyPerson .

The answer, although initially unintuitive, lies in the interpretation / semantics of the ObjectAllValuesFrom OWL construct in first-order logic (actually, Description Logic). In essence, the ObjectAllValuesFrom construct refers to the logical universal quantifier (∀) , and the ObjectSomeValuesFrom construct refers to the logical existence quantifier (∃) .

Answer 2

I am facing the same issue when reading the “OWL 2 Web Ontology Language Primer (Second Edition - 2012)”, and I'm not sure if Sharkey's answer clarifies this issue.

On page 15, when introducing the universal quantifier ∀, the book says: “Another property restriction, called universal quantification, is used to describe a class of individuals for which all related individuals must be instances of this class. We can use the following statement to indicate that someone is a happy person, exactly if all their children have happy faces. " [I omit the OWL instructions in different syntaxes, which can be found in the book.] I think that a more formal and perhaps less ambiguous presentation of what the author claims is

(1) HappyPerson = {x | ∀y (x HasChild y → y ∈ HappyPerson)}

I hope that every reader understands this notation, because I believe that the notation used in the answer is less clear (or maybe I'm just not used to it).

The book continues: "... There is one specific misconception regarding the universal limitation of the role. As an example, consider the axiom of happiness above. Intuitive reading suggests that in order to be happy, a person must have at least one happy child [my note: in fact, the definition says that every child must be happy, and not just at least one, so that his / her parents are happy. This, apparently, is the author’s laps]. However, this is not so: any person who not from the right point "of the hasChild property is a member of the class of any class defined by universal quantification by hasChild. Therefore, in accordance with our above statement, each childless person will be qualified as happy ..." That is, the author claims that (suppose '~' for logical NOT ), Considering

(2) ChildessPerson = {x | ~ ∃y (x HasChild y)}

then (1) and the value ∀ mean

(3) ChildessPerson ⊂ HappyPerson

This is not like me. If this were so, then every child, since he / she is a childless person, is happy, and therefore only some parents can be unhappy people.

Consider this model:

Faces = {a, b, c}, HasChild = {(a, b)}, HappyPerson = {a, b}

and c is dissatisfied (regardless of the near world or the assumption of an open world). This is a possible model that falsifies the thesis of the author.