Search code examples
first-order-logicdescription-logic

Understanding universal restriction of a concept in Description Logic (DL)

I am trying to understand the following Passage from a DL tutorial in terms of First Order Logic (FOL).

Passage

To represent the set of individuals all of whose children are female, we use the universal restriction

∀parentOf.Female (16)

It is a common error to forget that (16) also includes those individuals that have no children at all.

I take (16) to mean "if an individual has children, then those children are all female". My FOL representation of (16) is:

∀x∀y(parentOf(x,y) → Female(y)) (1)

My rational for this translation is that the implicit variable x represents the set of individuals being defined by the role parentOf. I assume x is universally quantified. The variable y represents female children, which I believe is called the successor of x in DL terminology, it is explicitly universally quantified in DL.

My FOL representation of "individuals that have no children at all" in FOL is:

∀x∀y ¬(parentOf(x,y)) (2)

My interpretation of the Passage, in FOL terms, is that if (2) holds then (1) holds. This is because the antecedent of (1) is false in this case.

Is my interpretation of the Passage correct?

Are my translations correct?

Solution

  • About your formula (1)

    If you say

    ∀x∀y(parentOf(x,y) → Female(y))

    or, equivalently

    ∀y((∃x parentOf(x,y)) → Female(y))

    you mean that every x can only have female children. But for stating this in DL you need concept inclusion, that is:

    ⊤ ⊑ ∀parentOf.Female

    that means "the range of role parentOf is included in concept Female".

    Concept and role inclusions are intensional assertions, i.e., axioms specifying general properties of DL constructs.

    Instead, DL's restrictions are not assertions, but constructs like concepts. So they are not used to state properties that are valid for every individual of the ontology. Like, when you say C⊓D, you are not stating that all the individuals of your ontology are instances of C and D, but you are simply "collecting" only the individuals that are instances of C and D at the same time.

    Therefore, the formula ∀parentOf.Female just wants to "catch" all the x such that, if x is parent of y, then y is Female. More formally, its semantics is the following:

    {x|∀y (parentOf(x,y) → Female(y))}

    About your formula (2)

    Analogously, the semantics of "individuals that have no children at all" is a set of individuals too:

    {x|∀y ¬parentOf(x,y)}

    or equivalently

    {x|¬∃y parentOf(x,y)}

    Indeed, you are collecting all the individuals that have no children, and not stating that all the individuals have no children.

    Conclusion

    You say right: "if (2) holds then (1) holds". The point is that neither (2) nor (1) (necessarily) hold.

    Since you are working with sets, you should not reason in terms of logical inference but of set inclusion.

    Therefore, the correct interpretation of the Passage is not

    if ∀x∀y ¬(parentOf(x,y)) then ∀x∀y(parentOf(x,y) → Female(y))

    but

    {x|∀y ¬parentOf(x,y)} is a subset of {x|∀y (parentOf(x,y) → Female(y))}