How to apply negation to a term?

Question

EDIT:

Maybe page 18 of this paper can help.

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This is a follow up from this question, where I am using ALCQ and I have this:

$\lnot(3 \le \exists \text{hasPet}.\text{Animal})$

How to apply the NOT operator? I thought about:

$3 \le \exists \text{hasPet}.\lnot\text{Animal}$

which actually works well for making the "every pet is an animal" false, but I am worried about the "he has at least 3 pets".

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My thought is driven by this example:

$\lnot(\forall \text{drinksCoffee}.\text{Frappe}) $

gives:

$\exists \text{drinksCoffee}.\lnot\text{Frappe}$

Answer 1

Comment

The expression ∀hasPet.Animal means:

the set of individuals that have pets that are animal.

It stay for a concept i.e. defines a set. See the interpretation:

$\{ \ a \in D^I \mid ∀b \ [(a,b) \in \text{hasPet}^I \to b \in \text{Animal}^I ] \ \ \}$.

Thus we can "negate" it: ¬(∀hasPet.Animal) getting the complement of the previous set:

$\{ \ a \in D^I \mid ∃b \ [(a,b) \in \text{hasPet}^I \land b \notin \text{Animal}^I ] \ \ \}$

that we symbolize as ∃hasPet.¬Animal.

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Usually the number restrictions: $\ge n$ R (at-least restriction) and $\le n$ R (at-most restriction), are "applied" to roles. The expression: $3 \le$hasPet stay for:

the set of individuals that have at least $3$ pets.

If so, what does it mean the expression ($3 \le$ ∀hasPet.Animal) where the number restriction is applied to a concept ?

I think that we have to read it as:

the set of individuals that have pets that are animal has at least three members.

I.e. the number restriction is on the members of the concept "hasPet.Animal" and not on the number of pets.