Semantics of Double Quantifiers in Description Logic Let's consider the following axiom from description logic where two
existential quantifies are applied to class $C$,
$\exists R.\exists S.C(a)$.
Where $R$, $S$ are relations (properties), and $C$ is a class
(concept).
Now, the quantifiers above may be interpreted in two ways:

*

*In the order of appearance of the relations:
$C$ class
instance $a$ is related via relation $R$ to an unknown individual
$x$. And then in turn $x$ is related via relation $S$ to a second
unknown individual $y$.

*In the order of proximity to the class: $C$ class
instance $a$ is related via relation $S$ to an unknown individual
$x$. And then in turn $x$ is related via relation $R$ to a second
unknown individual $y$.

As is found in the literature (from use-cases), the first
one appears to be the correct interpretation (semantics) of the double
quantifiers.
Could somebody please explain the reason behind the accepted
interpretation? For some, the accepted one may appear to be a bit
counter-intuitive.
Discussion on double mixed quantifiers like,
$\exists R.\forall S.C(a)$, $\forall R.\exists S.C(a)$ should make the
discussion comprehensive.
 A: 
Why the quantifiers above are interpreted in the order of appearance of the relations: $C$ class instance $a$ is related via relation $R$ to an unknown individual $x$. And then in turn $x$ is related via relation $S$ to a second unknown individual $y$?

The issue is the specific way the existential quantifier is interpreted in DL:

$(∃R.C)^{\mathcal I} = \{ x \in \Delta^{\mathcal I} \mid ∃y. (x,y) \in R^{\mathcal I} \text { and } y \in C^{\mathcal I} \}$,

where $\Delta$ is the domain of the interpretation and $\mathcal I$ the interpreting mapping.
Thus, $C$ is a concept (a "class term"), but also $(∃R.C)$.
Now we can try to unpack the example:

$∃ \text {hasParent}.∃ \text {hasFriend}.\text {Human}(\text {HarryPotter})$.

The concept $∃ \text {F}.\text H$ will mean $\{ x \mid ∃y. (x,y) \in F \text { and } y \in H \}$.
In (quite) plain text, the formula above specifies a concept: the class of those $x$ such that there is some $y$ such that "x is a Friend of y" and $y$ is Human.
If we now apply the outer existential quantifier, we are simply "specifying" (i.e. restricting) further the concept: $∃ \text {P}.(∃ \text {F}.\text H)$ will now mean $\{ x \mid ∃y. (x,y) \in P \text { and } y \in (∃ \text {F}.\text H)  \}$.
Now we can unpack it, replacing the meaning of the $∃ \text {F}.\text H$ concept in it:

the class of those $x$ such that there is some $y$ such that "x is a Parent of y" and $y$ is "a $(∃ \text {F}.\text H)$",

and in turn:

the class of those $x$ such that there is some $y$ such that "x is a Parent of y" and "y is a Friend of z", for some $z$ that is Human".

In conclusion, IF my reading is correct, the existential quantifier is used in DL in a different way that in usual FOL and in the example above we cannot swap them: a Parent of a Friend of Harry Potter is not necessarily a Friend of a Parent of HP.
