Using ∃ instead of ∀ in some statements

Question

In Description Logics sometimes we use ∃ to express "all those concepts" such as in the example below :

"the concept PlasticRoof is defined as the intersection of the concept Roof and the concept of all those things that are made of plastic"

PlasticRoof ≡ Roof ⊓ ∃madeOf.Plastic

We used here the existential restriction ∃ but what the statement says is that "all those things that are made of plastic". Why don't we use the universal restriction ∀ here since "it expresses that a propositional function can be satisfied by every member of a domain of discourse."

Answer 1

The distinction is between concepts that represent classes (or sets) of individuals and roles, i.e. (binary) relations between individuals.

Thus, Roof and PlasticRoof are concepts [ in FOL: $\text {Roof}(x)$ and $\text {PlasticRoof}(x)$ ], while madeOf is a (binary) relation [ $\text {madeOf}(x,y)$ ].

If we denote with $R^I$ the interpretation of the relation $R$, in Description Logic the semantics of $(\exists R.C)$ is : $\{ x \mid \exists y \ (x,y) \in R^I \text { and } y \in C \}$.

Thus, $(\exists R.C)^I$ is a set, and so $\exists R.C$ denotes a concept.

This is why the expression $D \sqcap \exists R.C$ is syntactically correct: it means the "intersection" of the sets denoted by two concepts, i.e. the set of those "objects" that satisfy the concept $D$ and the concept $\exists R.C$.

Thus, the formula amounts to the following "equivalence":

$\text {PlasticRoof}(x) \equiv \text {Roof}(x) \land \text {madeOf}(x, \text {Plastic})$ .