Here the Meanings of a description logic is given by an interpretation. This interpretation consists out of a Discourse and an Interpretation function. The Interpretation Function itself interprets all atomic Concepts and all atomic Roles or Properties will be when interpreted as a Subset of the Cartesian Product of the Domain of Discourse with itself.
So one tries to map the description logics to a Set Theoretical model (Set Theory). In Set Theory one has a basic set which is the domain of discourse and one maps the atoms, Properties and Roles to Subsets of this domain of Discourse.
For example the Top Concept that represents all Individuals that are in the Domain of Discourse, i.e. the most upper class of which everything is Subclass of it. In a Set Theoretical way the Interpretation of the bottom concept must be the empty set. Then one can interpret all these operators, for example the Complement or Negation can be interpreted in a way that the complement of A, the interpretation of not A is the domain of Discourse without A.
For the universal quantification of Role
R with the Range
C means one considers all Individuals of the Domain of a from the Discourse of where all individuals of
b that are connected with the Individuals of
a via the interpretation of the Role or Property
R. For them it holds
b must be member of Class
On the other hand in the existential Quantification of
R the Range is arbitrary, so one considers all Individuals of
a from the universe of Discourse where there exists
bfor all pairs
b that are represented within Relation, the atomic Property of R.