I'm struggling to understand the universal quantification operator in description logic. Here is an example of the problem:

given the DL concept

$\forall R.A $ (where $A$ is an atomic concept and $R$ is an atomic role)

and an interpretation $\mathcal{I}$ such that $\Delta^\mathcal{I} = \{a\}$ and $R^\mathcal{I} = \emptyset$ and $A^\mathcal{I}$ is empty or not , is it true that $(\forall R.A)^\mathcal{I} = \{a\}$?

More generally speaking, is it always the case that if $R^\mathcal{I}$ is empty then $(\forall R.A)^\mathcal{I} = \Delta^\mathcal{I}$? Is it a vacuous truth?


In Description Logic quantification restrictions define new concepts. The semantics of $\forall R.A$ defined so that $x\in (\forall R.A)^{\mathcal I}$ if and only if for each $y\in\Delta^{\mathcal I}$ where $(x,y)\in R^{\mathcal I}$ we have $y\in A^{\mathcal I}$.

With this definition one can check that your understanding is correct: for an empty role $R$ $\top \sqsubseteq \forall R.\bot$ (all things are related to no other thing through $R$). I dare say that few practical ontologies ever use empty roles as constructors.

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