We define a new notion of truth for logic programs extended with labelled variables, interpreted in non-Herbrand domains. There, usual terms maintain their Herbrand interpretations, whereas diverse domain-specific computational models depending on the local situation of the computing device can be expressed via suitably-tailored labelled models. After some introductory examples, we define the theoretical model for labelled variables in logic programming (LVLP). Then, we present both the fixpoint and the operational semantics, and discuss their correctness and completeness, as well as their equivalence.