A disjunctive Datalog knowledge base can have several incomparable minimal Herbrand models without a least Herbrand model. Take the single disjunctive fact
Both and are Herbrand models. Any least Herbrand model would have to be contained in both, hence contained in . But the empty Herbrand interpretation falsifies the disjunctive fact. This is a contradiction. Taking the intersection of all models therefore fails because that intersection need not itself be a model.