 Since the consequence relation is defined by a fixed-point condition, there are default theories that have no extension at all, and other theories that have multiple, mutually inconsistent extensions. For example, the theory consisting of the fact \(p\) and the pair of defaults \((p\) ; \((q \amp r) \therefore q)\) and \((q\) ; \(\neg r \therefore \neg r)\) has no extension. If the first default is applied, then the second must be, and if the second default is not applied, the first must be. However, the conclusion of the second default contradicts the prerequisite of the first, so the first cannot be applied if the second is. There are many default theories that have multiple extensions. Consider the theory consisting of the facts \(q\) and r and the pair of defaults \((q\) ; \(p \therefore p)\) and \((r\) ; \(\neg p \therefore \neg p)\). One or the other, but not both, defaults must be applied. 