extensions. Thus, the Nixon diamond example theory has two extensions, one in which Nixon is a pacifist and one in which he is not a pacifist. Consequently, neither nor are skeptically entailed, while both of them are credulously entailed. As this example shows, the credulous consequences of a default theory may be inconsistent with each other. Alternative default inference rules The following alternative inference rules for default logic are all based on the same syntax as the original system. Justified differs from the original one in that a default is not applied if thereby the set becomes inconsistent with a justification of an applied default; Concise a default is applied only if its consequence is not already entailed by (the exact definition is more complicated than this one; this is only the main idea behind it); Constrained a default is applied only if the set composed of the background theory, the justifications of all applied defaults, and the consequences of all applied defaults (including this one) is consistent; Rational similar to constrained default logic, but the consequence of the default to add is not considered in the consistency check; Cautious defaults that can be applied but are conflicting with each other (like the ones of the Nixon diamond example) are not applied. The justified and constrained versions of the inference rule assign at least an extension to every default theory. Variants of default logic The following variants of default logic differ from the original one on both syntax and semantics. Assertional variants An assertion is a pair composed of a formula and a set of formulae. Such a pair indicates that is true while the formulae have been assumed consistent to prove that is true. An assertional default theory is composed of an assertional theory (a set of assertional formulae) called the background theory and a set of defaults defined as in the original syntax. Whenever a default is applied to an assertional theory, the pair composed of its consequence and its set of justifications is added to the theory. The following semantics use assertional theories: Cumulative default logic Commitment to assumptions default logic Quasi-default logic Weak extensions rather than checking whether the preconditions are valid in the theory composed of the background theory and the consequences of the applied defaults, the preconditions are checked for validity in the extension that will be generated; in other words, the algorithm for