 Lukaszewicz (1988)  presents a modified definition of extension that avoids the previous two problems: it is defined in an iterative way and it warrants the existence of an extension. In a nutshell the idea is to keep track of the used justifications in the procedure. A default is applied in case its precondition is implied by the current beliefs and its conclusion is consistent with the given beliefs, its own justificiation, and each of the justifications of previously applied defaults. For normal theories Lukaszewicz’s definition of extensions is equivalent to the definitions from above. 