The growing size of the multiprocessor system increases its vulnerability to component failures. The fault diagnosis is the process of identifying faulty processors in a system through self-testing, and the diagnosability is an important parameter to measure the reliability of an interconnection network. As a new measure of fault tolerance, conditional diagnosability can better evaluate the real diagnosability of interconnection networks. In this paper, we derive the conditional diagnosability of the multiprocessor systems in terms of Complete Josephus Cubes CJCn (n ≥ 8) under the comparison model.