An important issue in diagnosis research is design methods for residual generation. One method is the Chow-Willsky scheme. Here, the Chow-Willsky scheme is extended as it becomes universal in the sense that, for both discrete and continuous linear systems, it Is shown to be able to generate all possible parity functions. This result means it can also be used to design all possible residual generators. It is shown previous extensions to the Chow-Willsky scheme are not universal, which is the case when dynamics controllable from fault exist, but not from the inputs or disturbances. Also included here are two new conditions on the process for fault detectability and strong fault detectability. A general condition for strong fault detectability has not been presented elsewhere.