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Using the PMC model of a multiprocessor system, a new, general quality of hybrid fitult (combinations of hard anid soft failing units) diagnosability is characterized which encompasses previously known results as well as provides a major extension. This new diagnosability, called t/ts/τ-diagnosability, serves as a measure of the extent to which collections of test results, called syndromes, contain information relative to faulty unit identification for various testing assignments. By analyzing a system's test assignment over tahges of the parameter values t, ts, and τ, the spectrum of diagnosis quality achievable in that system can be established. Fundamental interrelationships among these parameters are developed. Optimal designs of t/ts/τ-diagnosable systems are presented. The main result of this correspondence can be viewed as a master fault diagnosability characterization which from the perspective of usefulness of the syndrome space for correct diagnosis both unifies and expands the domain of hybrid fault diagnosabilities.