Diagnosis is a crucial task in safety critical systems. In particular, diagnosability is the property ensuring that any failure can be determined within a finite delay. Early works on diagnosability of discrete event systems (DES) establish intermediary models from the analysed model and then call specific procedures to check diagnosablity based on these models, while some recent works try to give a diagnosability formulation as a model-checking problem. The present work attempts to contribute to this trend by developing a single framework able to handle both of the diagnosability issues: how to model the problem? and how to decide it? Namely, a generic operative formulation of diagnosability using the µ-calculus logic is established, which allows resolving the diagnosability issue within a single formalism. We also show how the developed formulation can be extended to deal with further diagnosability issues. Besides, using the same formulation, an online diagnosis algorithm is developed.