Systems are subject to faults in their components, affecting their overall behaviour. In a "black-box" system, such faults only become apparent in the output when appropriate inputs are given, which poses a number of satisfaction and optimization problems regarding both testing and diagnosing. This work addresses such problems developing models with multi-valued logics that it formalizes and generalizes to multiple faults. Such logics extend Boolean logic by encoding dependencies on faults, thus allowing the modelling of an arbitrary number of diagnostic theories. The effectiveness of constraint solving over finite domains and sets is shown on several problems.