We have previously introduced an inference-based framework for decentralized decision-making, comprising of multiple observers, each with its own partial observation, where inferencing over the ambiguities of the self and the others is used to issue local decisions, and a global decision is taken to be the one possessing the minimum ambiguity level. In this setting, we previously introduced the notion of N-inference.-diagnosability to characterize the existence of a disjunctive decentralized diagnosis scheme so that any fault can be detected within a bounded delay, using at most N-levels of inferencing, by one of the diagnosers. While the disjunctive scheme relies on one of the diagnosers making the failure decision, the dual conjunctive scheme relies on none of the diagnosers making the nonfailure decision. It is known that the two schemes are incomparable, and in this paper we extend our earlier work to provide a more general framework, introducing the notion of N-inference diagnosability, capturing both disjunctive and conjunctive schemes. We also develop a method for verifying N-inference diagnosability, as well as discuss several of its useful properties.