Decentralized decision-making requires the interaction of various local decision-makers in order to arrive at a global decision. Limited sensing capabilities at each local site can create ambiguities in a decision-making process at each local site. We argue that such ambiguities are of differing gradations. We propose a framework for decentralized decision-making (applied to decentralized control in particular) that allows computation of such ambiguity gradations and utilizes their knowledge in arriving at a global decision. Each local decision is tagged with a certain grade or level of ambiguity, with zero being the minimum ambiguity level. A global decision is taken to be the same as a "winning" local decision, i.e., one having the minimum level of ambiguity. The computation of an ambiguity level for a local decision requires an assessment of the self-ambiguities as well as the ambiguities of the others, and an inference based upon such knowledge. For the existence of a decentralized supervisor, so that for each controllable event the ambiguity levels of all winning disablement or enablement decisions are bounded by some number N (such a supervisor is termed N-inferring), the notion of N-inference observability is introduced. We show that the conjunctive-and-permissive (C&P) boolean OR disjunctive-and-antipermissive (D&A) co-observability is the same as the zero-inference observability, whereas the conditional C&P boolean OR D&A co-observability is the same as the unity-inference observability. We also present examples of higher order inference-observable languages. Our framework does not require the existence of any a priori partition of the controllable events into permissive/antipermissive sets, nor does it require a global control computation based on conjunction/disjunction of local decisions, exhibiting that our ambiguity-based approach is more efficient.