Shared memories that provide weaker consistency guarantees than the traditional sequentially consistent or atomic memories have been claimed to provide the key to building scalable systems. One influential memory model, processor consistency, has been cited widely in the literature but, due to the lack of a precise and formal definition, contradictory claims have been made regarding its power. We use a formed model to give two distinct definitions of processors consistency: one corresponding to Goodman's original proposal and the other corresponding that given by the implementors of the DASH system. These definitions are non-operational and can be easily related to other types of memories. To illustrate the power of processor consistency, we exhibit a non-cooperative solution to the mutual exclusion problem that is correct with processor consistency. As a contrast, we show that Lamport's Bakery algorithm is not correct with processor consistency.