We study the problem of detecting global predicates in distributed systems which may be opposed to process crashes. Process crashes introduce a new form of uncertainty---the problem of reliable failure detection. We first show that this turns the previously observer-invariant notions of possibly and definitely into observer dependent notions and thus renders these predicate transformers useless in faulty environments. Then we introduce a new pair of predicate transformers called negotiably and discernibly tailored to the new system model in the following sense: Building on the notion of possibly, we show that negotiably retains the same type of ``pessimistic'' semantics and thus can be used to detect the violation of special types of safety properties. In analogy to definitely, we show that our notion of discernibly has similar ``optimistic'' semantics and thus can be used to detect certain forms of liveness properties. We give several examples for predicates to which our new predicate transformers are useful and present algorithms to detect negotiably and discernibly in asynchronous distributed systems.