Distinguishing trace-based system properties into safety properties on the one hand and liveness properties on the other has proven very useful for specifying and validating concurrent and fault-tolerant systems. We study the adequacy of these abstractions, especially the liveness property abstraction, in the context of secure systems for two different scenarios: (1) Denial-of-service attacks and (2) brute-force attacks on secret keys. We argue that in both cases the concept of a liveness property needs to be adapted. We show how this can be done and relate the resulting concepts to related work in the areas of concurrency theory and fault-tolerance.