Semantics of temporal logic over truncated paths (i.e. finite paths that correspond to prefixes of computations of the system at hand) have been found useful in incomplete verification methods (such as bounded model checking and dynamic verification), in modeling hardware resets, and clock shifts and in online and offline monitoring of cyber-physical systems. In this paper we explore providing semantics for temporal logics on other types of incomplete paths, namely incomplete ultimately periodic paths, segmentally broken paths and combinations thereof. We review usages of temporal logic reasoning in systems biology, and explore whether systems biology can benefit from the suggested extensions.