This paper presents a theory -- the PNF calculus -- which extends Allen's interval algebra into handling causal propagation of the states of temporal intervals through a network of time-connected intervals. The basic goal is to assert whether an interval is happening or not based on evidence dynamically provided by other intervals. The PNF representational structure enables the design of a fast algorithm which deduces those temporal implications of the state of some intervals based on the state of temporally-related intervals. The paper also contains basic results on PNF theory, and a conjecture about the completeness of the approach. We also discuss how to employ the PNF calculus as the foundation of methods for human action recognition, considering actions represented as a collection of time intervals corresponding to its sub-actions, events, and detectable states of physical objects.