Some events recur, and some happen only once. Galton refers to the latter as “once-only” events [1]. In a first-order logic of events that makes a type-token distinction, the possibility of concurrent occurrences of the same event renders the characterization of the intuitive once-onliness not very intuitive. In particular, the paradigmatic case of the n^th occurrence of a recurring event is shown to be not necessarily once-only. Counter-examples give rise to a classification of events based on the temporal relations among their occurrences. The problematic cases turn out to be those events that involve an indefinite individual; we call these indefinitely-specified events. We consider two options. The first is to restrict our event ontology, as has been implicitly done in most logics of events, to events that are definitely-specified. The second is to admit all sorts of events into our ontology and distinguish those that are definitely-specified from those that are not by statements in the object language. We opt for a representation of events as functional terms in the logic, and those terms denoting indefinitely-specified events seem to inevitably contain variables. Such non-ground terms turn out to be semantically problematic. To smoothly resolve these problems, we adopt Shapiro's logic of arbitrary and indefinite objects in which indefinite individuals are denoted by special terms [2]. Thus, indefinitely-specified events are naturally represented by functional terms with at least one argument denoting an indefinite individual.