Terminus post quem
Terminus post quem ("limit after which", often abbreviated to TPQ) and terminus ante quem ("limit before which", abbreviated to TAQ) specify the known limits of dating for events. A terminus post quem is the earliest time the event may have happened, and a terminus ante quem is the latest. An event may well have both a terminus post quem and a terminus ante quem, in which case the limits of the possible range of dates are known at both ends, but many events have just one or the other. Similarly, terminus ad quem ("limit to which") is the latest possible date of a non-punctual event (period, era, etc.), while terminus a quo ("limit from which") is the earliest. The concepts are similar to those of upper and lower bounds in mathematics.
For example, consider an archaeological find of a burial that contains coins dating to 1588, 1595, and others less securely dated to 1590–1625. The terminus post quem for the burial would be the latest date established with certainty: in this case, 1595. A secure dating of an older coin to an earlier date would not shift the terminus post quem.
An archaeological example of a terminus ante quem would be deposits formed before a historically dateable event, such as building foundations that were partly demolished to make way for the construction of a city wall. If it is known that the wall was finished in 650, then the foundations must have been demolished in 650 or earlier; all that can be said from the evidence is that it happened before the known event.
Other examples of things that may establish a terminus are known dates of death or travel by persons involved, a particular form of heraldry that can be dated (see pastiglia for example), references to reigning monarchs or office-holders, or a placing relative to any other events whose date is securely known. In a modern context, dated images, such as those available in Google Earth, may establish termini.
A terminus ante quem non differs from a terminus post quem by not implying the event necessarily took place. 'Event E happened after time T' implies E occurred, whereas 'event E did not happen before time T' leaves open the possibility that E never occurred at all.