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Terminus post quem and terminus ante quem ("limit before which") specify approximate dates for events. A terminus post quem is the earliest time the event may have happened, and a terminus ante quem is the latest.
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.
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, based on the latest securely dated coin. A secure dating of another coin to a later date would shift the terminus post quem.
An archaeological example of a terminus ante quem would be deposits formed before or beneath a historically dateable event, such as a building foundation partly demolished to make way for the city wall known to be built in 650. It may have been demolished in 650, 649 or an unspecified time before - all that can be said from the evidence is that it happened before that event.
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.