A foundation was the bottom layer of a structure, without which it couldn't stand. (VOY: "Flashback") Metaphorically, it referred to anything that formed the basis for a belief or society.
According to the Vulcan elder at the monastery at P'Jem, represents the Stone of J'Kah the foundation of all the Vulcans believe. A life of order and control through logic. (ENT: "The Andorian Incident")
In 2268, Pavel Chekov located a foundation under the surface of Sigma Draconis VI, indicating that buildings had once been present above ground. (TOS: "Spock's Brain")
In 2370, a holographic gangster claimed to be "laying a foundation" in Keystone City when he placed a brick in a special slot. (TNG: "Emergence")
In 2373, while attempting to regain self-control, Tuvok reminded himself that "logic is the foundation of function." (VOY: "Flashback")
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