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Given the two definitions:

arbitrary - based on random choice or personal whim, rather than any reason or system.

arbitrator/arbiter - an independent person or body officially appointed to settle a dispute.

What is the relationship between the two words? An arbitrator makes a choice when settling a dispute so there is obviously a link there, but it is not necessarily a choice that is random or on a whim. When we use the term arbitrary as in "an arbitrary length" the randomness portion of the definition is more important than the choice portion. Any idea how that came to be?

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Arbiter in Latin originally meant something like "attendee"; it was connected with the word bētō, "to go". Then it became "spectator" then "witness" and eventually "judge", which is more or less what it means today.

Arbitrary meant, sensibly enough, "dependent on the opinion of the arbiter" and did not descend to its current meaning of "capricious" or "random" until the early 17th Century.

I don't know if this was satire that became fixed in language, or a reflection of the somewhat haphazard judgment of sports judges, or something else.

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