By definition arbitrary can refer to choices, actions, etc. made manifest not by means of any underlying principle or logic, but perhaps by human whim, or some decidedly illogical formula. For example, rearranging, for no reason, the letters of the alphabet so the letters appear in a staggered fashion, i.e., ac-bd-eg-fh. If there were a purpose in doing so, (e.g. to make a newer, better alphabet) then it would not be considered arbitrary. For instance, fans of J. R. R. Tolkien consider his tengwar to be less arbitrary in both shape and order than the Latin alphabet.

Etymology

The word arbitrary comes from the Latin arbitrarius (see Arbiter) and in Middle English could be found also as arbitrarie.

Sources Online Etymology Dictionary and Dictionary.Com

Philosophy

Philosophers have for centuries argued the merits of arbitrary thought and practice. Should we make our choices, live our lives, conduct our business and create by our own simple willful judgements alone, perhaps based on our feelings on a given subject, or should we use our capabilities of logic, artfulness and creativity to determine our thoughts and actions?, is an example of this sort of inquiry.

Ayn Rand is a more modern example of a philosopher who has gone to some lengths in pondering these questions. Rand wrote several books and papers in which she questioned prevailing philosophies, and formulated her philosophy of "Objectivism", whose main premise is that humans should act based not on subjective judgements, or how they "feel" (more or less her terminology) about something, but what is objectively (i.e. based on reason and logic) just, worthwhile, moral, or true.

Law and Politics

One can hardly discuss "arbitrary" as a concept without discussing the justice system. The word arbitrary comes from arbiter, as in someone who is tasked to judge, i.e. a Judge. Arbitration is a term used often to refer to an alternative way to work out disputes in an organization etc., or a disputed claim between individuals without having to go to court.

In some countries, a prohibition of arbitrariness is enshrined into the constitution. See Voting in Switzerland#Citizenship for an example in which this prohibition even overrides democratic decisions.

Etymology

Willkür | Arbitrary