Arbitrary is a feature of a choice, decision or judgement which is sovereign, without any reasoning needed to support it.
A very good example of arbitrary judgement is a dilemma, whether a glass is half-empty or half-full. In popular use the answer distinguishes optimists from pessimists (more examples of arbitrary mindset). The arbitrary choice is rooted in idividual mindset (as a complex product of personal cognitive history) rather than in the patch of reality that triggers it. As we all keep rationalising our choices, there is usually a great deal of reasoning around it, but it is what it is — rationalisation.