I didn’t see this posted in this thread yet, so I’ll do it now. Clayton has written two excellent posts titled What Law Is and A Praxeological Account of Law. I suggest you read them because you will understand where he is coming from better.
Just as language is what arises from humans trying to communicate with each other, law is what arises from humans trying to settle disputes nonviolently. If you and I have a dispute, there are only two ways it can be settled, through mutual settlement or by violence. Common law and customary law are what arise from a decentralized system of law. There need not be any binding arbitration for either of these to produce law. Statutory law is what arises when there is a monopoly on the legitimate use of force in any given area. When Clayton refers to arbitration, he is referring to the process of mediation, where two or more participants use a mediator to find a mutually agreeable solution to their dispute. Mediation even exists in the modern world despite there being statutory law. In other words, despite having a state, there are people who are willing to settle disputes without a binding solution already! So I would not dismiss his statements as people do actually settle disputes voluntarily - often without mediation.
Ditto to you. Just because you cannot imagine people settling disputes voluntarily is not a reason to denounce it. And, as I previously stated, people already settle disputes voluntarily in a wide range of disputes.
It is possible for people to agree to abide by a decision made by a neutral third party, but this does not mean they will do so. A dispute is resolved only when all parties have decided it is resolved. This is true even today, in Statist America. For example, a murderer is not punished to the extent that the family of the victim would be satisfied. A member of the family then kills the murderer. Only then is the dispute resolved. This kind of thing does happen, perhaps not commonly with murder, but vindictive ex-spouses keying their ex’s car, or something to that effect.
No. Clayton is not defining judge in a different way then you. A judge is someone whose decision is final. A mediator is someone who helps two people reach a mutally agreeable solution. Clayton is using arbitrator in the sense of mediator. So, he is asking you to to imagine how disputes would be settled if there were no judges but arbitrators (mediators).