Summary: What happens when Christians disagree? If it is reasoning from the Scriptures about the Word, that's one thing. But when believers quarrel often it can be a sign of immaturity. When they try to resolve those disputes in front of the world, the gospel is t
In the last chapter Paul talked about how to deal with big problems in the church involving open, unrepentant sin. Now as we move into Chapter 6, he talks about handling smaller problems between believers. In a way, this is a continuation of the flow from Chapter 5. There he is telling the Corinthians that they ought to know better about blatant, unrepentant sin and should have the discernment to deal with the serious infection. In the same way, they ought to be mature enough to deal with disagreements without having to sue each other! The key here, as with much of what Paul talks about, is how your actions as a believer either positively or negatively affect the propagation of the gospel of Jesus Christ.
There are several possible reasons that it is unadvisable for Christians to take each other to court. 1) A secular judge doesn't understand kingdom values. 2. Often we go to court for revenge, not remedy. And 3) Christians fighting each other take the focus off of the gospel and put it on the church. Paul is basically saying that there is a distinction between someone who has been regenerated and someone how has not. This involves dealing with sin—we can't just go on being like (or worse than) the world and we also can't solve our problems when it is convenient by worldly means. You are different, so act like it! This theme continues on through Chapter 8.
This isn't to be confused with what Paul said in Chapter 5 vs 13. God will judge those in the world as to whether they receive Christ, but we will help Jesus judge, or rule the world upon his return.
Paul reminds the Corinthians that they will have a much larger role when Jesus comes back. 2 Timothy 2:12 (quickview)  "if we endure, we will also reign with Him". As Christians we will sit on thrones judging the world. We will help Jesus rule this planet. Doesn't that suggest a certain amount of discernment and maturity? Judges are not supposed to be swayed by opinion, emotion, or pressure. They are to apply the law fairly to all. We will be called upon to apply the Law fairly—adjudicating matters of great import.
So why not start now? If we get into squabbles with our brothers and sisters over small things, how can we hope to judge grave matters later on? We need to develop these skills now. So don't take matters to a secular judge, when we need to use kingdom principals first.
It's not clear what Paul means by this—whether we will take part in condemning Satan and his angels or that God's angels will somehow work for us as we help Christ reign over the earth. But his point is the same—in light of what you will be doing shouldn't you be able to handle average ordinary arguments?
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Outside judges have "no standing" in the church because they judge according to worldly standards and are not part of the body of Christ per se. They wouldn't understand or even be able to apply the character of Jesus to a situation. Paul means to shame the Corinthians because they couldn't seem to find even one respectable leader among them with enough wisdom to maneuver the dispute and help the parties to settle it. Or perhaps they worried that their lack of Christ-like character might be called out.