Paul opens up this chapter with the words, "I appeal to you therefore, brothers". Later on, he again states, "For by the grace given to me I say to everyone among you..."
Paul wasn't afraid of being firm. He "is not ashamed of the gospel of Christ". And he tells us himself in Philippians 3, "I myself have reason for confidence in the flesh also. If anyone else thinks he has reason for confidence in the flesh, I have more: circumcised on the eighth day, of the people of Israel, of the tribe of Benjamin, a Hebrew of Hebrews; as to the law, a Pharisee; as to zeal, a persecutor of the church; as to righteousness under the law, blameless."
But "whatever gain" he had, he "counted as loss for the sake of Christ". "Accordingly, though I am bold enough in Christ to command you to do what is required, yet for love's sake I prefer to appeal to you" (Philemon 8).
Paul isn't afraid to get to the point with those he is instructing. But he doesn't do it without love.
There are times when a firm command is needed. But so often we jump straight from noticing sin to zealously commanding a stop to something that bothers us. But Paul shows us, by his example, that sometimes, an appeal is all that is needed. He appeals to them- and then he gives them things to aim for. Instead of saying "You people are ALWAYS doing this, just stop!", he says, "By the mercies of God, strive to always do this".
This is hard for me. This reminder is mostly for myself. I rush through my day telling my younger siblings to do x, y, and z, and am not nearly as encouraging as I should be. I am not a naturally patient person. I know I could learn much from Paul's example.
Yes, there are times that call for more than an appeal... but after the appeal has been made. Have you ever notice that when you tell someone "I hate when you do...", usually what happens is an argument? But when you encourage someone somewhere- "I noticed you did the dishes without having to be asked tonight. That was great!"- often, they'll do it again without you having to go any farther about how they sometimes might not do what they should. Encourage with love. Appeal with love. And then- then- confront with love if need be. And always through the authority of Christ's word, not our own preference.