Paul, a prisoner for Christ Jesus, and Timothy our brother, To Philemon our beloved fellow worker and Apphia our sister and Archippus our fellow soldier, and the church in your house: Grace to you and peace from God our Father and the Lord Jesus Christ. (Philemon 1-3)Paul and Timothy... to whom? To Philemon, and Apphia and Archippus and the church in Philemon's house. Yet the letter is known as "Philemon." It isn't called "Philemon, Apphia, Archippus, and the house-church." Why not?
Simple: it is so called because the apostle mostly addresses himself to the individual, Philemon. This is emphatically plain in the Greek text. After the plurals in verse 3, Paul reverts to the singular pronoun in verse 4, as he addresses himself to Philemon individually. This use of the second person singular pronoun is continued in vv. 5-8, 10-14, 16, 18-21, and 23. Nor are most of the omitted verses real exceptions, as Paul employs the second-person singular verb in vv. 15, 17, and 22.
The apostle is writing about a personal issue: the business involving Philemon and his runaway slave, Onesimus. Yet you could say — are pretty much compelled to say — that the apostle address Philemon on this personal matter in an open letter.
In fact, when you think about it, one could argue that all of Paul's letters are open letters. Isn't that so? This is clearly the case in those addressed to churches. But even in letters addressed to individuals, plurals come back in (cf. final words of 1 Timothy 6:21, of 2 Timothy 4:22, and of Titus 3:15, in Greek).
As in the other letters, so it is plainly in Philemon. After all, though the body of the letter (vv. 4-24) is addressed to the man Philemon, the opening and close are addressed to the church. Paul addresses Philemon, but he brings in the assembly as well.
Why? Well, it's hardly rocket-science, is it? This had not been a private affair. It wasn't as if no one knew about Philemon and his runaway slave. Nor did it involve that body of believers alone, at this time. Onesimus was currently traveling with an entourage on church business under Paul's direction (cf. Paul's open letter to the church at Colosse, 4:7-9). Word was spreading. And so Paul addressed Philemon, though with the expectation that others read along, over his shoulder as it were.
Where? I don't know. I wouldn't suppose a formal church meeting, though that is certainly a possible setting for reading an apostle's epistle. However, as Carson and Moo say, Philemon "falls somewhere between the simple private letter and [the?] public letter intended for a broad audience" (An Introduction to the New Testament, 588).
In this way, not only did Paul address the individuals, nor only the individuals within the churches. He addressed them each and all as before the Lord, and as in community with the saints at large. So Philemon in particular needed to keep this in mind. He needed to remain conscious that what he did about Onesimus did not just affect himself and his slave. Philemon was doing what he was doing in the presence of God, and he was doing it under the gaze of the saints. What he did would affect them. Paul made him conscious of that fact, allowed it to exert its pressure in addition to the loving force of his words.
So how did this all turn out? The epistle is open-ended, in a sense, in that we don't know (from it) how Philemon responded to Paul's entreaties on Onesimus' behalf.
Yet it isn't really all that open-ended, is it? Does not its very presence in the Canon suggest that, rather than withdrawing in regal, offended silence, or exchanging offended notes about Paul with his likeminded fellow slaveholders, Philemon received Paul's open letter in humility and good faith?
Paul did the right thing before God.
Onesimus and Philemon did the right thing before God.
The result has brought blessing to the church for two millennia.