Part of the background to the interpretation of this verse is the whole context of 1 Corinthians. Those who see it in the local geographical and time context argue for the practice of men to uncover their heads and women to cover theirs as being due to the manner of life in Corinth in the 1st century.
I would argue for a wider context. Unto the church of God which is at Corinth, to them that are sanctified in Christ Jesus, called to be saints, with all that in every place call upon the name of Jesus Christ our Lord, both theirs and ours: (1Cor 1:2 KJVS)This opening statement makes it very clear that Paul had much more than the local Corinthian assembly in mind when he wrote. This is not actually 'the letter to the Corinthians'; it is the 'letter to the Corinthians and all, who in every place, call upon the name... That is, this letter always had a wider audience in the Spirit's intentions.
Another interesting verse is If any man think himself to be a prophet, or spiritual, let him acknowledge that the things that I write unto you are the commandments of the Lord. (1Cor 14:37 KJVS) Please notice that it is not the 'one thing' that Paul claims are the commandments of the Lord but "[u]all the things[/u] that I write unto you". This letter was written, partly, in answer to a letter received by Paul from Corinth:Now concerning the things whereof ye wrote unto me: It is good for a man not to touch a woman. (1Cor 7:1 KJVS) and at several points during the next part of the letter Paul used the Greek preposition 'peri' which in this context means 'concerning' or 'with reference to'. 1Cor 8:1,4; 12:1; 16:1,12. Our reference to 'Coverings' in right at the centre of this.
He begins in...Now I praise you that ye remember me in all things, and hold fast the traditions, even as I delivered them to you. (1Cor 11:2 ASV)... by commending them for keeping the 'traditions' ie the deposit of truth that he has committed to them. He claims, in For I have received of the Lord that which also I delivered unto you, That the Lord Jesus the same night in which he was betrayed took bread: (1Cor 11:23 KJVS)These are not locally developed traditions but traditions which Paul 'received' from the Lord and 'handed on' to the church at Corinth. The church at Corinth is not 'a one-off' but is part of the pattern which Paul 'handed on' to all.
Having commended them for "holding onto' the traditions (1 Cor 11:2) Paul immediately introduces the issue of the relationship between male and female and the differences in responsibility between the two. It is not difficult to follow his train of thought. He moves between 'headship' and the 'uncovered head' of the man and then onto the covered head of the woman. There are some difficulties in honest interpretation caused by the fact that the Greek language does not have a word for 'husband' or 'wife' but uses the idiom 'her man' or 'his woman'. He says that the man ought not to cover his head and that the woman ought to have 'authority on her head because of the angels'.
The verse in question... But if any man seem to be contentious, we have no such custom, neither the churches of God. (1Cor 11:16 KJVS)... is admittedly a difficult on. It is an a level with Paul's comment to the Thessalonians Remember ye not, that, when I was yet with you, I told you these things?(2Th 2:5 KJVS) The difference between ourselves and the Corinthians is that for them 1 Corinthians was an 'addition' to what Paul had already taught them, whereas for us we have to try to work out what kind of questions Paul is answering. Can we create a scenario?
Paul refers to some who might be 'quarrelsome' (contentious). There were some who were apparently not happy with the custom that was already established in the church at Corinth. At this point I would have to disagree with the Charity Ministry position and that of Watchman Nee. Their line seems to be that this is a general truth and that women ought, AT ALL TIMES, to have their head covered. I do not think the evidence supports this. The context is not the local market place but a meeting in which prayer and prophecy are manifested; ie the church meeting. This has relevance to the 'local custom' issue because it was not local custom for Greek women to cover their heads in worship, nor was it custom for the men to do so. Corinth was a Roman city and Roman men did cover their heads in worship although not always the women. There were no doubt those from a Jewish background in the meeting too. The Jews had strict rules and biblical commandment that when in closest proximity to God the head must be covered. So we have some interesting people in the meeting...
1. some believed that the men ought to cover their head but not necessarily the women - Romans
2. some believed that neither men nor women ought to cover their heads - Greeks
3 some believed that both men and women should cover their heads in approach to God. - Jews
Paul, and the early church, were usually very sensitive about the Jewish conscience. It must have been of some significance for Paul to 'ban' the Jews in the meeting from covering their heads. It was most likely to cause offense to their conscience. If it were not important it is most unlikely that Paul would have made these statements.
The Corinthians have exhibited an independent spirit in their self-centred spiritual exercises and, I would suggest, someone has suggested that there is no need for the women to cover their heads. Paul's answer is to that suggestion. It is not arbitrary but is part of the pattern of Paul's teaching in local churches. But as God hath distributed to every man, as the Lord hath called every one, so let him walk. And so ordain I in all churches. (1Cor 7:17 KJVS) See here too how conscious Paul is of the other churches while writing to Corinth. We can see this 'churches-consciousness' at several points in 1 Corinthians 1Cor. 7:17 (KJVS) But as God hath distributed to every man, as the Lord hath called every one, so let him walk. And so ordain I in all churches.
1Cor. 11:16 (KJVS) But if any man seem to be contentious, we have no such custom, neither the churches of God.
1Cor. 14:33 (KJVS) For God is not the author of confusion, but of peace, as in all churches of the saints.
1Cor. 14:34 (KJVS) Let your women keep silence in the churches: for it is not permitted unto them to speak; but they are commanded to be under obedience, as also saith the law.
1Cor. 16:1 (KJVS) Now concerning the collection for the saints, as I have given order to the churches of Galatia, even so do ye.
1Cor. 16:19 (KJVS) The churches of Asia salute you. Aquila and Priscilla salute you much in the Lord, with the church that is in their house. It seems very likely that in their personal estimation and in their estimation of the state of their own 'church' the church at Corinth was independent and considered themselves to be somewhat superior to other churches. I think this context provides the best-fit hypothesis for Paul's injunction that 'all the churches have no other custom'.