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The purpose of this essay is to provide an analysis of the famous passage from 1 Cor. 11:3-13:
3 Now I want you to realize that the head of every man is Christ, and the head of the woman is man, and the head of Christ is God. 4 Every man who prays or prophesies with his head covered dishonors his head. 5 And every woman who prays or prophesies with her head uncovered dishonors her head -- it is just as though her head were shaved. 6 If a woman does not cover her head, she should have her hair cut off; and if it is a disgrace for a woman to have her hair cut or shaved off, she should cover her head. 7 A man ought not to cover his head, since he is the image and glory of God; but the woman is the glory of man. 8 For man did not come from woman, but woman from man; 9 neither was man created for woman, but woman for man. 10 For this reason, and because of the angels, the woman ought to have a sign of authority on her head. 11 In the Lord, however, woman is not independent of man, nor is man independent of woman. 12 For as woman came from man, so also man is born of woman. But everything comes from God. 13 Judge for yourselves: Is it proper for a woman to pray to God with her head uncovered?
In another essay, Glenn Miller of the Christian ThinkTank provided an analysis of this passage noting that it was actually incongruous with an "anti-woman" sentiment. We will now provide an update based on findings in Bruce Winter's contextual study in After Paul Left Corinth [121ff]. After summarizing Winter's findings, we will give Miller's comments a once-over and make any related comments.
Apologists and commentators have suggested that there is something culturally relative about these commands, and they are quite right. Something missed here is the argument about veils for women is that Paul FIRST (11:4) speaks of veiling for men. This "veil" was actually the toga drawn over the head, which was done while praying or offering a libation to the gods. Even emperors did this. The act was limited to those who performed a leading role, such as sacrificing or praying, and such functions were by class function reserved for the socially elite.
Of course in the Christian church there were no elite in Christ; all could pray or prophecy to Christ. Those who have read our material on stratification and place in the ancient world may by now guess that elite Corinthians would be very upset over the idea of a poor person or lower class person doing this.
At the same time as Roman Corinth was founded, something else entered into the picture -- a new sort of "liberated woman" concept, but not the equal rights sort for all; rather, for women in high social positions only, to cliam "the indulgence in sexuality of a woman of pleasure."  This was a reaction to Roman male abuses of casual sexual companions.
Winter ties together these two matters thusly. Paul's reference here is strictly to married women, per the custom of the day that married women only wore veils among women. Thus Paul's reference to headship is between husband and wife, not men and women generally.
But the problem was that women were praying without their veil, and in this society, to go without a veil with another man present implied that you were his casual sexual companion. It was indeed like wearing a T shirt that said, "I am a prostitute." Paul's accusation is thus against those who abuse the freedom of the "new Roman women" and thereby dishonor their husbands by making it appear as though they are accompanied by a prostitute.
With this in mind, let's proceed to the core what is offered by Miller:
Women were obviously allowed to prophesy in church
Women were obviously allowed to pray audibly in church
"Prophet" was an official position and was "2nd in rank" in the church, behind apostles and before teachers (I cor 12.28-29)
The issue in the passage is some obscure point about head-coverings--NOT about women speaking in the church--and about people being contentious about it (v.16).
We have already noted above that v.10 says a women should have (exercise?) authority over her own head. ["Paul means that she should exercise wisely her right to decide whether to cover her head in a way that will honor her husband..." BBC:in loc.]
Winter adds that Paul "simply means that the woman is obliged to wear on her head that which signified to all and sundry that she was married." 
Now, even though the passage SUPPORTS women's speaking roles and 'authoritative speaking' roles, some have seen in the reference to 'headship' a basic male-over-female hierarchical subordination structure, as being ordained of God. Let me be quick to point out that EVEN IF THIS WERE SO, it would IN NO WAY negate the obvious fact that women were allowed (indeed, encouraged, when done in proper fashion) to function in worship. That fact remains unchanged in our text.
But what about the 'head' thing? Perhaps another digression is warranted, given the controversy surrounding it.
Some of the basic points first:
"head" does NOT mean the same thing we mean by it in Western culture. From the standpoint of anatomical function, in Paul's day it was the 'heart' that made the decisions, guided life, etc. "Head" was much more the 'adornment department' of the body! In other words, when people wanted to make decisions, they used their heart; when they wanted to get all "gussied up" ["dressed up", for you colloquially-deprived readers ;>) ], they used their head (e.g. hair, makeup, jewelry). So, in the literature, the word translated 'head' here often shows up as 'crown' or 'excellence'. [Hence, its usefulness in the passage of I Cor 11.]
The root notion was that of 'source', and from this usage it was applied to people--Zeus, Pharoah, the progenitors of the Twelve Tribes, Christ-with reference to the Church, man (Adam)--with reference to woman (Eve)....
If an author wanted to make a point about AUTHORITY, he would use two specific words--exousia ("authority"; Matt 28.18, Rom 13.1-3) and/or archon ("ruler"; Rom 13.3). He only used 'head' when dealing with issues of origination, completion, consummation.
In the passage under discussion, the only mention of the word 'authority' is in verse 10--and it is the women who possesses it!...
And finally, from a methodological standpoint, we could see this from the 'headship' passage in Ephesians. In linguistic studies, when you have a word which you do NOT know the meaning of, you try to decide from the invariable redundancy clues in the passage. If we didn't know what 'head' meant in Ephesians 5, and tried to figure out from the clues, we would decide that it meant something like 'servant'--one who saves, grooms, cleans, dresses, completes, protects, etc. We would NEVER come up with 'authority' from the actions and attributes of Christ in THAT passage! (He obviously has authority over His Bride, but it is not remotely in view in that passage.) But the literal notion of "that which completes" or "a major source of change" (i.e. "head"!) makes quite a lot of sense here. Simple inductive Bible study--without starting with a loaded meaning of 'head'--would yield something much more akin to 'active change agent' than 'ordained authority'...
Thus, I have to conclude that 'head' does NOT entail authority, but rather is used to focus on organic union (e.g. Christ/Church, Husband/Wife) and source/completion (e.g Christ/New Creation) motifs. The lexical data is simply too overwhelming at this point AGAINST the equation of the two.
So, I may not know what I Cor 11 means--relative to women wearing headgear other than hair at church, but I can tell from the passage what it does NOT mean! Women were obviously allowed to pray and prophesy in church, and were not commanded to 'be silent' at all. There is absolutely no restriction on women's roles (in worship at least) in this passage.
Miller is more correct than he realized. as Winter shows, there is more to this passage than has been conceived, and it does not reflect an inequity in the church, but rather, a matter of honor and shame that would be correlative to a woman today wearing seductive clothes to church!
6/2016: One other option deserves notice. Lucy Peppiatt in Women and Worship in Corinth suggest that this passage, like the later controversial "women, shut up" passage inb Ch. 14, is a quotation of a Corinthian view that Paul is refuting. Peppiatt makes an excellent case for this and it is worth examination.