What does 1 Corinthians 11:6 mean?
This verse concludes Paul's thought from the previous verse. Apparently, some Christian women in Corinth were rejecting the cultural norm to have their heads covered. In that society, women with heads uncovered in public were signaling their sexual availability, or association with idol worship. Apparently, some women of the church in Corinth were failing to cover their heads while praying and prophesying in church. Paul says bluntly that this practice brings shame on their husbands, fathers, or the male head of their household.The gap between millennia and cultures can make this reasoning unclear. A modern parallel might be a woman attending church services in extremely revealing clothing, or lingerie. Acceptable as those are in the right context, the surrounding culture perceives those as sexually-suggestive choices. While it seems extreme to imagine someone going to church in their underwear, the point Paul makes is exactly that. Sincerity and "freedom" don't override how others perceive that appearance, which would be an embarrassment to the person's spouse.
This may be another issue where Paul is commanding the Corinthian Christians to set aside what they perceive as freedom, or a "right." The higher concern is the spiritual good of others. In some cases, however, even arguments for "freedom" fall short of countering the harm of how certain behaviors are perceived. In this case, Paul does not allow room for them to decide. The dishonor created is too great. The implication of that style choice, in that culture, is too overt for Christian women to brush off or ignore.
Paul compares this to the dishonor for a woman of having her head shaved. This was sometimes used as a humiliating form of public marking on those who'd committed certain offenses. Presumably, even the women comfortable with uncovered heads in a church service would have felt disgraced to have their hair shaved off. Paul tells them, then, to just keep their heads covered during their services, for the sake of their metaphorical heads.