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First of all, the Bible doesn’t address directly the question of who is to partake of the Lord’s Supper. However, as we look at the Scriptures we can learn who is intended. Jesus instituted the Lord’s Supper in Matt. 26:26-29. He said the cup was His blood, or represented His blood which was shed for many for the remission of sins. So, first we see that it would be those who have been cleansed by the blood of Christ who would be taking the Lord’s Supper. Further He said in verse 29 that He would not drink of this fruit of the vine from now on until that day when He drank it new with them in His Father’s kingdom. Again, it would be with those in His Father’s kingdom, or with Christians.
In Acts 2:41, there were 3,000 baptized and it was them that continued in the apostles’ doctrine and fellowship, in the breaking of bread, and in prayers. The context of breaking of bread would be the Lord’s Supper. Then in Acts 20:7 it says the disciples came together to break bread. So, here we have the church, baptized believers, coming together to partake of the Lord’s Supper. In I Cor. 11, we see the church came together (verse 18) and they were partaking of the Lord’s Supper. Now they were abusing the partaking and Paul had to correct them, but it was the church involved. The Biblical references we have indicate that it was Christians who partook of the Lord’s Supper. I see no good it would do those who have not obeyed Christ’s commandments to partake of the Lord’s Supper. Of course, that would be for the Lord to decide.
Now let me address what is practical. It seems to me that we should pass the bread and fruit of the vine to the people in attendance. While it is intended for the Christians present, if there are those present who are not Christians and they choose to partake of the elements that is up to them. We shouldn’t try to be a police force and tell them they can’t do so. It is between them and the Lord. To not offer the Lord’s Supper on Sunday morning because some non-Christians are present is foolish in my view. I don’t think Judas being present at the Last Supper or the fact that whole families ate the Passover meal has any relevance to this issue. But, we each are accountable individually for our actions and the fact a non-Christian partakes of the Lord’s Supper should not be a concern any more than we are concerned about the many other things people do which the Lord does not approve.
I hope this is of some help to your understanding. If there are other specific questions, please don’t hesitate to let me know.