The Christian celebration based on the Last Supper is known by various names: [Holy] Communion, Eucharist, Breaking of Bread, the Lord's Supper, the Lord's table, the Mass, and by a term which is possibly unique to Christadelphians, the Memorial Meeting (although, as with many Christadelphian traditions, this may have its roots in the Restoration movement).
Paul uses the terms communion and eucharist (thanksgiving) in relation to this celebration, but the celebration itself he calls the Lord's supper, and the Lord's table.
The mention of the Lord's table is intended I believe to direct our thinking back to our Lord's pattern of behaviour recorded in the Gospels and not simply to one event (the last supper). Jesus' practice of regularly and frequently eating in company was a feature of his ministry which was highlighted in the Gospels.
Luke in particular makes a point of recording how often Jesus accepted invitations to dine out (5:29; 7:36; 10:38; 11:37; 13:26; 14:1; 19:5-7). A particular criticism by His opponents was that he ate with tax-collectors and sinners (Mark 2:15-16; Matt 11:19; Luke 7:34, 39; 15:1-2; 19:1-10).
The Pharisees and Essenes had very strict rules about the purity of the meal table. This purity was maintained not only by the preparation of the food itself but by who was permitted to share in the meal. For example, the Essenes specified that anyone "paralysed in his feet or hands, or lame, or blind, or deaf, or dumb, or smitten in his flesh with a visible blemish" had to be excluded from the meal table. In contrast Jesus words in Luke 14:13-21 appears to be directed at them when He said "when you give a banquet, invite the poor, the crippled, the lame, the blind, and you will be blessed". He follows this with a parable about the kingdom of God in which the poor, the crippled, the blind and the lame are invited to the feast in the kingdom of God.
The Pharisees excluded from their meal tables people who were "sinners" and this seems to have been a bone of contention with Jesus who openly dined with people who were outside their boundaries of acceptability. We need to ask "who were the 'sinners'?"
The word translated 'sinner' literally means a law-breaker. To groups like the Pharisees the term came to include people who did not observe their strict interpretation of the Law. The term was used to describe Jews who practiced their Judaism differently from the Pharisees' faction.* The term tax-collector is frequently associated with sinners because they were regarded as typical sinners. Matthew (himself a tax-collector) records two sayings of Jesus which demonstrate that tax-collectors were regarded as being in the same class as Gentiles, outside the covenant people (Matt 5:46-47; 18:17).
By dining with people who were regarded as being impure, or outside Israel - the covenant people - Jesus showed His concern for people who were excluded and marginalised. He challenged the boundaries set up by the religious leaders of His day which excluded even fellow-Jews from their meal tables, and demonstrated an inclusiveness in His table-fellowship which was motivated by His concern for others rather than by rigorous enforcement of a strict interpretation of the law. Unlike the Pharisees Jesus' table-fellowship was not fenced around to mark off the insiders from the outsiders and there was no barrier to be overcome before one could enjoy his company.
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* James D.G. Dunn, Jesus Remembered, p530.