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Few LDS realize that the word "missionary" which they use so much is a Latin translation of the Greek word "apostle!" Nor do they know that the Greek word "apostolos" is not always transliterated as "apostle." For example, in II Cor. 8:23, it is translated as "messengers." Thus, in the broad sense of the word, Christians do have living apostles today. But, in the more limited sense of Christ's twelve apostles, elorsonly "twelve apostles of the Lamb" and they are with Him now. Their names will be in the twelve foundations of the wall of the New Jerusalem (Rev. 21:14). Any apostles on earth now cannot be Christ's twelve apostles. Some question, "What about Judas Iscariot who betrayed Christ?" Acts 1:15-26 explains that Judas was replaced according to a prophecy made by King David in Ps. 69:25. The one who took Judas' place with the apostles had to be one who had been with Jesus and the other apostles from the time of Jesus' baptism until He ascended into glory (Acts 1:21-22). No one today meets that qualification. No Bible text says that any other apostle was ever replaced. The apostle James' death is recorded in Acts 12:2, but nothing is said of a successor for him.
Surely if Christ had meant for the offices of the twelve apostles to be passed on to others, He would have left a list of qualifications for them. But, their calling and work were so unique that no man today could do what they did (e.g. be eyewitnesses of Christ's entire earthly ministry - Acts 1:21-22; 10:39, 41-42; II Peter 1:15-18; I John 1:1-5; be personally called by Christ - Matt. 4:18-22; Mark 1:16- 20; Luke 6:13; John 1:43; Eph. 4:11; be taught the gospel by Christ during His earthly ministry - Matt. 20:17; Mark 4:34; Luke 6:20).
The LDS also have ignored the order in which apostles and prophets are mentioned in Eph. 2:20 and 4:11. The order is "apostles and prophets," just like it is in I Cor. 12:28: "God set some in the church, first apostles, secondarily prophets, thirdly teachers..." But, the LDS have a prophet first over the entire Church. Then they have apostles in a secondary leadership position. Since the LDS claim to be the "restored" church, should they not have "apostles" first? When the LDS Church began, Joseph Smith was called "the first Elder" not the first apostle. (P. of G.P. J.S. History 1:72).
LDS use Eph. 4:11-13 to claim that the purpose of apostles today is to help the church come together "in the unity of the faith." However, discord in the LDS Church has resulted in over 200 splinter groups from the original organization, so the LDS apostles have not brought all to the "unity of the faith." The Mormon booklet, Denominations that Base their Beliefs on the Teachings of Joseph Smith, the Mormon Prophet by Kate B. Carter, as well as the book Divergent Paths of the Restoration by Steven L. Shields, names many of these sects and gives a brief sketch of their history and beliefs.
http://www.utlm.org/onlinebooks/mclaims7.htm#claims