I'd intended to get this out much earlier; however, yesterday was hectic beyond belief. My apologies to one and all.
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Yesterday, August 15th, we celebrated the Solemnity of the Assumption of the Blessed Virgin Mary, a holy day of obligation. In the Eastern tradition, it's also known as the Dormition (Latinization of the Greek Koímēsis, "falling asleep") of the Theotókos ("God-bearer").
Both the Eastern and Latin traditions hold that Mary was taken bodily into heaven, in the same manner as had her Son. The Eastern tradition maintains that she died and was resurrected on earth before the assumption; the Latin tradition leaves it an open question. Certainly, when Ven. Pius XII promulgated the apostolic constitution Munificentissimus Deus on November 1, 1950, he didn't insist that the Blessed Mother remained alive right up to that point; in fact, he alluded to her death several times.
However, like many Marian teachings, Protestants, especially Evangelicals, hold that because the Assumption isn't spoken of in Scripture it must not be a doctrine requiring Christian assent, let alone faith. Now, we've already spoken of the many problems inherent in sola scriptura; if you need reminding, then start with Part I. Suffice it for now to say that authority within the Catholic Church is a three-legged stool, resting not only on Scripture but also from the apostolic tradition and the teaching magisterium of the Church, which reconciles the other two.
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The earliest traditions surrounding the passing of Mary were written about the fourth century, though that's no reliable guide as to the origins of the tales. The oldest of these seems to be the Liber Requiei Mariae (Book of Mary's Repose), a link to which I was unable to find, followed by the "Six Books" Dormition narratives, which is further summed up in a manuscript apocryphally attributed to St. John the Evangelist (aka John the Theologian). Professor Stephen J. Shoemaker of the University of Oregon has many of these early manuscripts available online here. Certainly, though, the feast of the Dormition has been celebrated in the East since at least the 400s, and in Rome since the reign of St. Sergius I (687-701).
Now, the assumption isn't inherently improbable. In Genesis 5:24, we find that Enoch the father of Methuselah "walked with God; and he was not, for God took him." The author of Hebrews, apparently working from the Septuagint, tells us, "Now before he was taken he was attested as having pleased God" (Heb 11:5); certainly walking with God would be a sign of favor. We also have the example of Elijah, who was taken to the Lord in a fiery chariot before the eyes of Elisha (2 Kgs 2:10-11). And at the moment Christ died, among other portents, "the tombs also were opened, and many bodies of the saints who had fallen asleep were raised, and coming out of the tombs after his resurrection they went into the holy city and appeared to many" (Mt 27:52-53).
Beyond the Biblical precedents, we have the knowledge of Mary's gift from God of permanent, enduring grace ―an enduring shield against sin. How do we know this? Because the angel who delivered the message of her bearing the Christ-child addressed her as kecharitōmenē (usually translated as "highly favored [one]"), the perfect passive participle of charitoō (Lk 1:28); the use of the perfect passive indicates an action suffered that began in the past and continues onward. Saint Jerome could not possibly have translated this idea better into Latin than gratia plena, "grace-filled". From this state of perfect, permanent grace the Scholastics deduced not only Mary's immaculate conception but also her preservation from the curse of sin spoken of in Genesis 3:16-19. For, as St. Paul reminds us, "For the wages of sin is death, but the free gift of God is eternal life in Christ Jesus our Lord" (Rom 6:23); how could she who was preserved from sin see corruption in the grave?
It should also be noted that, while the apocrypha themselves are not historically reliable, that doesn't change the inherent probability of Mary's resurrection and assumption. Thomas Mozley, an Anglican scholar, wrote, "The belief was never founded on that story. The story was founded on the belief. The belief, which was universal, required a definite shape, and that shape at length it found." As the late "Father Mateo" argued, the argument from fitness is implicit in Scripture (e.g. Hebrews 2:10, 7:26); properly used, it doesn't prove what's in doubt but rather persuasively supports that which is already given to be true.
Scholars and saints from the late Fathers on have seen Mary symbolized and prophesied in Psalm 132:8: "Arise, O Lord, and go to thy resting place, thou and the ark of thy might." But the modern theology of the Assumption rises from the symbolism of Mary as the New Eve, following Genesis 3:15: "I will put enmity between you and the woman, and between your seed and her seed; he shall bruise your head, and you shall bruise his heel." This symbolism wasn't unknown to the Church Fathers; Tertullian first drew the parallel near the end of the second century.
Since Mary as the New Eve and Mother of the Church (cf. Rev 12:1-6) parallels Jesus as the New Adam (cf. 1 Cor 15:21-22) and Head of the Church (cf. Col 1:18), her resurrection and assumption is hardly repugnant to the apostolic tradition, nor does it take away from Jesus' unique role as Son of God and Son of Man. Certainly the reformer Heinrich Bullinger believed in the assumption: "For this reason we believe that the Virgin Mary, Begetter of God, the most pure bed and temple of the Holy Spirit, that is, her most holy body, was carried to heaven by angels."
Pius XII's apostolic constitution was actually the fruit of centuries of internal discussion and development. The first requests to define the Assumption as belonging to the deposit of faith, according to Dr. Ludwig Ott, came in 1849; at the first Vatican council (1870), over two hundred bishops signed a petition addressed to Bl. Pius IX requesting the dogmatization. In May 1946, Pius XII sent a query out to the bishops of the world: "Do you, venerable brethren, in your outstanding wisdom and prudence, judge that the bodily Assumption of the Blessed Virgin can be proposed and defined as a dogma of faith? Do you, with your clergy and people, desire it?" The response was an almost unanimous "yes". In short, the definition of the Assumption wasn't the whim of one power-crazed theological tyrant but rather a well-considered, even inevitable, result of the thought of the universal Church.
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In defending the doctrine of the Assumption, you should first make sure that you've completely demolished the sola scriptura objections. Certainly Scripture was not intended to be an omnibus history of the Church up to AD 70; nor was it intended to be a catechism of the earliest Christians or a complete record of their beliefs, binding on conscience or otherwise. In fact, sola scriptura itself isn't recorded in Scripture!
Second, be sure you emphasize that, while the traditions surrounding her death weren't recorded for two or three hundred years after the fact, the belief in her assumption into heaven preceded the traditions themselves. As such, the belief goes back almost to the beginning of the Church, and is of a piece with the reverence the early Christians had for the Theotókos, the Mother of God.
Third, many Protestant objections stem from what may be called a "fixed-sum view of love", such that any reverence or devotion to Mary is assumed to be taken away from Jesus. This is a false and unrealistic conception; people aren't born with a limited supply of emotions that must be rationed as strictly as possible. We recognize the Blessed Mother as creature, not Creator; her blessings and favors are given her by God, not inherent in her being. So long as we remember these distinctions, there are almost no honors we can give her that exceed the bounds of propriety.
- Although the Assumption was declared a dogma of the Church only recently, it has been part of Christian faith since almost the beginning.
 In the United States, the obligation is abrogated when the 15th falls on a Saturday or Monday.
 Mozley, Reminiscences of Oriel College and the Oxford Movement (1969), 2:368; cit. in Father Mateo, Refuting the Attack on Mary: A Defense of Marian Doctrines (1993, 1999), p. 29.
 Refuting the Attack on Mary, pp. 29-30.
 In Douay-Rheims, Pss 131:8 has "Arise, O Lord, into thy resting place: thou and the ark, which thou hast sanctified;" in the old Vulgate, et arca sanctificationis tuae (the new Vulgate has fortitudinis). The difference seems to be in the Septuagint Greek backing DR, which uses the verb hagiasmatos, which refers to making or preserving sanctification. I leave it to the professionals whether the Masoretic or Septuagint text captures the intent of the psalmist better.