David's child
Whether David's expectation of reunion with his dead child implies infants who die are saved, read against the biblical account of inherited sin and guilt.
2 Samuel 12:15-23
15Nathan departed to his house. Yahweh struck the child that Uriah’s wife bore to David, and it was very sick.
16David therefore begged God for the child; and David fasted, and went in, and lay all night on the ground.
17The elders of his house arose beside him, to raise him up from the earth: but he would not, and he didn’t eat bread with them.
18On the seventh day, the child died. David’s servants were afraid to tell him that the child was dead, for they said, “Behold, while the child was yet alive, we spoke to him, and he didn’t listen to our voice. How will he then harm himself, if we tell him that the child is dead?”
19But when David saw that his servants were whispering together, David perceived that the child was dead; and David said to his servants, “Is the child dead?” They said, “He is dead.”
20Then David arose from the earth, and washed, and anointed himself, and changed his clothing; and he came into Yahweh’s house, and worshiped. Then he came to his own house; and when he requested, they set bread before him, and he ate.
21Then his servants said to him, “What is this that you have done? You fasted and wept for the child while he was alive, but when the child was dead, you rose up and ate bread.”
22He said, “While the child was yet alive, I fasted and wept; for I said, ‘Who knows whether Yahweh will not be gracious to me, that the child may live?’
23But now he is dead, why should I fast? Can I bring him back again? I will go to him, but he will not return to me.”
What each tradition says
Rabbinic / Midrashic
The Jewish Encyclopedia argues that it is untenable to hold that the Hebrews of biblical times believed their own nature was corrupted through the fall of the first man. It notes the surprising absence of reference to the fall story and theme in the Old Testament, and on that basis rejects the contention.
The Jewish Encyclopedia (1905), The Jewish Encyclopedia (1905), art. "Fall of Man" — On the argument from the Old Testament
Patristic
Ambrose says David lost the infant he loved. He wept for him while still alive but did not long for him when dead. While the child was sick David entreated the Lord, fasted, and lay in sackcloth; when he heard the child was dead, he changed his clothes, worshipped God, and took food.
Ambrose of Milan, On the Decease of His Brother Satyrus (Nicene and Post-Nicene Fathers, Second Series, Vol. 10) — Book II, §25 (on David's dead child)
Catholic
According to Aquinas, it became necessary to baptize children. Through birth they incurred damnation through Adam, but through a second birth they might obtain salvation through Christ.
Thomas Aquinas, Summa Theologica (Fathers of the English Dominican Province trans., 2nd revised ed.) — III q.68 a.9 — Whether children should be baptized?
Orthodox
Gregory of Nyssa argues that infants' early deaths do not imply some grievous misfortune, nor make them equal to those who purified themselves by every virtue. Rather, God's far-seeing Providence cuts short the immensity of sins in those whose lives would be so evil.
Gregory of Nyssa, On Infants' Early Deaths (Nicene and Post-Nicene Fathers, Second Series, Vol. 5) — The treatise's final summary of its argument
Reformed / Protestant
John Gill reads David's words as going to the child in death and the grave, and to heaven and eternal happiness, where David hoped and believed the child's soul was. This shows, Gill says, that Old Testament saints believed in the soul's immortality and a future state of eternal life and bliss, not annihilation at death.
John Gill, John Gill's Exposition of the Whole Bible — On 2 Samuel 12:23
Evangelical
Wesley explains "I shall go to him" as going into the state of the dead in which he is, and into heaven. He does not doubt that he shall find him there.
John Wesley, Wesley's Notes on the Bible (Notes on the Second Book of Samuel) — Note on 2 Samuel 12:23