Pilgrim, you said, “Again, the very fact that death exists, of which some unborn infants also experience, goes to prove that they too are sinners by nature, and thus liable to judgment.” I agree that infants are totally depraved and have a sin nature. Infants who die in infancy do temporarily live in a fallen world, and they suffer physical death as a result. To say, however, that some of the ones who die in infancy are condemned to hell because of their sin nature is speculative. If some of them go to hell, then why not all of them? What makes the difference between one dying infant and another? Baptizing the infant? Being a son or daughter of an elect person?

Boettner commented on the Westminster confession that has been previously discussed:

“It has often been charged that the Westminster Confession in stating that ‘Elect infants, dying in infancy, are regenerated and saved by Christ’ (Chap. X, Sec. 3), implies that there are non-elect infants, who, dying in infancy, are lost, and that the Presbyterian Church has taught that some dying in infancy are lost. Concerning this Dr. Craig says: ‘The history of the phrase “Elect infants dying in infancy” makes clear that the contrast implied was not between “elect infants dying in infancy” and “non-elect infants dying in infancy,” but rather between “elect infants dying in infancy” and “elect infants living to grow up.” ’ However, in order to guard against misunderstanding, furthered by unfriendly controversialists, the Presbyterian Church in the U. S. A. adopted in 1903 a Declaratory Statement which reads as follows: ‘With reference to Chapter X, Section 3, of the Confession of Faith, that it is not to be regarded as teaching that any who die in infancy are lost.’ . . . Concerning this Declaratory Statement Dr. Craig says: ‘It is obvious that the Declaratory Statement goes beyond the teaching of Chapter X, Section 3 of the Confession of Faith inasmuch as it states positively that all who die in infancy are saved. Some hold that the Declaratory Statement goes beyond the Scripture in teaching that all those dying in infancy are saved; but, be that as it may, it makes it impossible for any person to even plausibly maintain that Presbyterians teach that there are non-elect infants who die in infancy. No doubt there have been individual Presbyterians who held that some of those who die in infancy have been lost; but such was never the official teaching of the Presbyterian Church and as matters now stand such a position is contradicted by the Church's creed.’ It is sometimes charged that Calvin taught the actual damnation of some of those who die in infancy. A careful examination of his writings, however, does not bear out that charge. He explicitly taught that some of the elect die in infancy and that they are saved as infants. He also taught that there were reprobate infants; for he held that reprobation as well as election was eternal, and that the non-elect come into this life reprobate. But nowhere did he teach that the reprobate die and are lost as infants. He of course rejected the Pelagian view which denied original sin and grounded the salvation of those who die in infancy on their supposed innocence and sinlessness. Calvin's views in this respect have been quite thoroughly investigated by Dr. R. A. Webb and his findings are summarized in the following paragraph: ‘Calvin teaches that all the reprobate “procure” -- (that is his own word) – “procure” their own destruction; and they procure their destruction by their own personal and conscious acts of such must live to the age of moral accountability, and translate original sin into actual sin.’ ”
http://www.reformed.org/calvinism/boettner/infants_boettner.html