I was born & raised in a paedo-baptistic family, and have changed to credo baptism. Since then, I have been "phased" by paedo arguments many, many times. I can honestly say that it is not because of any hostility to the position that I haven't re-changed my mind. But I am still a credo-baptist because:
1. The command to circumsize was wider in scope than infant children of believers, or even those to whom the promise was made:
Ishmael, Abraham's servants, and Abraham's servants' children were none of them the "child of promise." Some of them were not children at all. Some of them were not biologically related.
2. The promises are wider in scope than the practice of paedobaptism, also:
Genesis 17:7 "And I will establish my covenant between me and you and your offspring after you throughout their generations for an everlasting covenant, to be God to you and to your offspring after you."
Psalm 103:17-18 "But the steadfast love of the LORD is from everlasting to everlasting on those who fear him, and his righteousness to children's children, to those who keep his covenant and remember to do his commandments."
The children talked about in these verses are not just biological infants, but biological children of any age, along with those children's children right on through to the end of the age. This is the thinking that was behind the "half-way" covenant: if somewhere back along the line there were believing ancestors, the children had a right to be baptized, whether their parents were believing or not.
Proverbs 3:33 "The LORd's curse is on the house of the wicked, but he blesses the dwelling of the righteous."
This would also include servants, and every person who was a part of the household.
3. The language of the promises is interpreted differently depending on to whom it is addressed:
Peter preached that the "promise is to you" to every person in attendance on the Day of Pentecost. Yet the only ones who were baptized were the ones who received the word with gladness. "The promise being to" language in this case is not taken as a grounds to presume salvation, but merely as a statement of the extent of the promise: it extends to you. It is also taken as a statement of extent "to them that are afar off." Yet it is taken as grounds to presume either regeneration or election when it is addressed "to your children."
4. Abraham was a father in two senses: a natural sense, and a spiritual sense:
I am a spiritual child of Abraham, but am still distinguished from the family, and from the covenant practices, that ran along natural lines.
5. Not all those who were circumcised were disciples:
The first circumsized child ever was not a disciple.
6. The household language of the NT is rare enough to lend credence to the possibility that everyone in the house believed and was baptized:
In the instance of the Phillippian jailor, there is reason to believe that everyone in the household was of a discerning age, because Paul preached to all (each one) of them. Also, in another household instance, the household was mentioned afterwards as those who were ministering to the saints. This would indicate that those in the household were old enough to do so. In the instance of Lydia, it is just as easy to assume that she was unmarried and had no children.
7. If the households included unbelievers, they would also reasonably have included unbelieving adults.
8. As has been mentioned by others, women being baptized indicates a difference in the economy of the covenant:
The heads of households & prospective heads of household were no longer the ones to whom the covenant sign was given. The new covenant sign emphasizes the individual in covenant, rather than the household.
[Edited on 3-23-2004 by a mere housewife]