In 2 Timothy 3:16-17, we read "All Scripture is given by inspiration of God, and is profitable for doctrine, for reproof, for correction, for instruction in righteousness, that the man of God may be complete, thoroughly equipped for every good work." Among the many good works Scripture thoroughly equips us for is baptism. Scripture tells us to baptise, and to be baptised.
Being such an important practice, the issue of baptism is discussed frequently, and in detail, in the New Testament. We see John the Baptist going ahead of Christ, baptising people and preparing the way of Christ. We see Christ Himself being baptised by this same John in order to fulfill all righteousness (Matthew 3:15). Jesus commands us to make disciples of all nations, baptising them and teaching them to obey all that Christ has commanded (Matthew 28:16-20). Baptism is said to be the answer of good conscience towards God (1 Peter 3:21), and is even listed alongside faith as a means of salvation (Mark 16:16). As a result, we see several examples of baptism taking place in Scripture.
Most notable about baptism is that every time we see it, it is always performed on a consenting, and more importantly believing adult. Nowhere in Scripture are we commanded to baptise those who, through immaturity of mind or spirit, cannot believe. Yet, for centuries, infant baptism has been a common practice. Even as early as the third century, Tertullian of Carthage states "...the delay of baptism is preferable; principally, however, in the case of little children." (1).
Though Tertullian's criticism of infant baptism is the earliest known record of the practice, it can be reasonably assumed that, but for divine inspiration, a practice precedes its criticisms. However, following the logic of 2 Timothy 3:17, we can rule it out as an original Christian practice with the formation of a simple syllogism:
P1: Scripture thoroughly equips us for every good work.
P2: Scripture does not equip us for infant baptism.
C: Infant baptism is not a good work.
In spite of this, infant baptism has been practiced for centuries, and remains popular today, even in many Reformed denominations. So what's the excuse? Well, while Scripture does not explicitly cover infant baptism, pedobaptists suggest that it is implied. There are no explicit examples of infants being baptised, but there are explicit examples of households being baptised. What is a household? Well, it's everyone who lives within a certain family dwelling. The mother, the father, maybe the grandparents, and... the infants.
An obvious criticism of this reasoning is that, statistically, it's unlikely that a household contains an infant too young to profess faith. Does yours? Perhaps it does. Maybe you've just had a child, or are expecting one. But how many people are in the same position? Think of your church. As a percentage, how many of them live with a baby under their roof?
Let's put it another way: Even when households are baptised in Scripture, this is often explicitly paired with their belief. In Acts 18:8, for example, we read "Then Crispus, the ruler of the synagogue, believed on the Lord with all his household. And many of the Corinthians, hearing, believed and were baptized." (Emphasis added). This is not surprising, as the criteria for a legitimate baptism in Scripture seems to be identical to the criteria for salvation: Belief. No, it doesn't explicitly say that, just as it doesn't explicitly say "baptise infants", either. However, if a pedobaptist can argue that infant baptism is implied by the baptism of whole households, I can present the following case against it.
In Acts 8:26-39, Philip is lead to an Ethiopian man, who happens to be reading the book of Isaiah. So Philip begins to explain the prophecies, of course leading the man to Jesus. They pass by some water, and the man asks to be baptised. "Then Philip said, “If you believe with all your heart, you may.” And he answered and said, “I believe that Jesus Christ is the Son of God.”" (v37).
The word "if" is a conditional one, strongly suggesting the existence of an else. For example, imagine a child requesting permission to leave the dinner table and play with his toys. The mother tells him that if he has eaten all of his broccoli, he may do so. The child then gets up and plays with his toys. What does this imply? It implies that he has, indeed, eaten all his broccoli. If he hasn't? Then he's being disobedient!
In Acts 8:37, Philip's "if" condition is that the Ethiopian eunuch believes with all his heart. The result of fulfilling this condition is "you may" get baptised. Now, what if the eunuch does not believe with all his heart that Jesus Christ is the Son of God? In this case, Philip's statement suggests the "else" is "you may not get baptised".
Now, if you were to ask a baby "do you believe Jesus Christ is the Son of God?", you will not receive an answer with similar conviction. You may hear a burble. You may hear a coo. You may even hear some very loud yelling and screaming. But before a child is even able to pronounce the word "Dada", how are they able to confess Jesus as Lord? Thus, if the criteria for baptism is a wholehearted belief in Christ, then a child, who cannot believe with any of their heart, cannot logically be baptised.
I contend this case is significantly stronger than the pedobaptist's. While there is no good reason to assume any household baptised in Scripture had a non-consenting infant, I can think of no good example of an "if" that is not paired with an "else". Therefore, using Scripture in the way it is intended, according to the logic of 2 Timothy 3:16-17, and taking everything the Bible has to say on the topic of baptism, Credobaptism is the only sustainable position, and pedobaptism is find severely wanting at best.
References
1. Tertullian of Carthage - De baptismo, Chapter 18, circa 200 A.D. (link)