This lesson is a study of the argument about the Greek word εις (eis). The question we answer is as follows: In Acts 2:38, does the Greek word εις mean (1) “in order to receive”, or does it rather mean (2) “because you have received”?
Here's the text: "Peter said to them, 'Repent and let every one of you be baptised in the name of Jesus Christ, for [εις] the forgiveness of sins, and you shall receive the gift of the Holy Spirit' " (Acts 2:38).
The English word “for” in the phrase "for the forgiveness of sins" translates the original Greek word εις.
εις —is usually pronounced “ice” or “ees”.
εις —is usually translated “into”, “unto”, or “for”.
The argument regarding baptism says that, in Acts 2:38, the phrase "for [εις] the forgiveness of sins"   means “because of the forgiveness of sins”. So the passage is made to say, "Repent and be baptised because of the forgiveness of sins".
This supposedly puts forgiveness before baptism, so that First one is forgiven by God and saved, and because of that, one is then baptised.
But why couldn't Peter mean that the forgiveness of sins is on offer and because of that, repent and be baptised to receive it? That would put repentance and baptism back before forgiveness, where they belong. Forgiveness by grace is the prospect; repentance and baptism (among other things) lead a person into that prospect.
Notice that Peter does not speak of baptism only, but also of repentance. "Repent and be baptised..." (Acts 2:38). If the argument is correct that forgiveness is granted before baptism, then forgiveness is also granted before repentance. But the scriptures say that one must repent before forgiveness is granted.
On another occasion, Peter said, "Repent and return so that your sins may be blotted out" (Acts 3:19).
Peter did not say, “Repent and return because your sins have been be blotted out.” He said, "Repent and return so that your sins may be blotted out". Peter put repentance before forgiveness. He made repentance a condition of forgiveness.
And if Peter has put repentance before forgiveness, that's where he has also put baptism, because he gave both repentance and baptism the same relationship to forgiveness viz "Repent and be baptised... for [εις] the forgiveness of sins..." (Acts 2:38).
We read that the men of Nineveh "repented at [εις] the preaching of Jonah" (Matthew 12:41).
It is sometimes argued that the word εις in this verse points backward because the preaching came before repentance. So the Ninevites repented because of the preaching of Jonah. They did not repent to get the preaching of Jonah. It is argued that in the same sense one is baptised because of the forgiveness of sins, not to get that forgiveness.
But that is not what is meant by Jesus. That is not a sound exegesis. Nineveh’s repentance involved a turning away from sin and turning to the preaching of Jonah. They repented in order to receive the benefits of Jonah’s preaching, to escape condemnation and to enter into the grace and salvation that Jonah preached.
As another example of εις, take Paul’s statement, "With the heart one believes unto [εις] righteousness, and with the mouth one confesses unto [εις] salvation" (Romans 10:10).
In that quote the belief results in righteousness or justification, and likewise the confession results in salvation" (Romans 10:10).
Translators generally render this passage as though a heartfelt and confessed faith leads to justification. In other words salvation is the outcome of a confessed faith. Translators don't render the verse to say that faith and confession are because of salvation, and come after salvation.
Jesus spoke of his, "blood of the new covenant, shed for many  for [εις] the forgiveness of sins" (Matthew 26:28). The second instance of the word "for" is eis in Greek. As we all know, Jesus did not shed his blood because forgiveness had been granted, but so that it could be. This is another instructive parallel to Acts 2:38.
However it is even more instructive to examine how Paul relates baptism to the death of Christ and our death to sin (Romans 6:3-5).
Paul says, "Don't you know that all of us who have been baptised into [εις] Christ Jesus have been baptised into [εις] his death? Therefore we have been buried with him through baptism into [εις] death, so that as Christ was raised from the dead by the glory of the Father, so we too might walk in newness of life" (Romans 6:3-4).
Now in this passage, translators put "into" for εις indicating that we meet and join Christ’s death and resurrection through baptism. We are "buried with him through baptism... so that we might [rise to] walk in newness of life" (Romans 6:4).
So even if Peter (in Acts 2:38) said, “be baptised because of [εις] the forgiveness of sins,” that would simply mean, “Be baptised because forgiveness is available. Christ’s death is your only hope of forgiveness. Because of that, be baptised into that death.”