FAULTY PREMISES OF PSA

As far as I can tell, the logic behind Penal Substitutionary Atonement goes like this:

1. God is just.
2. Therefore, God cannot let sin go unpunished. 
3. For a human to be forgiven means sin goes unpunished.
4. To win our forgiveness, Jesus - though perfectly innocent - had to go to the cross and bear the punishment which we deserved.
5. Now that God has meted out His punishment on His Son, He can forgive us.

Point #1 is obviously supported by Scripture, so PSA gets off to a good start.

But then the wheels come off immediately at Point #2. Can you show me where Scripture teaches that every sin must be punished by God or else his justice is compromised? (Instead, the Bible says that "mercy triumphs over judgment", James 2:13.)

Point #3 is true, but this is simply the definition of "forgiveness"! If the forgiven sin must still be punished - even vicariously as Point #4 would have it - then that offense was not in fact forgiven. 

It was repaid. 

If I say, "I forgive you for lying to me" and then turn around and punish you for lying to me, did I really forgive you? And how in the world would it make sense for me to punish someone else for your lie and then call that "justice"?

Points 4 and 5 are faulty conclusions because they are built on faulty premises. 

What does Scripture say?

Do you think this passage from Isaiah 30:18 sounds like the kind of hardcore justice which the God of PSA exhibits? -

Therefore the Lord waits to be gracious to you, and therefore he exalts himself to show mercy to you. For the Lord is a God of justice; blessed are all those who wait for him.

And what can PSA say in the light of Psalm 103:10-14? -

He does not deal with us according to our sins, nor repay us according to our iniquities. For as high as the heavens are above the earth, so great is his steadfast love toward those who fear him; as far as the east is from the west, so far does he remove our transgressions from us. As a father shows compassion to his children, so the Lord shows compassion to those who fear him. For he knows our frame; he remembers that we are dust.

It's fine to spend time hypothesizing about how the cross works, but any resulting theory needs to be grounded in Scripture and it needs to go no further than Scripture allows. 

“Love, not anger, brought Jesus to the cross. Golgotha came as a result of God’s great desire to forgive, not his reluctance.”

― Richard J. Foster