In Act IV scene 2 Elizabeth is brought to John in hopes that she can convince him to confess. Even Reverend Hale hopes that she can encourage him to lie in order to save his life.
Alone in the cell, the two discuss their failings in their relationship. Elizabeth tells John that she cannot judge him and begs him to forgive her coldness. This creates a conflict for John. He feels that his is not a good man, he knows he has committed sins, and so he believes confessing would be just another lie and would save his life. He is conflicted: should he lie, destroy his name, and live, or should he refuse to confess and face execution?
In the end, he decides the name he leaves behind for his sons is more important, and he rips up his false confession.