What does John Proctor try to make himself believe in his conversation with
Elizabeth?

Hale tells Elizabeth that LIFE is God's most precious gift and nothing is worth throwing your life away. He says it may be that  a liar less than someone who gives up his life. She wants it to be his decision only. Elizabeth can't risk choosing for him if her choice is something he won't truly be able to live with forever. John feels that "his honesty is broke" because of his affair with Abigail, and therefore he is a sinner.

"Nothing is spoiled by giving them this lie that were not rotten long before" means John has already sinned, so he might as well keep sinning since he considers himself already ### and going to # down there . John clearly wants Elizabeth's opinion as to whether or not he should lie/confess. She doesn't tell him what to do because she wants John to make his own decision. It is HIS life and he has to live with his decision. John also wants her forgiveness, and she tells him that he needs to forgive himself. She does take on some of the blame, though

Throughout their conversation, it is clear that John wants what from Elizabeth? Why doesn't she give it to him?

John clearly wants Elizabeth's opinion as to whether or not he should lie/confess. She doesn't tell him what to do because she wants John to make his own decision. It is HIS life and he has to live with his decision. John also wants her forgiveness, and she tells him that he needs to forgive himself. She does take on some of the blame, though. She feels that her cold behavior prompted John to have an affair. She admits to keeping a cold house, not knowing how to "show" (express) her love, and counting (considering) herself so plain that no good love could come to her. (Because of her low self-esteem, it was hard for her to let him love her.)

He wants to live and be with his family.

idk i just need the points