When it comes to forgiving one another, it can be difficult at times. Some of your good-hearted people like the Nice Guy, All-Loving Hero, and the Wide-Eyed Idealist would be always willing to let bygones be bygones, but as for the Anti-Hero, Knight Templar, and the Jerkass, not so much. Then, there are some people who actually can (or will) forgive and can still give others the cold shoulder, which is this trope.
In short, this is more of a cynical cousin of Easily Forgiven. Instead of following the adage "forgive and forget", these type of characters take the "forgive" part only. No matter how forgiving the character is, he still feels genuinely hurt by the transgressor's actions or words, which explains the harsh attitude. Maybe the character is still distrustful towards him and believes that the transgressor might do the same thing all over again; then again, this reason may be justified. Perhaps the person has a bit of a short temper and takes the little things too seriously (like having his foot being accidentally stepped on), which makes it impossible to forgive the other completely. When it comes to this trope, just because people forgive doesn't necessarily mean they will move on.
Usually, other variants would show that one character will come to sincerely apologize to the other and they will still call them out of their wrongs with responses such as "you've hurt my feelings" or "you should be sorry". They may hear that they are trying to change their ways, but is still not good enough to make their completely happy again. These kind of forgivers would always focus on the bad qualities of others, and if they already forgiven them, they may give out warnings of threats or other ultimatums to them should the action be repeated. As far as forgiving others and not forgetting their act goes, this is seen as a character still finding it difficult to truly move on, just a character having bad trust issues, or just a character being an insufferable jerk.
It's worth noting that this trope isn't always found on the cynical side of the Sliding Scale of Idealism Versus Cynicism. To put this trope positively, forgiving others doesn't mean tolerating what the other has done. It can be a process to heal all negative emotions that the victimizer has caused. Then again, just because the person has forgiven the other doesn't mean they have to be friends with them. Also, the wronged person would still be satisfied to see him receive his Laser-Guided Karma for good measure.
The Atoner may not care if the forgiver is still not going easy on him, since the only thing that matters to him is that he is pardoned for his wrongs. If he does care for his approval again, though, then he will work hard to make that person happy with him again, or at least improve himself morally. If the forgiver does indeed become happy, it can be heartwarming moment. If not, then this can make a rather complicated story, and it may lead to the implication that the character has never really forgiven them at all.
Pretty much Truth in Television, but examples are unnecessary. Compare Flippant Forgiveness (the sarcastic variant), My Fist Forgives You, and Restrained Revenge (both tropes are about getting even with the wrongdoer while forgiving him, usually physically). Contrast Rejected Apology, Reformed, but Rejected, and Heel Face Door Slam which demonstrate that the character is NOT forgiven at all no matter how sincere he is in his apology and turning his life around.
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In A Certain Magical Index, most people who know of Accelerator's past, and especially his involvement in the Level 6 Shift project, acknowledge that he's trying to be a better person, but certainly are not about to forgive what he's done. Accelerator doesn't expect them to; indeed, he can't forgive himself. The only person who seems to have wholeheartedly forgiven him is Last Order.
In Fullmetal Alchemist, Scar was given this sort of advice from his master in regards to avoid dwelling himself in revenge too often or he will turn into a beast:
"Enduring and forgiving are two different things. You must not forget the unjustness of society. As a human being, you must hold the event in contempt. Yet, you must endure. You must put an end to the chain of hatred!"
In Kotoura-san, Hiyori was uneasily forgiven for bullying Haruka and nearly getting Manabe killed and as a side effect broke Haruka in the process, although this is because Haruka can read her mind and see that she's genuinely sorry for what she's done. \\Also, Manabe knows Hiyori was the one responsible for the attack, and was actually shocked and suspicious when she starts hanging out with them. Haruka's grandfather also implied quite clearly he knew what Hiyori had done and, while not mistreating her, did make it clear she wasn't off the hook with him. (He made a "Haruka-land" theme park for his granddaughter, with her and friends as mascot characters. Hiyori's was locked in a burning cage.)
Inverted with Habia from The Lion King Adventures. He is forgiven for murdering Tama, but he himself never forgets it. Word of God states that he did eventually get over it as he grew up.
In The Newest Challenger, Sasuke's Chidori through the chest on Naruto. It's brought up by Naruto when Sasuke is first introduced to his friends, and while Naruto still remembers it he recalls it with a mere shrug of the shoulders, compared to the reaction of many of his friends, one of whom tried to attack the past perpetrator on the spot.
Brendan: You never had any interest in underdogs. But I was your son.
Paddy: You are my son, Brendan.
Brendan: Am I?
Paddy: Yeah, you are. I'm just asking you if can find... find a little bit of space in your heart to forgive me a little bit.
Brendan: Yeah? All right, I forgive you.
Brendan: But I do not trust you.
Brandon and Tommy, however, have reconciled at the end of their fight.
Steven tries to avert this in the film The Wisdom Of Crocodiles:
Anna Labels: It's been a difficult day. I'm sorry. Problems at work. There's no excuse for taking it out on you. Is it alright for Saturday with Martin and Karen?
Steven Grlscz: Sorry? Oh yes.
Anna Labels: Good.
Steven Grlscz: Actually can we not do this? I can't do it this quickly. Accept apologies when I'm still angry. You can't say something like that and then say sorry. Do you see?
The story of "The Man and the Serpent" from Aesop's Fables starts with a serpent biting a farmer's son to death, which angered the man that he had decided to cut his tail off with an axe. The serpent seeks revenge against him stinging all of the cattle of his farm, causing a huge loss. The farmer then seeks for reconciliation with the serpent, but the serpent relents, saying that the farmer will never forget the death of his son and he won't forget the loss of his tail. The story ends with the statement "Injuries may be forgiven, but not forgotten."
Lucky from Survivors had been betrayed by his sister Bella because she did not speak up for him when he was banished from the pack. When he comes back, he ignores her trying to talk to him before finally forgiving her...but he never truly forgives her for what she did.
In the Discworld short story "The Sea and Little Fishes", Granny Weatherwax punctuates the day's Paranoia Gambit by finishing off with a nice grandmotherly speech about forgiving trespass (in the context of having been slapped by one of the other witches who'd turned hysterical by her uncharacteristic niceness) before returning to her regular self, effortlessly winning the Witch Trial with a magnificent display of fire magic, and adding "...I never said nothin' about forgettin'".
From the Burn Notice episode "The Hunter", Sam and Madeline had a conversation about Sam blowing up her living room in "Lesser Evil":
Sam: Maddy, I thought we've forgotten about that.
Madeline: Forgiveness is a process...
From The Fresh Prince of Bel-Air episode "The Script Formerly Known As...", Philip was rather reluctant when Vivian told him to accept both Will and Hilary's apology for bringing a dismissed juror of Philip's trial to Hilary's talk show.
ICarly: The episode "iStillPsycho" shows Nora, who was released from prison, coming to apologize to Carly and the gang for what happened in "iPyscho". Even though they rebuked her for her psychotic behavior, they immediately become in good terms with Nora. For now...
In the Grand Finale of M*A*S*H, Margaret reopens old woulds about Charles supposedly touching his nose while in surgery; even before Margaret actually comes out with, Charles states, "Oh, no you don't! We settled that a long time ago!"
Discussed and played with a little in the pilot of StargateSG1. Jack O'Neil(l)'s son Charlie died after accidentally shooting himself with Jack's sidearm before the movie, and Jack has this to say about it:
Jack: I think, in her heart, [my ex-wife] forgave me for what happened to our kid. She just... couldn't forget.
Daniel: And what about you?
Jack: I'm the opposite. I'll never forgive myself. But sometimes I can forget. Sometimes.
John Watson in Sherlock communicates this to his wife, Mary, after it's revealed in the worst possible way that she used to in fact be a professional assassin, and had lied to Watson about her entire past through all their relationship, but he has nevertheless decided to forgive her.
Watson: [during a tearful hug with Mary] All this does not mean that I'm not still basically pissed off with you. I am very pissed off, and it will come out now and then.
In the Walking Dead episode "The Grove", after Carol confesses to Tyreese for killing Karen and David and offers her gun to him, Tyreese forgives her but tells her he will never forget what she did.
Papa Roach's "Blood" pretty much deals with this trope. This line in particular:
I will forgive, but I won't forget
And I hope you know you've lost my respect
The Corrs' "Forgiven, Not Forgotten" is about a woman grieving over the loss of her lover who committed suicide. She's forgiven him for leaving her behind, but is still emotional over this.
As you can see in this old strip, Garfield seems willing to forgive Jon for trying to have him declawed, but gives him a very unsubtle reminder not to try it again.
The Bible: Exodus 34:7 gives the implication that God is good, kind, compassionate, and merciful, yet this doesn't mean that He will let the guilty go unpunished for their sins.
At the start of Assassins Creed IV, Edward usurps the life and identity of a treacherous former Assassin named Duncan Walpole. By completing Walpole's assignment he compromises the Brotherhood, violating a major tenet of the Creed. Then he kills several Assassins in his brief time among the Templars. Caribbeean Brotherhood partially forgive Edward and only spare his life because Mary Read vouches for him and for his skill in battle, his possession of the Eagle Sense and his nature as a Wild Card, equally damaging to Templars as he is to their cause. It is only years later and much Trauma Conga Line before Edward becomes Older and Wiser and finds his place among the Brotherhood.
In Umineko: When They Cry, Yasu/Beatrice forgives Natsuhi for throwing him/her off the cliff, saying s/he has tormented her long enough. S/he will never forget that Natsuhi is responsible for his/her miserable life, but as long as she regrets what she did to him/her, s/he won't hold any grudge against her.
An aversion in Deathwatch: as your armor's history, you can get the option "To Forget is to Forgive. Never Forget." It grants an automatic Hatred against whichever enemy killed your armor's past owner. This being Warhammer 40K, however, you're probably not doing it right until you have Hatred for everyone.
Shadow Of The Templar has Simon beating the shit out of Jeremy and then having a highly dub-con sex with him. Also, the team never passes on an opportunity to remind everyone of Simon throwing a mug at Johnny and giving him a concussion.
Seto Kaiba: I can't believe the five of you tried to take over my company.
Mokuba: And don't forget how they kidnapped me!
Seto Kaiba: Yeah... that too.
Gansley: Mr. Kaiba, please let us explain. We were just pretending to work for Pegasus!
Seto Kaiba: I instantly forgive you. But I'm still pretty sore about that whole trying to kill me thing.
Avatar: The Last Airbender : Katara seems to have this sort of attitude toward Zuko after she and the rest of Team Avatar accepted him into the team after grudgingly acknowledging his Heel-Face Turn. She does fully forgive him at the end of the series, though.
The animated adaptation of Franklin: In the episode "Franklin Forgives", Franklin was advised by Bear to forgive and forget in regards of forgiving Harriet for knocking over Goldie's, Franklin's goldfish's, bowl. Franklin response was that he doesn't want to forget Goldie.
Hawkgirl from Justice League, definitely. In fact, some members of the League do not forgive her after it revealed she was The Mole, at least for a while, and even after she saves the day after her recuperation with Dr. Fate, she is heckled by an angry mob. Even much later, she still has hundreds of detractors among the public, as seen by the hate websites aimed at her, although her teammates all do eventually. (It takes Wonder Woman the longest, but she eventually does.) In fact, it was very hard for her to forgive herself for what she too, going on a self-imposed exile to punish herself at first.
In The Simpsons episode "$pringfield, or, How I Learned to Stop Worrying and Love Legalized Gambling.":
Homer: You know, Marge, for the first time in our marriage, I can finally look down my nose at you. You have a gambling problem!
Marge: Hmmm, that's true. Will you forgive me?
Homer: Oh, sure! Remember when I got caught stealing all those watches from Sears?
Homer: Well, that's nothing, because you have a gambling problem! And remember when I let that escaped lunatic in the house 'cause he was dressed like Santa Claus?
Homer: Well you have a gambling problem!
Marge: Homer, when you forgive someone, you can't throw it back at them like that!
Homer: Aw, what a gyp.
Homer: Remember when I -
Homer: Oh yeah, I forgot already.
Another episode involves a flashback to Homer and Marge meeting as kids, and Marge livid with Homer when realizing he is actually a boy who stood her up for a date. When he reveals the insane chain of events that had actually stopped Homer arriving in time, Marge calms, but is still sore about her experience, despite Homer constantly hounding her whether she's let it go afterwards.