I had a post Asking for forgiveness, some days ago, in which I outline my theory about teaching children to apologize.
In a nutshell, that stems from the realization that the proper purpose of an apology is to invite forgiveness. That is what guides me as I figure out how to teach my kids to apologize and to forgive people, and it guides the selection of the words that we use.
Anyway, today commenter Barbara C. asks a couple of good questions:
1. How do you handle it if the "injured" party just does not feel capable of forgiving…if the hurt feels too big to let go of easily?
2. How would you handle those who seem to apologize for their own benefit (i.e. "by expressing the feelings that are inside him, so that the speaker feels his true feelings has been heard")? They do it more to relieve their own burden even if they know that in the process they will be putting a larger burden on the other person.Of course, maybe I am taking what is meant to be a lesson for children and applying it to my adult problems….(sigh)
Here's my answer, tidied up a bit for better bloggery.
Obviously each question is really two questions: what if it's your kid? and what if it's the other kid? And maybe there's also a subtext of what if it's me, the grownup, dealing with this from other grownups?
bearing blog 
1a. When my child can't or won't respond to an apology with "I forgive you." Truthfully, my child usually easily say the words "I forgive you" because, I think, they are so well practiced at it. They know that the sky doesn't fall down if they say it before they "feel" it. They know that life goes on, and usually the words have helped.
But if it was one of my children who felt they could not forgive another, I would take that as a sign that the kids need additional intervention before the discussion is over — maybe my child is reasonably afraid that the behavior won't stop, etc., in which case it is probably time for some redirection to different activities. You can forgive someone and still decide you're done playing with them for the day. We would stay there and explore it further. On the other hand, if my child is just being obstinate, well, I don't allow that for the young ones. If you can say "I forgive you," you do.
1b. When my child seeks forgiveness and doesn't hear it.
More commonly, it's my child who has apologized and asked for forgiveness, and another child doesn't say the words. I've been through this one a lot.
2a. The "fake apology" coming from my kid. Well, young children rarely give "fake apologies," right? The "I'm sorry you took what I said the wrong way" kind? They sometimes refuse to apologize, and they sometimes say "I'm sorry" when they don't mean it — and that last is something I wholeheartedly support!
Fake apologies, which have the words "I am sorry" or "I regret" in them but point the sorrow or the regret the wrong direction because they are not grounded in a desire for forgiveness but instead in a desire to continue making a point, are the domain of older kids and adults. People who want to save face.
I have yet to hit the teen years, but I imagine that if I hear one of those coming out of my tween's mouth, I will take him aside and explain that there is nothing wrong or unusual about feeling that you have been misunderstood or wrongly accused, but that the fake apology is never appropriate. If you really desire to be forgiven (and even if the other person is wrong about you, you should desire his forgiveness because forgiveness is good for him and good for your relationship) you will find a way to express that desire sincerely.
Maybe you will have to suck it up and say "I am sorry" and let the person think it is an admission of wrongdoing. Maybe it is not advisable to admit wrongdoing (there are sometimes legal consequences after all) and if no apology you can offer is accepted, at that point you just have to let it go and try (silently) to forgive *him* for refusing to forgive *you.*
2b. Other people's fake apologies. "If someone fake-apologizes to you," I guess I will tell my kids, "the ball is in your court."
You have the choice to accept it as if it were a sincere apology. This is called "taking the high road." It is difficult, but you can have some satisfaction because it is an exercise in humility. It means you let the other person have the last word, and you let it stand for what it is. Sometimes the exact choice of words can be a little tricky, though, because what makes a fake apology fake is that it does not, actually, ask for forgiveness, and so "I forgive you" is a non-sequitur.
(Try it: "I'm sorry you took what I said out of context." "I forgive you." Doesn't work, does it? You see why the fake apology is so insidious? It deprives both people of forgiveness. The only logical response is… "…Um… I'm sorry I took what you said out of context, too?" Or… "I forgive you for being so unclear that I couldn't tell what the context was?" Logically, the argument continues.)
Anyway, "I accept your apology" might work.
There is an alternative response, particularly if you care what the other person thinks of you. You have the choice to treat it as an opportunity for more dialogue, chock full of I-statements ("I get the sense that you feel I have misunderstood you. Do you want to tell me more about that?") Past history and expected future interaction are the guide to which approach makes sense, and you can stop at any point and accept the apology — such as it is — on the theory that it is the best you will get.
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One more thing: I think it is totally appropriate to apply lessons for children to adult problems. That is exactly what we are supposed to do as we grow up: apply what we have learned in the past to whatever is going on right now. It works really well if the lessons were good. And I think that the purpose of apologies does not change with age.