The words, "I'm so sorry you are experiencing what you are, " can be the most powerful balm to hurt or upset feelings because those words coommunicate an energy of empathy and validation. Yet, we often insist, instead of apologizing, on justifying or explaining our points of view in a way that does not really acknowledges the validity of what the other person is experiencing.
Apologizing doesn't mean that you are wrong and the other person is right. Apologizing is simply a way of communicating that you care more about what the other person is feeling and experiencing than about proving a point or "being right."
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Along the same lines, agreeing to disagree is expression of validating that others are entitled to their own opinions and ideas that may be different than your own. It is an expression of respect and maturity.
Yet, when emotions are high, agreeing to disagree in a positive way can be one of the most difficult things to do. During such moments, you might need to rely on the first two tips, breathing and engaging touch, to remind yourself that you love and respect the person with whom you are experiencing a conflict.
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When you can remember that love, often times the intensity of your emotions will diminish enough to find that space where you can feel safe to allow yourself into that space of differences, which can be used to help you both grow versus being the source of conflicts.