An apology without contrition is like a contract without a signature; it expresses an intent, but it doesn’t engage the necessary action. Apologies only work at the “heart level”; they are ineffectual at the “head level”. Trying to rationally parse exactly where we are wrong is never satisfying to the other person. You must beware of a partial apology; if it is partial, it is not an apology. People confuse completely wrong with completely sorry. I may not be wrong in every way, but I must be completely sorry for the way in which I am wrong.