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Agreeing to disagree elevates disagreement as closure, an end. It's not.
Agreeing that we disagree is to say that we both acknowledge hearing each others ideas.
And, it provokes the idea that we need to identify that disagreement.
That's why Republicans use the wrong form. It is designed to let them run and hide from the truth rather than dealing with truth. They want to run back to their authority figures on radio and Fox listening to lengthy half-truths and rolling lists of unfinished unchallenged attacks. Attacks that make them feel powerful until they realize that they cannot answer challenges except by changing the topic quickly over and over again. Or, by invoking a false ending.