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To discuss opposing views, with the goal of resolving a dispute. A noun is sometimes used between "argue" and "out." If you two ever hope to reconcile, you need to argue this out now. As a lawyer, I have listened to a lot of couples argue out the terms of their divorce.
argue something out
to settle something by discussing all the important points. We are going to have to argue this out some other time. Must we argue out every single detail of this contract?