If you have even ventured in any argument, then it's inconceivable that you would not have encountered the phrase “let us agree to disagree”. What it means is that both parties decide that they no longer want to continue the argument to persuade other party to change its opinion and that both decide to end the argument without reaching the conclusion knowing that differences exist. What it might really mean is that one who proposes to ‘agree to disagree’ thinks that other party is so stupid that talking to that it is like talking to a piece of wood and time can better be spent elsewhere than trying to bring some sense into other party. Or in simpler words, discussion is pointless.
Logic is, after all, logical. There are facts and there are premises. There are unstated premises called assumptions. Following well established rules of deduction, conclusion arises from facts, premises and assumptions. If both parties to a argument are rational and logical, then both must agree to conclusion, given that they agree to facts, premises and assumptions. If they don't, they can construct another argument whose conclusion will be that piece of disagreement. Hence going sufficiently backward, logically speaking, one can reach a point when both parties agree to common starting point.
Hence, a logical argument must end at some point in time and at that time, both party must subscribe to same point of view, which could be original position of either party, or third completely different position. There cannot, however, be a position when two parties can ‘agree to disagree’ in purely logical argument except when they don’t agree on some fundamental assumption. This deadlock typically arise when one party deviates from logic and doesn't admit to conclusion arising from agreed upon premises, facts, and assumptions. Hence, the moronic implication of real meaning of agreeing to disagree.
Actually, I've not been completely fair. Sometime time constraint or mere futility of proving a point may prohibit both parties from continuing to construct arguments to prove premises and assumptions. However, most typical use of the phrase is not because of time limitation.