History is filled with bloody examples of differences that could not be resolved -- enmity so deep that no group could accept a solution other than 100% extermination of the other, today called "genocide."
A variation of this is trial by combat; duel to the death. This assumes either that the strongest, bravest, or wiliest of two contenders must surely have had the right position, favored of God. Or that enmity was so great that neither contender would accept any outcome from which an opponent lived, so one of the two had to die.
Most of us in industrial societies think that we have escaped this level of barbarity. Don't bet on it. Yugoslavia was considered a civilized area until it dissolved in the Balkan Wars of the 1990s. In the United States, death threats are not unusual by sports partisans maddened to insanity just by some decision or incident involving their team, and in Britain football hooliganism is legendary. Civilization is a thin veneer.
And public "debates" or unending political campaigns are marked by endless varieties of ad hominem attacks, propaganda smears, and internet rumor campaigns. These methods prove so effective that they are standard features of campaign strategy.
The difficulty of agreeing on better ways to come to agreement is as big a challenge as any in Compression -- maybe as big a challenge as all our resource shortages and dangers combined.