I hear this statement from many otherwise intelligent people. It is not wisdom it is merely a truism. Violence is always responded to with violence, even among Quakers.
A modicum of thought will reveal to anyone with an open mind that all wars end. Europe was at war for a thousand years; but isn't today. The same was true for internal wars between different parts of China. What happened to the Cold War? All wars end and the truism about violence begets violence doesn't offer any explanation of why wars end. The phrase is inane. 
Gandhi would be the worst example, if people knew their history. Not a peace lover, that historic sham, Gandhi threatened to have India join Hitler if he weren't promised independence. So much for non-violent theory.
All wars seem to end for different reasons. Some because too much blood is shed, others because of new offensive or defensive weapons, still others because of new alliances and still others when the outcome is evident.
Violence begets violence and after the violence there are new an unpredictable outcomes.