Combatting violence with violence is illogical. It magnifies/propagates/exacerbates. It doesn't solve. What would?

Just because it's not a solution to the woes of the world, doesn't make it illogical. The goal of fighting violence with violence is to dominate our enemies so our civilisation can survive, not solve the world's problems.

It takes two to make peace. Usually the only alternative to violence is appeasing our enemies, which is just perceived as weakness and draws an attack. That exacerbates the problem too. The only way to prevent violence is to project strength and thus discourage attack.