My brother doesn't immediately respond with violence, but I get your point. The situations I listed with him were the more extreme ones. I don't exactly fault him for breaking someone's jaw if that kid threw a big rock at the back of his head. That could lead to serious injury. However, his reaction to the kids upending his lunch over him does seem to be a little extreme in some people's eyes(?), but I forgot to clarify that the leader kid and his group had been teasing and poking the sleeping lion that is my brother for days before that. My brother usually has a lot of patience when it comes to other children, but he's also a little unpredictable (with how harshly he would retaliate). Trust me, my father has actually given him a three strike rule, and after the third strike he's free to hit back. We know how much harm he is capable of, especially me since I've witnessed it and have been on the receiving end sometimes. He's not the shrieking, flailing type. I only know when he's really mad because he's usually a happy, talkative kid, but when he's mad, he gets really quiet and just stares at the offender. He's not panicky at all, when he hits you when he's mad he really, seriously means to harm you, and when someone like that is trained with just how to hit others it honestly is scary.Just a brief note... everybody saying that fighting back is the answer - I agree. In principle.
But it has to be controlled violence. It isn't about going berserk and dislocating some kid's shoulder. Going hysterical won't help. I'm pretty sure Nydanna's kid doesn't want to be seen as "the crazy scary one." And as Hellis very astutely noted, getting a reputation for being a hardass actually encourages violence against you, often - you're a living, breathing video game boss for anyone who considers themselves a tough guy to take down as an ego boost and to show off. My brother suffers from this, being a fairly heavily built, 6'4" guy with MMA training. He's a target.
You need to teach them how to throw a controlled punch. How to defend themselves capably and in control. That actually negates a lot of the perceived issues with "encouraging violence" in your kids - that way, you're teaching them that control is needed if you ever use violence for self-defence. It's not about a zerg rush of adrenaline and power and shit.
Again, just a personal opinion. It worked very well for me. I was targeted by a lot of bullies and had to stick up for my brother a lot, and keeping my cool and landing a few good punches as if I knew what I was doing always worked better than going crazy. Going crazy got me in trouble too, often, because they saw me as hysterical, unstable, and violent if I did that. And confidence and control scared off the bullies way more than some screaming, terrified kid flailing everywhere. As has been noted, they're insecure themselves - they don't wanna go up against someone confident and calm.
But honestly, would anyone fault him punching someone if they almost caused him serious injury? The reason me and my brother were put into taekwondo in the first place was that my father wanted us to be able to defend ourselves. Even our trainer has seriously told us to only use it in self defense, and arguably my brother only hits back after he has been hit. Now that he is a junior black belt and I'm damn proud of him, he knows that he has to set an example for the other kids who are training in the art like he is. Nowadays, my brother is still rather cheeky, but he's usually well-meaning to others. His way of speaking and his guts do cause some to be annoyed or aggressive towards him sometimes but he doesn't usually mean offense.