There's no right to defend your property against threats
The hard part about the NAP is defining what the initiation of force means. The NAP states that no one has the right to initiate force against the property of others. But what is included in "initiation of force"? Many say that fraud, threats of force and the actual first use of force apply. But I don't agree. Fraud, yes, because two parties agree to a transaction and one party refuses to fulfill his obligation after his counterpart fulfills his. That's the same as theft. One party gives and the other does not compensate. Equally, the actual first use of force applies, for obvious reasons. Property is actually violated, thus compensation is required.
Threats of force, on the other hand, are different. A threat is hard to define. If someone pulls out a gun, points it at your head and says they're going to kill you, that's a threat. If a guy on the other side of the world tells you on the internet that he's going to fly to your country, go to your house and slap you, that's also a threat. So do you have the right to use force to defend yourself in both of those situations? If you do, why? No one in either scenario has violated your property. If you use force to defend yourself, who violated whose property first? One problem is intent. You don't know what the guy with the gun is actually planning to do. Maybe he's not actually planning to kill you or even hurt you at all. You can't prove what his intent is, so you would be the force initiator if you took action against him and you would owe him for actually violating his property. Another problem is the fact that what constitutes a threat is different for everyone. The definition of "threat" is subjective.
Every time you drive on the road where other people are also driving, the threat of injury or death is present. Does that mean you have the right to use force to defend yourself against that threat? There may be some who say yes. Most would say no. But who's right? How could either side prove that they're right? That would be impossible, and that's the problem with subjectivity. It could be proven that you used force against the threatener, but it couldn't be proven that they used force against you.
Every time you drive on the road where other people are also driving, the threat of injury or death is present. Does that mean you have the right to use force to defend yourself against that threat? There may be some who say yes. Most would say no. But who's right? How could either side prove that they're right? That would be impossible, and that's the problem with subjectivity. It could be proven that you used force against the threatener, but it couldn't be proven that they used force against you.
For most, this is an unsatisfying conclusion, at best. For me, too. The practicality of it, though, isn't as bad. Now, let's say that you successfully defended yourself against the gun wielder in the above example. You violated his property and you are at fault. He takes you to court to seek compensation. It's a pretty clean-cut case and the judge rules against you. The plaintiff had asked for $5000 in damages. The judge, though, being a reasonable person, and being aware of how it will look if he grants this $5000 judgement, only awards the plaintiff $1. You may be able to get away with refusing to pay the compensation since the final arbiter, the public itself, will probably be sympathetic to your case and ignore your refusal, whereas, otherwise, you may face some level of ostracisation (which is a whole other topic).
I had hoped that there would be a clever, logical way to frame this so that the right to defend against threats would be revealed, but that search has been fruitless. If property has not been violated, no actionable force may be employed in a legal sense. There is, however, a point at which philosophy meets reality, and that point, in this case, somewhat alleviates the dissatisfaction of this conclusion.
Comments
I personally would not move a muscle, out of terror. So, this person is controlling me, violating my freedom to be sure I will not die if I walk away. What do you think.
The gun example doesn't illuminate your point as well as it could, so I came up with this one: imagine the man has two swords, blades crossed, with your neck between them and your back to the wall. If you move, you will be cut. He hasn't actually touched you, but he has physically restricted your movement because you know that movement will damage your property.
The difference is that in the gun example, there's a threat. You don't know what his true intent is, if the gun is loaded, or even if it's a real gun. You aren't physically restricted. You can move wherever you want without any certainty of danger. In that case, the principle remains the same: he hasn't violated your property and there's no certainty that he will. This is like everyday life. Everyone, at all times, is somewhat of a threat. That doesn't mean that you have the right to use force against them.
In the swords example, there is no threat. It's a certainty that your property will be damaged if you use it. The initiation of force has occurred and your property is being controlled by force. You are, as you illustrated in your question, unable to exert control over your property without certain damage.
So, I think that the principle remains, but I had to rethink it. Thanks for that question!