The general objection to the NAP among philosophers is that it basically moves the goalposts one step back: instead of trying to work out right and wrong, it says all wrong are things which cause aggression. The debate then becomes, what is aggressive. This is itself something incredibly difficult to answer. Does insulting someone break the non-aggression pact? Does causing harm incidentally to one's actions (e.g. pollution, doctrine of double effect)? Who and what are we entitled to be aggressive to in the first place (criminals, the mentally ill, attackers, in marriage, sexual relationships, children as corporal punishment), etc.? What do we do if someone breaks the NAP? Note: all of these problems are the same problems we deal with when we do normal moral philosophy. The issue is we're just changing the words a bit.

Where it does seem to give answers to issues (e.g. in the trolley problem), it is not clear why we ought to follow the non-aggression principle. The idea you cannot instigate force might be believed by a deontologist, but it won't be believed by a consequentialist, and it's not at all clear why we ought to believe one over the other due to the NAP. Discourse ethics like that of Habermas gives a good argument for some variant of NAP but he goes well beyond what libertarians want from the principle (he's a self-identifying socialist after all). NAP doesn't really have 'arguments' for its position, but rather is just coincidentally believed in some form by some other people who have non-NAP-related arguments in its favour.

Finally, its answers in many cases just aren't intuitive, or rather, seem to be so out of line with common sense that philosophers don't use them. There's a general principle in both Hobbes and Rawls, which is that you should test beliefs against your intuitions, and the more out-of-line your argument is with common sense, the more likely you've made a mistake somewhere on the way. NAP typically concludes things like the government ought not exist except as a voluntary nightwatchman (and even then, I think that's pushing the principle too far), and furthermore implies anything that leads to third-party harm invalidates a contract (e.g. pollution, social degradation, etc.) These conclusions are so out of whack with most people's intuitions that it makes NAP difficult to substantiate.

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In formulation, for Mill's harm principle, force can be used to limit the liberty of a person against their will to prevent harm to others. In the libertarian non-aggression principle, though this principle is subject to disagreement of definition among libertarians, any use of force, other than in self-defense, is wrong.

The libertarian non-aggression principle is usually taken to entail an argument against state monopoly on force in general - where Mill's principle is concerned with when such force is justified - as well as other state actions considered as forms of aggression, such as taxation.

The problem is that once you try to define "aggression" in any helpful way, it turns out that everything turns on the definition of "aggression," and if you can solve that, you've already constructed a moral theory with way more detail than just the NAP, so the NAP turns out to be unimportant.

So for instance, do the following things count as aggression?

• Stealing someone's stuff.

• Trapping someone in a jail.

• Polluting a river.

• Telling someone "if you don't work for me, I'm going to kill you."

• Spoiling all the food except the food you own, and then telling someone "if you don't work for me, I'm going to let you starve to death."

• A natural disaster spoils all the food except the food you own, and then you tell someone "if you don't work for me, I'm going to let you starve to death."

• Yelling at someone.

• Yelling at someone when they're trying to sleep.

• Lighting off fireworks when other people are trying to sleep.

Answering these sorts of questions and more are very important questions, which the NAP doesn't help us with.