My thinking: Anarchists want to forbid the existence of the State.

However, in a society with no state, someone can achieve such a predominance over their peers that they can forbid anyone else from the use of violence, in effect founding a state. (This can happen for example because this person has forced everyone else under their rule with through violence).

If this is to be forbidden then there must be an organization with enough authority to enforce the lack of state. This organization would have to prevent anyone from violently forcing everyone else under their rule, so in effect, this organization would have to obtain the monopoly over the use of violence, and thus by definition it would be a state.

Anarchism, therefore, seems to me to be a self-refuting proposition, since only a state has enough authority to forbid the founding of a state.

Is anarchism a self-refuting proposition?

Possible weaknesses I can see in this:

* By the same line of argumentation we can argue that there must be a world government instead of many independent national governments, and this proposition seems controversial;
* What I'm saying here seems to boil down to proposing the necessity of the existence of a state *from the definition of a state*, and - as far as I'm aware - making propositions regarding real life from definitions of words is an error. Such an error was made, for example, in the famous Anselm's ontological argument.