Anarcho-Capitalism isn't real Anarchism

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We often hear this from arnarcho-communists, that anarcho-capitalism "isn't real anarchism." (Some wrongly use this as an argument against the philosophy itself, but that is besides the point of this article.) Is this true? And if it is, should we celebrate it?

Anarchism is defined as the lack of a ruler. From this, anarcho-capitalism can't be defined as anarchism, as it advocates the individual's legitimate legal authority to be a ruler of his private property (in his person and material things alike.) But this would certainly also be the case with anarcho-communism, as it too advocates the individual's legitimate rulership of his own body (though not on material things.)

Due to this, the definition of anarchism always differ from each interpretation provided. To avoid the most confusion, definitions of the State are often used instead. Anarcho-capitalism defines the State as "a person, or group of persons, having the monopoly of the initiation of force, or absolute decision making, in a given geographical territory." Anarcho-communists define it as "a person, or group of persons, enforcing private property." They see any act of private property enforcement, as an act of governing -- of the State being present -- and thus not anarchistic.

With the anarcho-communist definition of the State, when applied to the anarcho-capitalistic philosophy, it is implied that the State is contained within the individual -- that is, a government exists in each person. Rephrase the anarcho-capitalist definition of the State, to fit this interpretation, and it would sound: "a person, or group of persons, having the monopoly of the initiation of force, or absolute decision making, on his/their private property." This is, of course, completely inline with the libertarian private property ethic -- in fact, it is the definition of the private property ethic: "the owner of any given piece of property, has the rightful legal monopoly of the initiation of force and absolute decision making on it."

The anarcho-capitalist and -communist definitions of the State are both correct -- they aren't contradictional with each other. Both definitions legitimately describe anarcho-capitalism as implementing the State in each individual person (this is often implied elsewhere, as anarcho-capitalist actively advocate that State power be decentralized down to the individual level.)

Shouldn't we celebrate this discovery? It certainly eases debates between right- and left-libertarians, as the definitions of anarchism and the State are often confused. (Personally, I've had some really frustrating times trying to get lefties to accept the anarcho-capitalist definitions.)

What do you think? Did I make a mistake in my reasoning anywhere? If so, please let me know.

I think there are some flaws in your definitions, resulting in a mistaken conclusion. To the market anarchist, property is properly defined by human action, namely Lockean homesteading and voluntary exchange. As such, the socialist necessarily claims superior ownership over individuals when exercising his beliefs, meaning he is a would-be ruler.

In exercising authority over one's property, one is not claiming rulership over others, instead merely clearly defining the sphere where others would illegitimately be claiming to rule him. This standard is reciprocal and universal. In contrast, the socialist in attempting to create a universal principle instead removes any means of measuring where individual authority rightfully exists, and opens the door to tyranny.

The State and government are not synonyms. Government is a group of people who claim the authority to usurp the rights of others through a territorial monopoly in violence, basically like in your description above, but the usurpation of the rights of others is the point of trespass, not the assertion of the rights that were usurped. This distinction cause sthe rest of your argument to fall apart.