To save his socialism, he gave up self-ownership, but his reasons for doing so are weak.
If self-ownership survives Cohen's half-hearted assault, the free market is not yet out of the woods. Cohen has another argument against libertarians, this one directed at Lockean theories of property acquisition. According to the Lockean theory, individual self-owners may, by mixing their labor with unowned land and other natural resources, come to acquire it. (Some people don't like the phrase "mixing your labor," but Lockean accounts don't depend on accepting it. The important notion is that you have to occupy unowned land, or do something to it, in order to acquire it.)
Cohen maintains that this theory fails just by itself to support property rights in land. It is, as it stands, incomplete. For the justification of property rights to be successful, an additional premise is needed. The premise in question is that land is initially unowned. If everyone starts off with rights to an equal share of the earth's surface and resources, the Lockean theory has nothing on which to operate.
We may grant Cohen his point, but it avails him nothing. Why should we assume that people begin with property rights of the kind he wants? He gives no argument that they do; and the assumption that property is at the start unowned is a reasonable one. Murray Rothbard with characteristic insight dissected the equal shares position: