Libertarianism, as that is understood by the political philosophers who call themselves libertarians, is based on two principles: self-ownership and non-interference. First, all persons own themselves.
(x)(Person(x) -> Owns(x,x))
Second, if x's claim of ownership of y would interfere with z's self-ownership (for example, x steals z's car), then x does not own y.
(x)(y)((Person(x) ^ (Ez)((Person(z) ^ Owns(x,y)) -> -Owns(z,z))) -> -Owns(x,y))
Call the conjunction of these two the general principle of self-ownership (GPSO).
Suppose Helen is an unwilling dependence worker: she does not want to care for Ben, her adult son with muscular dystrophy. However, without Helen's care, Ben is trapped in bed, and cannot exercise self-ownership. Hence, if Helen owns herself, Ben does not own himself. On the other hand, if Ben does own himself, it is because Helen was somehow forced, against her will, to care for him. So then Helen does not own herself. In either case, the second conjunct of the GPSO then implies that the first conjunct is false. This can be formalised:
(UDW) (Ex)(Ey)(Person(x) ^ Person(y) ^ (Owns(x,x) <-> -Owns(y,y)))
UDW + GPSO |- -GPSO
Strictly speaking, this does not show that the GPSO is inconsistent. But it does show that it is inconsistent with a fairly obvious fact about humanity: without care, often from unwilling dependency workers, dependent individuals cannot exercise the autonomy on which libertarianism rests. Given this fact of dependency, GPSO has to be rejected.