Because the child did not enter into a contract with you. Even in a libertarian society, only free agents who are capable of rational thought are capable of entering into a contact, as no coercive contracts are considered binding. For this same reason it is not a legitimate contact if you hold a gun to someones head while they sign themselves into slavery. As a child is not a rational being it can not legitimately enter into a contract of self slavery with you. I am not sure how that would be dealt with. Clearly a child is incapable of entering into a legitimate contract, so it can not be born into slavery. However you are not required to care for the child of your slave if this is not part of the original contract signed by the slave. This would result in the child dying of malnutrition eventually if you forbid the slave from caring for it. Some anarchists believe that children are the property of their parents until they are old enough to make a claim to their own independence, following this line of thought you would legitimately be able to kill the baby as it would be the property of your slave. I disagree with this though, I think that children own themselves as much as adults do, but as they can not fend for themselves it brings up some interesting problems. I do not believe that you have the right to kill your slaves baby, but perhaps you do have the right to order your slave to not care for its child. Certainly you do not own the baby as a slave. I suppose I would hope that some charitable defense agency would act on behalf of such babies by taking them from you and raising them to adulthood, in the hopes that the adult will pay them back for saving its life. As the baby is not owned by you and also not owned by its parent, I believe it would be acceptable for a third party to take parental control of the child, especially as that would be in the best interests of such a child and indeed is what one would likely imagine the child would choose if it had the ability for rational thought. You can own the product of your slaves labor, but that does not include its children. Your freedom ends where another persons begins. You are free to contract people into slavery and people are free to contract themselves into slavery, however one person can not contract another person into slavery. Thus a parent can not contract its child into slavery anymore than I can contract you into slavery. People and livestock are different. As you do not own the child, putting it down would be murder imo. I certainly would not object to a defense agency assassinating you anyway. Not caring for the child would be more acceptable, even though that would certainly result in its death. I believe that you could rightfully force the slave to abort the fetus if it has not developed substantially. However, after the child is born you can either care for it or ignore it. Better yet, some agency could offer to pay you in order to enter onto your property long enough to remove the child (as you do not own the child you can not strictly speaking sell it to them, but you could sell the right to obtaining the child from your land). Except that assumes that you own your slaves children and I would disagree with this so the point is moot. The most you could do is deny care to the child and let it die of dehydration. You will not have hordes of slaves, because if you raise the children up to the point that they are rational beings, if they choose freedom and you detain them you will be holding prisoners. At that point it will be moral for defense agencies to storm your property and release your prisoners, as well as to hold you to some form of justice for false imprisonment. If you do not raise the children up, they will die very quickly after they are born. So you can not force your slaves to produce massive armies for you. Slavery must involve coercion. There is no such thing as a person who willing chooses to be a slave. If you deny people the right to sell themselves then you claim to own that right yourself. Essentially you claim that you must own the right to selling humans, which you deny to everybody, in order for a free market to exist.