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Would it be unconstitutional for a state to pass a law that makes it illegal, subject to criminal sanctions, for anyone to have more than one child? If so, on what grounds would it be unconstitutional?

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Being the good leftist that I am, I think we can find these kinds of protections (the kind that allow citizens to decide to have as many children as they like) in a penumbra of amendments - especially the 1st, 5th, 9th, and 14th Amendments. The First Amendment (specifically the Free Exercise Clause) would probably provide the weakest protections; again, though, some notion of substantive due process (derived from the 5th/14th Amendments - particularly the 14th Amendment, as it applies to states) would likely invalidate such child-limiting laws.

I can already hear the objections of Justice Scalia now...