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The Head of State in many monarchies are largely ceremonial figures. The validity of the regime depends more on the actual underlying government, whether autocratic, oligarchic, democratic or anarchic.
Even if Adam had dominion over the world, his heirs had no right that continue that dominion
If God's positive donation of 'all the beasts of the earth and birds in the sky and etc' were made personally to Adam, it would revert back to God on Adam's death
Assume that everything did not revert to God on Adam's death.
Then we have the case of having to figure out which of Adam's children inherit everything. This leads us to major objection 2.
It is implicit in the bible that Adam's dominion is in-part based his right (as a person) to take care of and provide for his own subsistence. All of Adam's heirs would have this same right.
Even if Adam's heirs had dominion over the world, there is no law of Nature nor positive law of God that determines which is the right heir (to dominion over the world) in all cases between generations
Even if we could find a perfect way of determining succession between heirs, 'the knowledge of which is the eldest line of Adam's posterity' is utterly lost
"Adam had not, either by natural right of fatherhood or by positive donation from God, any such authority over his children, nor dominion over the world as is pretended"
But Bellarmine said "If many men had been together created out of the Earth, they all ought to have been Princes over their Posterity. "
Something to the effect of "God gave adam dominion over eve... because the kids he made with eve were half eve... he must have dominion over them"
Locke found this Fillmer's response to Hobbes' Leviathan
“To confirm this natural right of regal power, we find in the decalogue,
that the law which enjoins obedience to kings, is delivered ill the terms,
Honour thy father; as if all power were originally in the father.”
The Decalogue also tells us to honor our mother
"why may I not add as well, that in the decalogue the law that enjoins
obedience to queens, is delivered in the terms of “Honour thy mother,”
as if all power were originally in the mother?"