You can have an objective morality in the sense of its rules following from its premises. However, selecting the premises cannot be objective.
For instance, if we could all agree that human flourishing (and agree on the definition of it) is a sound basis for morality, we could afterwards haggle over whether this or that action promotes or degrades or is indifferent to human flourishing; and we could use science to measure the actual effects of various actions.
However, someone who doesn't agree with that basis, or doesn't agree with our definition of it, would be outside of this moral system, and could conceivable have their own which could also objectively follow from their axioms.
For instance, if we could all agree that human flourishing (and agree on the definition of it) is a sound basis for morality, we could afterwards haggle over whether this or that action promotes or degrades or is indifferent to human flourishing; and we could use science to measure the actual effects of various actions.
However, someone who doesn't agree with that basis, or doesn't agree with our definition of it, would be outside of this moral system, and could conceivable have their own which could also objectively follow from their axioms.
