An institution whose outputs are structurally biased toward powerful private actors cannot satisfy the beneficence conditions invoked in P1–P4, defeating the consequentialist legitimation strategy from within.

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Defeating (an argument) from within(describing how the bias problem undermines the legitimation strategy): Showing that an argument fails by using its own logic and assumptions against it, rather than attacking it from the outside.
Legitimation strategy(as something being defeated or undermined): A method or argument used to convince people that an institution deserves their trust and authority—that it has the right to make decisions affecting them.
P1–P4(as conditions cited in the passage): References to earlier propositions or premises (numbered 1 through 4) in the same argument that the reader would need to look back at to fully understand.
Structurally biased(describing how institutions can be unfair by their very nature): When the rules or design of a system itself favors certain groups over others, not because of individual choices but because of how it's built.
beneficence(Used as the practical expression of the ideal of love of humanity): The practice of doing good to others, presented here as the means through which love of humanity is expressed
consequentialist(Shared position of Russell and Moore): One who believes that the rightness or otherwise of an act is in some way dependent on consequences.

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