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Answered by the great John Henry Cardinal Newman.Blue Horizon #37
How do we know with certainty that Christ actually intended the successors of Peter have this charism is a whole other question
John Henry Newman’s proof from necessity:
“A revelation is not given, if there be no authority to decide what it is that is given….The absolute need of a spiritual supremacy is at present the strongest of arguments in favour of the fact of its supply. Surely, either an objective revelation has not been given, or it has been provided with means for impressing its objectiveness on the world. If Christianity be a social religion, as it certainly is, and if it be based on certain ideas acknowledged as divine, or a creed….and if these ideas have various aspects, and make distinct impressions on different minds, and issue in consequence in a multiplicity of developments, true, or false, or mixed…what power will suffice to meet and to do justice to these conflicting conditions, but a supreme authority ruling and reconciling individual judgments by a divine right and a recognized wisdom?..philosophy, taste, prejudice, passion, party, caprice, will find no common measure, unless there be some supreme power to control the mind and to compel agreement.” An Essay on the Development of Christian Doctrine, 1845, Rev. ed. 1878, Part I, Ch II, Sect. II, 12-13].
newmanreader.org/works/development/chapter2.html
[Cited in *Apologetics and Catholic Doctrine, Archbishop Michael Sheehan, revised Fr Peter Joseph, The Saint Austin Press, 2001, p 146-7].